NEAR-RINGS

NORTH-HOLLAND
MATHEMATICS STUDIES 23
Near-Rings
The Theory and its Applications
GUNTER PILZ
Institut fur Ma the ma tik
Johannes-Kepler- Universitat L inz
Linz, Austria
Revised edition
NH
gp
1983
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM · NEW YORK · OXFORD

® North-Holland Publishing Company, 1983
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording
or otherwise, without the prior permission of the copyright owner.
ISBN: 0 7204 0566 1
First printing 1977
Revised edition 1983
Publishers:
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM · NEW YORK OXFORD
Sole distributors for the U. S. Λ. and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY, INC.
52 VANDERBILT AVENUE, NEW YORK. N.Y. 10017
PRINTED IN THE NETHERLANDS

TO MY
BELOVED WIFE
GERTI

INTERDEPENDENCE GUIDE
The numbers indicate the ones of the paranraphes; 7a is §7,
section a), and so on. Full lines mean heavy, dotted lines
slight dependencies . (§9 J is a mere collection of results.)

PREFACE TO THE SECOND EDITION
Since, the арреалепсе of, the f,iAbt edition of, this book, a substantial питЬел
of, papens and nesults on пеап-пл.пдб came, out and new pants of, the theony юеле
bonn. Hence I uxu veny pieaiecf юкеи Nonth-Holland оЦелеа me the. possibility
to pnoduce an updated, nevised, соплесХ-ed and extended vension of, this booh
on nean-nings, which jib still the only one in this lield [but two excellent
otheA texts але in pnepaAation]. At that time, I didn't know the ammouni о I
wonk I had accepted.
This ediXion contains a tnemendous numben of, minon additions and conxcctiom.
hfcten. penfjonming these changes, I neatly know the big асЦелепсе between
"countable infinite" and "uncountable finite" пош. In lad, most of, the nesults
discoveAed alien the £4ASt edition але in some way inconponated on. at least
touched in this edition.
Alio, ffiWi moie chaptens юеле added. They сопселп педиХал пеал-nings, tame
nean-nings, ЫсеШлаИгел пеал-nings and the connections between пеал-nings
and automata. The chapter on polynomial пеал-nings иш substantially enlanged.
The list of, пеал-nings >(, small onden шал extended by adding stnuctunal
informations and ьотг пеал-nings on the non-cyclic abeiian дпоирл of, опАел & and
on Α.. Alio, thib edition contains Z2Z nemankable [counten.]example* of, пеал-
Плядь. Most extensions of, this edition але foinly woven into the text; hence
I do hope that this second edition is not just a sptit extension of, the fj*ASt
edition.
The "neligious wot" , if, light on. lefjt пеал-nings але "betten", is btilt un-
bettled. I do think that night пеал-nings one night, but the book ends with
a conciliatonu chapten using lefjt nean-nings.
Many thanks go to Nonth-Holland fon theiA оЦел and the most pleasant coopena·
tion. Many colleagues contnibut _ . mal important nemanks, pointing out
ennons and neading роли of, the new tnanuscnipt. Uajon contributions came [in
alphabetic ondtn] (лот G.Bet&ch ITiibingen, СеЛтапу), J.O.FMeldnum [Edinburgh,
Scotland], S.V.Scott [Auckland. New Zealand], V.S.So [Taichang, Taiwan) and
H.J.Ueinert [Clausthal, Germany). Also, many thankb go to G.Koller and A.
Kutzler for their excellent typing job. They incorporated the auditions so
blulfolZy into the text that nobody.who only want* tv read these additions,
can discover them. Last, but not at aXX least, I deeply thank my wife Gerti.
She helped me most.

IX
FROM THE PREFACE TO THE FIRST EDITION
Ыеак-лЛпдб оке <je.neAaJU.zin Kingi. Roughly spoken, а пеак-King is a "ling
[H,+,.)t wheAe * is not necesia/UZiy abelian and with only one distAibutive
lauf\
Неал-кЬлдб aJiise in a natuKol way·, take the bet М(Г) oi all mappings Ojf α
дкоир (Γ, + ) Into itself, define addition * point-wisely and о ал composition.
Then (М(Г), + ,·) is a пеак-King. Even i& Г is abelian, only one diAtKibutive
law it> alsaayb lul^illed·. ($+g)«fe » ioh+g»h holds by the definition o£ l+g
while ion. &o[g+h) - &og+6oh we would have to assume that £ is a homomoKphism.
AnotheA example is supplied by the polynomials w.K.t. addition and
substitution.
A well-known Kesult in King theoKy says that eveAy King can be embedded
into the ling Е(Г) о& all endomoKphisms oi borne abelian дкоир Г. Von пеал-
Kings we pKove (/.86) that eveAy пеак-King can be embedded into Μ(Γ) (,οη.
боте дкоир Г. Hence one might view King theoKy аь the "Ипеак theoKy oi дкоир
mapping*,", while пеак-Kings provide the "поп-Ипеал theoKy". SuKpKisingly,
a lot о l "Ипеал Kesults" can be tKans^eKKed to the genenal case a&teA suitable
change*. Ψοκ instance, the "atoms" o& King theoKy, the pKimitlve Kings, «еле
descKibed by the famous density theoKem ol N. Jacobson Цок King* with
minimum condition: ЫеааелЬикп-Ак£йг theoKem on simple Kings). Fok пеак-Kings
simiXaK Kesults conceAning pKimitive пеак-Kingi weKe obtained via the woKk o<
seveAal authoKi [but the pKoois але totally аЩелгий)'. the Kole atf Homp(V,V)
^ок Kings is played by М(Г) ок боте Kelated type* in the пеал-King cast
[4.52, 4.54).
HistoKlcally, the iiKSt step towoKd пеак-Kings was an axiomatic KeseaAch
done by Vickson in /905. He showed that thenz do exist "fields with only
one distributive law" (* neax-fields). Some yeans latex these пеал-iields
ihowed up again and pKoved to be useful in cooKdinatizing cextain important
clones oi geometKic planes [KecalZ that VescaKte'6 method o& cooKdinatizing
the "usual" plane by the iield о l кеа1 питЬель was one o& the most successful
steps in geometKy). It was lassenhaus who was able to deteKmine all finite
пеал-iields [8.34). Nowadays, пеак-iields але a mighty tool in choKacteAizing
doubly tKansitive gKoups [S.44), incidence gKoups [&.6S] and FKobenlus дкоирл
(8.8/). Since the sum oi two endomoKphiAms oi a non-abelian дкоир (Г, + ) is
not an endomoKphism in geneAal, the bets Е(Г) o& all finite sums and
difaieAences o^ endomoKphismi oi г weKe consideAed. blith Kespect to addition

χ
Preface
and composition, these Ε(Γ) 's але nean-rings belonging to the class о& the.
" distnibutively generated" near-rings.
Many роли o& the. well-established theony oh rings were transferred to near-
rings and now near-ring-specific ieatun.es меле. discovered, building up a
theony oi пгап.-нлпд{, step by step. Up to now, about 550 papers on near-rings
(and near-fields) with about 8000 радел appealed in print, but theme. exists
no book on this subject.
Tki6 book tries to unl^y the theory and its terminology and to give a
systematic and well-assorted account oh the pnesent itate o& the theony.
Some nemarks але to be made:
[a] Generally, I avoided to give pnoofa ion. theonems which але either not
along the main stream о& discussion ол. але long ones which contain
special method!, seemingly applicable only in this context, cannot be
simplified by previous nesults and involve many other (e.g. geometrical)
details, but але neadilu accessible in the literature.
(b) Several nesults fallowing &nom universal algebna ok inom the theony o&
gnoups with multiple operators оке cited, but not proved in order to
devide nesults which але spexil^ic ion near-rings and thoie which але
not.
Near-ring theony i& &ал away &лот being а теле collection o£ trivial nesults
concerning лоте "pathological" systems without any application to otkeA
branches oi mathematics, Apart farm the applicationi conceAning axiomatics
and geometry mentioned above, special clones ojj finite near-rings [the finite
"ρΙαηαΛ" near-rings) give new and highly evident classes о& balanced
incomplete block designs already with small parameters (8.117-8.124). Moneover,
these planar near-rings can be used to characterize fnobtnius groups, hence
also finite gnoups with ^ixed-point-faee automonphism gnoups (8.96, 8.97).
1^ Г i£ a finite, invariantly simple non-abeZian длоир, Ε (Γ) is "primitive"
and there fane eveAy selfamap o& Τ fixing zero is the "sum o(> endomonphism"
[exact farmulatlon in 7.47). AnotheA version oh the density theonem 4.52
shows that the. density property is (in the пеал-ring-case) something like
an interpolotion pnoperty,giving the nesutt that ii а near-ring N (with
some additional properties) o& mappings on a gnoup г "inteApolates" at
гело and at tuio otheA places then N "inteApolates" already at arbitrary
(finitely) many points (4.65). Also, near-rings might be the appropriate
tool to develop a "non-abelian homologieal algebna" (9.264) and show up
again in algebnaic topology (9.262), functional analysis (9.261) and in

Preface
χι
cateaoniet, with gn.oup objecti (9.265). finally, the authon hopzi that neax-
xingi and "nean-xing modules" (= Ы-дпюир&) will pnove. to be tueiul ion. о
питЬел o& theonlei which txy to genexotize "tinean." пелиЫл to the. "non-
tineax ca&e", ion. instance in the. theoxiei oi automata, and dynamical bybtemb
(лее §9 i)), to make the pnovexb
"li you. txy to non-lineanize,
you witl iind the пеап.-пл.пд& nice"
come txue. Tnom the xing-theoneticat point oi view, many bizonxe iituationi
аплле. in nzan.-ni.ngi. Ton example, not evexy le.it ideal ib a iubnean-nlng.
Howevex, thexe оке ievexal impLLcite application!, oi nean.-nJ.ngi to пЛпд
theoxy [ion. the nzax-ning-neiult& ihow what i& coniined to xingi and what
iA not) and to univexial algebna [becauie a high pexcentage oi deiinitiom,
and neiulti oi the пеах-пллд theony соплу oven, to univexial algebna).
On quotation!,·. ReiexenceA to othex iectiom, one done e.g. by "2d)" meaning
"§Z, section d)" on by "2d3)" abbneviaXlng "§2, iecttim d), numbex 3)".
Uumbexi ioilowing потел oi authou xeiex to the bibliognaphy at the end oi
the book, li only the authon'i name i& given, oil papenj, oi thib authon
cited in the bibliognaphy one meant. Thi& bibliognaphy ihould be iaixly
complete a& ion оь nean-nlngi one concerned. Ton neax-iieldt, and netoted
■iubjecti we only tUt thoie рарепл which dinectly iniluenced the material
in tha, book. Thii bibliognaphy wa& compiled in ionmex yeaxi by J.Clay,
G.Bet&ch, J.Malone, H.Heathexly and the authon. Namei in bnacketi neiex to
the liit oi "Supplementaxy wonki" which containi the поп-man-xing-papexi
cited in tkib book.
Sevzxal neAulti in thu, monognaph axe new on in a new land hopeiully
impnoved) ionm without ipecial notice.
In the beginning oi рп.оо& thene i& no nepetition oi the obiumptiom {to &ave
ipace). "■»" and "<=" mean that the difiection indicated i& txeated at moment
(in pnooi-ь oi equivalmceA).
hleon.-ni.ngi have the AMS-clab&iiication титЬел 16A76, neon.-iieldi alio 12K05.
It ii, a pleasuxe to thank Un.. E. Tnedxikaon oi the editonial ьЫЦ oi the
Nonth-Holland Publishing Company and the neviewex ion a plea&ant coopexation
and a lot oi ui&iul iuggeitiom,. Many thanki go albo to Мпл. Hoipodax ion

XII
Preface
кек excellent typing job and to G. %oMch, V.-S. So, H.E. Ee.lt, J.V.P.
MeldKum and to M.L. Holcombe {ok leading paxti, o{ the. manuicKipt and
providing иле{и1 hints and important comments. Mo it o{ all I have to thank
my wi{e. {ok hex patience and endurance in living with an abient-minded-
huiband in the pott угалл.
And now good tuck and much {an with пгал-fiingi!
Linz, Austria; GunteK Vilz
August 1976 {{isist edition)
Запиакы 19&3 [bzcond edition)
Remark: A "'№еал-King Hwbt&ttex" comes out once ok twice а. уеак, containing
information about the. Kecent developments in the theoxy o{ near-
Kingh. I{ you want to obtain copies, wKite to A. Obwatd ok to the
authoK o{ this book.

xiii
CONTENTS
Interdependence guide vi
Preface to the second edition vii
From the preface to the first edition ix
§ 0 PREREQUISITES 1
PART I: NEAR-RINGS FOR BEGINNERS
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS 6
a) Fundamental definitions and properties 7
1) Near-rings 7
2) N-groups 13
3) Substructures 14
4) Homomorphisms and ideal-like concepts 15
5) Annihilators 20
6) Generated objects 23
b) Constructions 24
1) Products, direct sums and subdirect products .... 24
2) Near-rings of quotients 26
3) Free near-rings and N-groups 29
c) Embeddings 33
1) Embedding in M(r) 33
2) More beds 37
d) Some axiomatic considerations 38
1) Miscellaneous results 38
2) Related structures 41
§ 2 IDEAL THEORY 4 3
a) Sums 44
1) Sums and direct sums 44
2) Distributive sums 49
b) Chain conditions 50
c) Decomposition theorems 53
d) Prime ideals 61
1) Products of subsets 61
2) Prime ideals 62
3) Semiprime ideals 66
e) Nil and nilpotent 69
PART II : STRUCTURE THEORY
§ 3 ELEMENTS OF THE STRUCTURE THEORY 74
a) Types of N-groups 75
b) Change of the near-ring 81
c) Modularity 84
d) Quasiregularity 89
e) Idempotents 91
f) More on minimality 95

xiv Contents
§ 4 PRIMITIVE NEAR-RINGS 102
a) General 103
1) Definitions and elementary results 103
2 ) The centralizer 106
3) Independence and density 110
b) 0-primitive near-rings 115
c) 1-primitive near-rings 120
d) 2-primitive near-rings 124
1) 2-primitive near-rings 124
2) 2-primitive near-rings with identity 126
3) 2-primitive zero-symmetric near-rings with
identity and a minimal left ideal 130
4) 2-primitive near-rings with identity and
minimum condition 131
5) An application to interpolation theory 133
§ 5 RADICAL THEORY 135
a) Jacobson-type radicals: common theory 136
1) Definitions and characterizations of the radicals136
2) Radicals of related near-rings 139
3) Semisimplicity 145
b) Jacobson-type radicals: special theory 149
1> ?o and ?1/2 149
2) }] 152
3) J2 152
c) The nil radical 160
d) The prime radical 161
e) Concluding remarks 163
PART III: SPECIAL CLASSES OF NEAR-RINGS
§ 6 DISTRIBUTIVELY GENERATED NEAR-RINGS 17 0
a) Elementary 171
b) Some axiomatics 174
c) Constructions of d.g. near-rings 176
d) Distributively generated near-rings with finiteness
conditions 178
e) "Free" distributively generated near-rings 180
f) D-groups and (N,D)-groups 182
g) Structure theory 184
§ 7 TRANSFORMATION NEAR-RINGS 188
a) М°(Г) 189
b) M(I') and Μ (Γ) 197
c) Е(Г), А(г) and Ι(Γ) 206
d) Polynomial near-rings 215
1) Polynomials and polynomial functions 215
2) R[x] 218
3) P(R) 219
4) Ideal theory in R[x] 220
5) F[x] 223

Contents xv
6) Γ[χ] and Р(Г) 230
7) Polynomials over Ω-groups 233
8) Concluding remarks 244
§ 8 NEAR-FIELDS AND PLANAR NEAR-RINGS 248
a) Near-fields 249
1) Conditions to be a near-field 249
2) The additive group of a near-field 251
3) The center and the kernel of a near-field 253
4) Dickson near-fields 254
5) Near-fields and doubly transitive groups 258
6) Normal near-fields and incidence groups 260
7) Planar near-fields 265
b) Planar near-rings 268
1) The structure of planar near-rings 268
2) Planar near-rings and BIB-designs 276
§ 9 MORE CLASSES OF NEAR-RINGS 287
a) IFP-near-rings 288
1) IFP-near-rings 288
2) p-near-rings 298
3) Boolean near-rings 300
b) Near-rings without 301
1) Near-rings without nilpotent elements 301
2) Near-rings without zero divisors 305
c) Affine near-rings 313
d) Near-rings on given groups 321
1) Multiplications on a group 321
2) Near-rings on simple and on cyclic groups 325
3) Near-rings with identities on given groups 327
4) Near-rings with other properties on given groups. 330
e) Ordered near-rings 333
f) Regular near-rings 345
g) Tame near-rings 350
h) Bicentralizer near-rings 361
i) Near-rings and automata 378
j) Miscellaneous 392
APPENDIX 4 04
Near-rings of low order 405
222 remarkable examples and counterexamples 426
List of some open problems 435
Bibliography 437
Supplementary works 464
List of symbols and abbreviations 465
Index 467

1
§0 PREREQUISITES
For the concept of sets we can use any one of the usual set
theories with the axiom of choice and using classes. In order
to avoid logical difficulties as much as possible, we use
statements about classes only as abbreviations of "less obscure
ones". For instance, if If denotes the class of all finite sets,
"Fe3 " is only an abbreviation for "F is a finite set".
"3χεΑ" stands for "there exists an χεΑ", "3 xeA" for "there
exists exactly one χεΑ" and "\/χεΑ" for "for all χεΑ".
Inclusion will be denoted by s, strict inclusion by «=. 0 will
A
denote the (an) empty set and 2 the power set of A; if A.
(ιεΐ) is a collection of sets, we will write the elements of
X A^ as (... ,a.,...) , where a-εΑ.. If all A.5A then X A.:=A
ι el ι ε0
and also Π Α·:=Α. Α\ Β is the set-theoretic difference. If
ie0
•v· is an equivalence relation on the set A, A/^ will be the
factor set of A w.r.t. ъ and ir:A -* A/^ will be the canonical
projection.
The sets of all natural numbers will be denoted by IN , the
natural numbers together with 0 by IN , the prime numbers by P,
the integers by Z, the rationals by Q, the reals by IR and the
complex numbers by С
If f is a function from A to В and if Α,'ΞΑ then f/д will be
the restriction of f to A, and f(A,) will denote the image of
A
Aj under f. В will be the set of all maps from A to B. If
B=A, i:B ■* A will be reserved for the inclusion map.
If A is any set containing something like a "zero element" 0,
A* will denote A\{0}.So e.g. 1N*=]N, while P*is not defined.

2
§0 PREREQUISITES
"Field" will always mean "skew-field". The symmetric
(alternating) group on η symbols will be denoted by S (A ,
respectively). The integers modulo η will be written as 7 and
represented by Zn = {0,1,. . . ,n-l).
We need an abstract version of "generated objects":
0.1 DEFINITION У^2А is called a Moore-system (Dubreil-Du-
brei1 -Jacotin) on A if
(a) kzji.
(b) Μ is closed w.r.t. arbitrary intersections.
0.2 PROPOSITION If Л is a Moore-system on A and if B^A then
[в] . := Π Μ is the smallest element of M{vt.r.t. ?) con-
л ЩсМ
Μ а В
t a i η i η g В.
0.3 DEFINITION Let the notation be as above.
(a) [s]// is called the element of Μ which is generated
by B.
(b) bzjl is called finitely nenerable (f.g. ) if there is a
finite subset В of A with [в]„= A.
0.4 DEFINITION A Moore-system β. is called inductive if
^contains the union of every chain of elements of M.
0.5 EXAMPLES
(a) 2A is an inductive Moore-system on A.
(b) The set of all subgroups of a group Γ is an inductive
Moore-system on Γ.
(c) The set of all closed subsets of a topolonical space Τ
is a Moore-system on Τ which is not inductive in
general .
We now turn to chain conditions.

§0 PREREQUISITES
3
0.6 DEFINITION A (partially) ordered set (A,<) is said to
fulfill the minimum condition if every non-empty subset
contains (at least) one minimal element.
0.7 PROPOSITION For a partially ordered set (A,<), the
following conditions are equivalent:
(a) The minimum condition.
(b) The descending chain condition (DCC) : every strict
chain a. > a„ > ... of elements of A terminates
after finitely many steps (or, equivalently, for each
chain a. ^ a „ ^
3 ηεΙΝ
= a
n + 1
)·
0.8 DEFINITION Linearly ordered sets with the minimum condition
are called well-ordered.
0.9 REMARK In replacing < by > , we get the concepts of
"maximum condi ti on", "ascending chain condition" (ACC)
and " i nverse wel1-order".
Every non-empty subset of an ordered set with the minimum
(maximum) condition has the same property.
0.10 PROPOSITION Let
Л
be a Moore-system on the set A.
(Μ.,'ξ) fulfills the ACC => every element of Μ is f.g..
If Μ is inductive, the converse also holds.
Proof. Let (M ,^) have the ACC and assume that some MzM
is not f.g. and generated by BsA. Take some
arbitrary Ь-εΒ. [ib,}]^ =:B. + M. Take some bpcM\B.
and form B„:= [{b. ,b„}] \- M. Continuing this
process, one gets an infinite chain Β,^Βρ^Β,Ε... of
elements of M. t a contradiction.
Now let Μ be inductive and suppose that every
element Μ of Μ is f.g.. Assume moreover that
M.<=M~=M.c=, . . is a strict infinite chain of elements
of M. Let M:= [) Μ. εβ. be generated by (say)
ieIN
{a,,...,a }. But there is some keIN with the
property that {a, ,. . . ,a }?Mk, so we get M^ = M, which is
again a contradiction.

4
§0 PREREQUISITES
Finally, ιt should be remarked that in general we use small
letters for elements, capitals for sets and script letters
for collections of sets.

PARTI
NEAR-RINGS FOR BEGINNERS
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
§2 IDEAL THEORY

§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
a) FUNDAMENTAL DEFINITIONS AND PROPERTIES
Near-rings are generalized rings: addition needs not be
commutative and (more important) only one distributive law is
postulated.
Examples of near-rings are
(a) the set Μ(Γ) of all mappings on an (additively written)
group Γ with pointwise addition and composition;
(b) the polynomials R[X] (R a commutative ring with identity)
under addition and substitution;
(c) an arbitrary additively written group with zero
multiplication;
as well as many others.
Similar to ring theory, "modules over a near-ring N" ("near-
modules" or "N-groups") will be introduced. They play an
important r51e in the theory of near-rings. This section contains
the basic definitions, examples and properties of near-rings
and N-groups, and of substructures and ideal-like objects in
these kinds of algebras.
Since near-rings and N-groups (with a zero-symmetric N) are
special classes of Ω-groups (groups with multiple operators),
a whole bunch of concepts and results is "a priori" available.
Compared with ring theory, some complications arise: an element
multiplied by о is not 0 in general, the characterization of
ideals is a little bit more complicated, ideals are not always
subalgebras, and so on.

1a Fundamental definitions and properties
7
1.) NEAR-RINGS
1.1 DEFINITION A near-ri nq is a set N together with two binary
operations "+" and "." such that
(a) (N,+) is a group (not necessarily abelian)
(b) (N,. ) is a semigroup
(c) V η.,Πρ,η-εΝ: (η,+η~).η3 = η,.η-, + η-.η, ("right
distributive law") .
1.2 REMARKS In view of (c), one speaks more precisely of a
"right near-rinn". Postulating
(с ' ) V η ^ , η ? , η.ε Ν: η. . (η ~ + η 3) = n..n~ + n..n~
instead of (c), one gets "left near-rings". The theory
runs completely parallel in both cases, of course; so one
can decide to use just one version.
Although left near-rings are more frequently used in the
literature up to now, we will use right near-rings:
•The left distributive law is in some way unnatural in
near-rings of functions (the most important examples)
and especially unmotivated in near-rings of polynomials
and formal power series.
•An ad-hoc-test done by the author showed that about
80% of the books in which rinq-modules play an important
role use left-modules, which are also more familiar from
the theory of vector spaces. In 1.18, we will see that
choosing left N-groups forces one to use right near-rings.
•The right distributive law is exclusively used in papers
on the closely related concept of composition rings
(which were systematically studied prior to near-rings!).
1.3 NOTATION Near-rings will usually be denoted by Ν,Ν',Ν, or
similar symbols. We abbreviate (N,+,.) by N.
Multiplication will in most cases be indicated by juxtaposition; so we
write nin2 instead of η,.η,,. In dealing with general
near-rings the neutral element of (N,+) will be denoted
by 0.|N{ will be the order of the near-ring N.

8
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
The term "near-rinq" will often be abbreviated by "nr.",
Throughout this monograph, the class of all near-rings
will be denoted by ΎΙ. If "N" appears, it will always be
a near-ring, without further notice.
1.4 EXAMPLES
(a) Let Γ be an additively written (but not necessarily
abelian) group with zero о ("omykron"). Then the
following sets of mappings from г into Г are nr.'s
under pointwise addition and substitution:
Й(Г):= {f:r-r} =ГГ.
М0(Г):= {f:r-r|f(o) = o}.
Μς(Γ):= {f:r+r|f is constant}.
М°С(Г):= {f6:r-.r|6crA f6(Y) = {° ;f V = o }<
(Evidently,Γ , Μ (Γ) and М°(Г) are isomorphic
groups).
Μ .(Г):= {f:T->-r|f is continuous} (Γ a topological
group).
Another related example is
Mdiff(IR):= {f:IR + IR j f is di f ferenti abl e} , while the
real functions having an indefinite integral do not
form a near-ring (they are not closed w.r.t.
composition).
For S?End(T) define
MS(T):= {f:r-r| VSeS: f°s = s°f}.
Evidently, M{.d}(r) = Μ(Γ) and М{б}(Г) = М0(Г),
where б is the zero endomorphism.
These Ms(r)'s will become very important in §4.
(b) Let Г be as above. Near-rings on Г are e.g.
(Γ,+.t) with yt6 = о for all γ,δεΓ;
(Γ,+,») with γ»6 = γ for all γ,δεΓ.
More generally, take some subset Δ of Γ and define
^Δδ:={ο !f Hi · Then <r' + -V fs a "ear-ring if
οφΔ.
(Multiplications of this type are called the "trivial
ones" in Malone (3), because they are exactly those
ones which can be defined on any group, making this
group into a near-ring.)

1a Fundamental definitions and properties
9
Now let G be a fixed-point-free automorphism group on
Γ (i.e. V geG \/ γεΓ: g(Y)=Y-=>(Y = ovg = id)).
Choose any subset {Β,|ιεΠ of the set of all
nonzero orbits of G on Γ (Betsch called these orbits
"1-orbits" and the other ones "0-orbits") ; moreover,
choose any set of representatives {b ^ ε Β ,· | i e I} =:B
and define γ·ηδ to be о if 5$ U B. and to be =9a(y)
B iel 1
if δ is in some В., where g5 is the unique
automorphism in G sending b. into 5:
b,·
π
.-.»..δ
-6
Ьз·
B1
Bn
с
*?
о
Then (Γ,+,·Β) is a near-ring as one sees by looking
at the different possible cases. These types of nr.'s
were introduced by Ferrero (5) and will prove useful
in the theory of planar and integral near-rings.
Anyhow, one sees that every group can be made into a
near-ring in various ways. See also Olivier (2).
(c) Let V be a vector space over some field F. Call as
usual a map V+V an affine map if it is the sura of a
linear and a constant one. The set ^ f,(V) °f a^
affine maps is again a near-ring (operations as in
(a)).
(d) Let R be a commutative ring with identity. Near-rings
are (R[x], + ,°) and (R [[x]], + ,°), where ° means
substitution. Another near-ring is formed by the set
P(R) of all polynomial functions on R with the
operations as in (a) (see §7d)).
(e) Of course, every ring is a near-ring.
1.5 PROPOSITION Vn.n'eN: On = О Л (-n)n' = -nn'.
Proof: as for rings.

10
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
1.6 REMARK As most of our examples show, nO = 0 and n(-n')
= -nn' do not hold in general. For instance, in Μ(Γ)
f°0 = 0 means that "f goes through the origin" and
fo(-f ) = -fof means that "f is an odd function".
One therefore defines for a near-ring N:
1.7 DEFINITION
(a) N : = {ηεΝ|ηΟ = 0} is called the zero-symmetric part
of N.
v-
the constant part of N.
(b) Nc:= {ηεΝ|ηΟ = η} = {ηεΝ|\/η'εΝ: nn1 = η} is called
N and N are itself near-rings (see 1.22 (a)).
1.8 EXAMPLES (M(r))Q = MQ(r); (М(Г))С = МС(Г).
1.9 DEFINITION Νε 7? is called zerosymmetriс (constant) if
Ν = Ν (Ν = Ν , respectively).
У) ( У] ) stand for the classes of all zerosymmetri с
(constant) near-rings.
1.10 EXAMPLES Elements of 7] are (notation as in 1.4) MQ(r),
Ms(r) if όε5, (Γ,+,·Β), every ring.
Μ (Γ)ε7? , while Μ(Γ) or R[x] are neither in ΎΙ nor
in 7), . Cf. Adler (1), p. 610.
1.11 DEFINITIONS The following concepts are defined as in ring
theory: left (right,-) identities, left (right,-) i η ν e r t i-
ble elements , left (right,-) cancellable elements , left
(right,-) zero divisors, idempotent and ni1 potent elements
Moreover, call dεN distributive if
\/η,η'εΝ: d(n + n') = dn+dn1. Let Nrf:= {deN [ d is
distributive} .
Let Ύ)γ be the class of all near-rings with identity
(usually denoted by 1).

1a Fundamental definitions and properties
11
1.12 EXAMPLES The identity function serves as an identity in
Μ(Γ) and Μ (Γ). Invertible in these near-rings are
exactly the bijective functions. 2x is an example of a
nilpotent element in 2.[x]. Cartan (1) characterized all
invertible elements in (F[fx]l) , F a field: 7 a.x1
0 1-1 Ί
has an inverse in (F[[x]]) (w.r.t.0) iff a,+0.
If N = Maff(V) then Nrf = HomF(V,V). If N is a ring then
N = Nd. It is clear that Nd=N0. If N has an identity 1
then 1εΝ„.
The next assertion stems from Berman-Si1verman (1).
Generalizations can be found in Kaarli (4), Lyons (4), Miron-Stefanescu
(1), Ramakotaiah-Reddy (1), Zand (1),(2).
1.13 PROPOSITION If θεΝ is idempotent then we get a "Peirce-
decompos i ti on":
Μ ηεΝ 3χΛε{χεΝIxe=0} 33xncNe: η = χ +χ,.
о ' ι ol
Taking e = 0 one gets
V ηεΝ 3 η εΝ 3 η εΝ : η = η +η .
о о с с ос
Hence (N,+j = (NQ,+) + (Ν ,+) and NQnNc = {0}.
Proof. η = (n-ne)+ne will do the decomposition job.
If η = xQ+Xi = χό+χϊ with χ ,χ'ε{χεΝ|xe=0} and
Ί
yle· xl
y\e cHe then ne = x,e = xie. But
x,e = y.ee = y,e = χ
xl = xi
V
1
and
xie
4
It follows that
and
= x
1.14 DEFINITIONS Let N be a near-ring.
If (N,+) is abelian we call N an abelian near-ring; if
(N,.) is commutative we call N itself a commutative near-
ring. If N = Nd> N is said to be distributive. If all
nonzero elements of N are left (right,-) cancellable, we say
that N fulfills the left (right,-) cancel 1ation 1 aw. N is
integral if N has no non-zero divisors of zero.
If (N*=N\{0},.) is a group, N is called a near-field
(abbreviation: nf.). A near-ring which is not a ring will
be referred to as a η ο η -r i n q. Similarly, a non-fi eld is a
nf. which is no field. A near-ring with the property that
N. generates (N,+) is called a distributively generated
near-ri ng (dgnr.).

12
1 THE ELEMENTARY THEORY OF NEAR-RINGS
1,15 EXAMPLES (Notation as in 1.4) Μ(Γ) is abelianiff г is
abelian. (Γ,+,ο) serves as an example of a commutative
and distributive non-ring, while (Γ.+,*) is integral.
In the language of 1.4(b), (Γ,+,·β) is integral iff all
non-zero orbits are "1-orbits". (Zo>+) with 0·0 = 0·1 =
1·0 = 1·1 =1 is a nf. All other nf's are zero-symmetric.
Let Γ be a group. If Γ is not abelian, the sum of two
endomorphisms is not necessarily an endomorphism any more.
But the set of all (finite) sums and differences of endo-
morphisms o^ Γ is closed under addition and composition
and forms a dgnr. Ε(Γ).
0,
1.16 HISTORIC REMARKS Near-fields were the first nr's considered
in the literature. In 1905, Dickson (1),(2) changed the
multiplication in a field in order to get examples of
"one-sided distributive fields" (= nf's) showing that the
second distributive law does not follow from the remaining
axioms for a (skew-)field. His "changed fields" are called
"Dickson nf's" (see §8(a)4)).
A couple of years later Veblen and Wedderburn started
to use nf's to coordinatize certain kinds of geometric
planes.
In 1936, Zassenhaus (1) determined all finite nf's: they
have order pn (ρεΡ, ηεΙΝ) and are (up to 7 exceptional
cases) Dickson nf's. In (2) he showed up the connection
between nf's and fixed-point free permutation groups.
Ore (1), Furtwa'ngl er-Taussky (1) and Taussky (1) started
axiomatic considerations in the thirties for what we now
call near-rings.
A first name for these structures was proposed in 1938
by Wielandt (1): "Stamm" (=tribe) ("stem" is still used
in the Italian literature). Wielandt also announced
structure-theoretic results in this note.
The first ones to use the name"near-ring"were Zassenhaus
in 1936 and Blackett and P.Jordan in 1950.
In 1932 Fitting (1) characterized those automorphisms
of (non-abel ian) groups, whose sum is an automorphism, too,
thereby implicitely starting to consider dgnr's.

1a Fundamental definitions and properties
13
Finally, the fifties brought the start of a rapid
development of the theory of near-rings.
Now we are going to define the analogue of the concept of a
module in ring theory: certain operator groups.
2.) N-GROUPS
1.17 DEFINITIONS Let (Γ. + ) be a group with zero о and let ΝεΤ?.
Let μ: ΝχΓ—>г . (Γ,μ) is called an N-group
(η,γ)-ηγ
("near-module over N" (but cf. the different meaning e.g.
in Karzel-Pieper (1) )) if
V γεΓ V η,η'εΝ : (η+η')γ = ηγ+η'γ Λ (ηηι)γ = η(η'γ).
If the meaning of μ is clear we write ..Γ for the N-group
above. Let N^f be the class of all N-groups. To simplify
the notation, ..r stands for N-groups throughout, without
further notice. See also Kuz'min (1).
1.18 EXAMPLES
(a) Let N be a nr. Then μ: ΝχΝ-
►Ν
(η,η')+ηη'
makes (N,+) into
an N-group, denoted by „Ν.
(b) Each (left) module Μ over a ring R is an R-group.
(c) Let Γ be a group. Then Γ is an M(r)-group μ,.,Γ
with μ: Μ(Γ)χΓ+Γ
(f.r) - f(y)
1.19 PROPOSITION Take ^Γ ε Η$ .
(a) \/γεΓ; 0γ = ο;
(b) \/γεΓ VneN: (-η)γ = -ηγ;
(c) V ηεΝ : no = ο;
(d) \/γεΓ V ηεΝ : ηγ
no .
Proof, (a) and (b): as for (ring-) modules,
(c) : no = nOo = Oo = о.
(d) : ηγ = ηΟγ = no.

14
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
1.20 DEFINITION ^Γ ε ц<% is called unitary if Νε ΎΙι and
\/ γεΓ: 1γ = γ.
Since 1?, У) , 71 , and all N^f are varieties in the sense of
universal algebra it makes sense to speak about a lot of things
(see also Prehn (1 )-(3)):
3.) SUBSTRUCTURES
1.21 DEFINITION
(a) A subgroup Μ of a nr. N with M.M
subnear-ri ng of N (notation: M<N).
Μ is called a
(b) A subgroup .Δ of ΝΓ with ΝΔ = Δ is said to be an
N
N-subgroup of Γ (Δ2..Γ).
+ )
1.22 EXAMPLES
(a) N and N are subnear-rings of N. Hence it follows
4 ' о с
from 1.13 that (N, + ) is a split extension of
its subgroups (N .+) and (N ,+). See Pilz (9),(10)
for the converse problem of constructing near-rings
out of a zero-symmetric and a constant one.
(b) If Nr is a (ring-) module then the N-subgroups are
just the submodules of Г.
Later on we will see that the subnear-rings of the M(r)'s
are in a certain sense already all near-rings. We know already
one procedure to get subnear-ri ngs of М(Г): the M<.(r)'s of
1.4. Two more methods are:
1.23 EXAMPLES
+ )
(a) Take a subgroup Δ of Γ. Μ (Γ): = {feM(r)|f(A) <= Δ} is
a subnear-rinq of Μ(Γ).
(b) Take a normal subgroup Δ of Γ.
ΜΓ/Δ(Γ): = {feM(T)|V γεΓ: f(Y+A) «= f(y)+A} is a
subnear-ring of Μ(Γ) (cf. Betsch (3)).
The term "N-subgroup of H" refers to N.

1a Fundamental definitions and properties
15
1.24 REMARK Wielandt (3) proposed a construction method for
subnear-rings of Μ(Γ) which gives the 3 kinds of subnear-
rings mentioned above as special cases.
The method is as follows:
Take any cardinal number a, form the direct product Γα
and a subgroup Δ of Γα . Each feM(r) can be considered
to be εΜ(Γα) if it is defined component-wise.
Let Μ .(Γ): = {feM(r)|f(A) έη Δ} < Μ(Γ). Then
Ot j й
(а) МД(Г)
Μ1.Δ<Γ>
<b> Μ{5ι>...,5α}(Γ) = Μα+1,Δ(Γ) with Δ={(γ,5ι(γ),...)5α(γ»|γεΓ}.
(с) МГ/Д(Г) =αΜ2>Ε(Γ) with E = {(γ,γ')|γ-Ύ'εΔ}
4.) HOMOMORPHISMS AND IDEAL-LIKE SUBSETS
1.25 DEFINITION Let N,N' be ε V and ^Γ,^Γ'ε^ ·
(a) h: N+N' is called a (near-ring) homomorphi sm if
\/ ιη,ηεΝ: h(m+n) = h(m) + h(n) л h(mn) = h(m)h(n).
(b) h: μΓ*νΓ* is called an N-homomorphism if
tf γ,δεΓ V ηεΝ: h(y+6) = η(γ) + h(5) Λ η(ηγ) = nh(y)
There seems to be no need for explicit definitions of
nr.-monomorphisms (Ну* Ν'), Hom(N,N'), Horner,Г'), Г ъ ..Г',
Ker h, Im h, and so on. If there exists a monomorphism N >-» N'
we say that N is embeddable in N' and write N5 N'. A similar
convention applies to N-groups.
1.26 EXAMPLE For all γεΝΓ: h : Ν+Γ ε HomN(N,r)
η->·ηγ
1.27 DEFINITION Let Νε У) and ΝΓεΝ<# .
(a) A normal subgroup I of (N,+) is called ideal of N
(I <1 N) if
а) IN * I
б) V η,η'εΝ V1εI: n(n'+i )-nn'εΐ.
Normal subgroups R of (N, + ) with a) are called ri ght

16
§1 THE ELEMENTARY THEORY OF NEAR-RINGS
i deals of N (R ά Ν), while normal subgroups L of
(N,+) with β) are said to be left ideals (L & N).
(b) A normal subgroup Δ of Γ is called ideal of „Γ
(Δ <3Ν Γ) if V γεΓ \/ δεΔ \/ηεΝ: η(γ+ό)-ηγεΔ .
Other names: N-kernel or submodule (cf. 1.33!). The
term "ideal" is motivated by (Kurosh) and is very
handy in formulating simultaneous statements about
N-groups and near-rings.
1.2B REMARKS The left ideals of N coincide with the ideals of ^N.
Moreover, one easily sees that a subgroup I of Μ (Δ of Γ)
is an ideal iff
n1 ξ nj (mod I)
n2 ξ η' (mod I) => Vn2 = ηϊ+η2 (mod T> Λ η1η2 = ηίη2 <modI)
Yj ξ γ} (mod Δ)
*2
(mod Δ) => γ1+γ2 Ξ ?1+*2 <mod Δ> Λ
л \/ηεΝ: ηγ« ξ ny^mod Δ), respectively).
So ξ (mod I or mod Δ) is a "congruence relation"
(cf. (Gratzer)) if Ι (Δ) is an ideal. If I <l N and
I + N, we write I <l Ν , etc.
In 1.27, (a ) β) and (b) can also be written as
\/ η , η ' ε Ν \/ i ε Ι : η(ι'+η')-ηη'εΙ and
\/γεΓ \/6εΔ V ηεΝ : η(6+γ)-ηγεΔ .
Factor nr's N/I (I ά Ν) and factor N-qroups Γ/Δ
(Δ aN Γ) are defined as usual (cf. any book on universal
algebra). If L й. Ν, then N/L is meant in the sense of
N-groups.
Clearly {0} and N are ideals of N as well as {o} and
Γ are ones of ,,Γ. These ideals are called the trivial ideals

1a Fundamental definitions and properties
17
1.29 THEOREM ("Homomorphism theorem").
(a) If I ^ N then the canonical map π: N+N/I is a
nr.-epimorphi sm. So N/I is a homomorphic image of N.
(b) Conversely, if h: N-*N' is an epimorphism then
Ker h <J N and N/Ker h = N' .
The corresponding statements hold for N-groups.
The proof is analogous to the one for groups, rinqs or
universal algebras, and hence omitted.
So ideals are just the kernels of (N-) homomorphisms.
As usual for "sophisticated" algebraic structures we get with
the usual proof:
1.30 THEOREM (so-called "2nd isomorphism theorem")
Let h: N-jfrN' be an epimorphism. Then h induces a 1-1-
correspondence between the
subnear-rings (ideals) of
N containing Ker h and
the subnear-rings (ideals)
of N' by A(<= N) - h(A):
Moreover, for all ideals I of N containing Ker h we get
N/I = h(N)/h(I).
If тг: N-*N/1 is the canonical epimorphism, we therefore
get for all ideals J of N containing I
N
"/
J/I
N/J .
Again the analogous statements hold for N-groups. Observe
in this case that for the last formula we have to assume
that J is also an N-group to make J/I meaningful
(cf. 1.33, 1.34).

18
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
1.31 DEFINITION A subnear-ring Μ of N is called invariant if
ΜΝ*ΞΜ and NMEM .
Invariant subnear-rings and ideals coincide in rings, but not
in near-rings:
1.32 PROPOSITION
(a) NQ :Sj, N, but not generally NQ <l N.
(b) N is an invariant subnear-ring of N, but in general
neither a right nor a left ideal.
Proof, (a) N is a left ideal: for all η,η'εΝ and
η rN„ we have (n+n -n)0 = nO+n 0-nO = 0 , so
oo v о ' о
n+n -ηεΝ , and [(n(n'+n )-nn')]0 = n(n'0+η 0)-ηηΌ =
= 0 , hence η (n'+n )-nn'εΝ .
N is not necessarily an ideal: N:= M(IR) , id.RF.N =
= MQ(IR) , 1_: Ш - IR eM(IR) , but id°j_ = _1фМ (IR) .
(b) N is an invariant subnear-ring:
\J η εΝ \/ ηεΝ : (nn )0 = nn and (n n)0 = η 0 =
= nn , which implies that ηη„εΝ„ and η ηεΝ„ .
с ее ее
N is not a left or right ideal in general, since
N is not always a normal subgroup of (N,+) :
Take a non-abelian group Γ and γ,δεΓ with
γ+δ 4= 6+γ . f : Γ-"ΓεΜ (Γ). Now (id + f -id)(o) = γ,
Y Χ-»γ c Y
but (id + f -id)(5) = δ+γ-5 4= Υ implying that
id+f -id^M (Γ) . So Μ (Γ) is normal iff Γ is abelian.
Υ с с
1.33 REMARK In general there is no direct connection between
N-subgroups and left ideals, as we have seen above. This
is the reason for avoiding the terms "near-modules" and
"submodul es": submodules would not be near-modules in
general, for ideals of N-groups are not necessarily
N-subgroups. So in general N-groups are not"Ω-groups"
("groups with multiple operators") in the sense of
(Kurosh) or (Higgins). This does not happen for zero-
symmetric near-rings (see also Prehn (1)-(3)):

1a Fundamental definitions and properties
19
1.34 PROPOSITION
(a) L at N -> NQL τ L
(b) N = N <—> each left ideal of N is an N-subgroup of N.
(c) Ν = Ν -> (Δ <L Γ => Δ <Ν Γ) for all Γ ε JJ.
Νϋ
Proof, (a) L <J N -> \/«.eL \/ηηεΝ : η Л = η (0 + ί,)-η Οεί.
о о
<-: {0} 4t L -> {0} <N N => N0 = {0} -> N = NQ
(b) ->: by (a)
<-: (0} si,
(c) is settled similarly
1.35 PROPOSITION
(a) V γε.^Γ: Νγ <Ν Γ.
(b) \/ Δ £Ν Γ : No = Nco ? Δ .
So No is the smallest under all N-subgroups of ..Γ. Throughout
this monograph we will write
No = Nco
Ω .
Of course, N = NQ implies Ω = {о}. By 1.19(d),
\j γεΓ: Ω = Ncy . Also, Ω ^Ν Nc .
1.36 DEFINITION
(a) Ν(ΝΓ) is simpl e: <=> Ν(,,Γ) has no non-trivial ideals.
(b) ΝΓ is called N-simple: <-> ..Γ has no N-subgroups
except Ω and Γ (cf. 1.35).
1.37 PROPOSITION If Ν(ΝΓ) is simple then all (N-) homomorphic
images are (N-) isomorphic either to {0} or to N({o} or Γ)
Proof: by 1.29.

20
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
1.38 EXAMPLES
(a) In §7 we will see that М(Г) (|г| > 2) and MQ(r) are
simple nr.'s (7.30, 7.33).
(b) See Blackett (4) for some more examples of simple nr.'s
of real functions.
(c) If N = NQ then N-simplicity implies (by 1.34(c))
simplicity for each мг£м^ ·
Since {0} is always minimal in the set of all ideals of N,
we define more interesting ones to be minimal:
1.39 DEFINITION A minimal ideal of N is an ideal which is
minimal in the set of all non-zero ideals. Similarly,
one defines minimal rinht ideals, 1eft ideals, N-subgroups
(minimal under all N-subgroups =)= Ω)» etc. .
Dually, one gets the concepts of maximal ideals etc. .
1.40 PROPOSITION ION is maximal in Ν (Δ <^Г is maximal in Г)
iff N/I (Γ/Δ) is simple.
Proof: 1.30.
Near-rings in which every (one- or two-sided) ideal f {0} is
maximal are studied in Ferrero-Cotti - Rinaldi (1),(2).
5.) ANNIHILATORS
We will need the "noetherian quotients" quite frequently:
1.41 DEFINITION Let Δ1·Δ2 be subsets of ΝΓ ε J% .
(Δχ : Δ2) : = {ηεΝ|ηΔ2 ? Δ^ .
Abbreviations: ({δ} : Δ2) = : (δ : Δ2), similarly for
(Δ:δ), (δ:Δ).
(ο:Δ) is called the annihilator of Δ.
If necessary, we indicate the nr. N involved by writing
(*1 : Δ2)Ν.

1a Fundamental definitions and properties
21
1.42 PROPOSITION Notation as above.
If Δ. is a subgroup (normal subqroup, N-subgroup, ideal)
of „Γ, the same applies to (Δ. : Δ~) in ,,Ν.
The proof is easy and therefore omitted.
1.43 COROLLARY
(a) \/ γεΓ : (ο:γ) ^ N
(b) \/ Δ <Ν Γ : (ο:Δ) <i N
1.44 PROPOSITION Let Δ,Δ. (i e I) be subsets of ΝΓ. Then
Π (Δ·:Δ) .{(14, : Δ) and U (Δ,:Δ) ? ( U Δ. :Δ).
ΙεΙ 1 ίεΐ i ε Ι 1 ε Ι 1
For η ε Π (Δ.:Δ) <-> \/ιεΙ : ηΔ ? Δ. <»> ηΔ s Π Δ,· <->
i εΐ 1 1 ΙεΙ
<—> η ε ( Π Δ· : Δ) and similarly for the union.
ίεΐ Ί
1.45 PROPOSITION Let Δ be a subset of ^Γ ε Ν^ .
(a) (ο:Λ) = Π (ο:δ)
6εΔ
(b) ΝΓ ^ νγ· -> (0:Γ) = (0:Γ·)
Proof: straightforward.
Consider the h 's of 1.26.
1.46 PROPOSITION Ker h = (ο:γ) , so Νγ - Ν/(ο:γ).
Proof: homomorphism theorem.
1.47 DEFINITION ΝΓ is called faithful if (ο:Γ) = {0}.
1.48 PROPOSITION ^Г faithful -> N с» М(Г).
Proof: Consider for each ηεΝ the map f :Γ+Γ . f εΜ(Γ).
γ*ηγ
Then h: N * Μ(Γ) turns out to be a near-ring homomor-
n * fn
phism with Ker h = {ηεΝ|fp = 6} = {ηεΝ|ηε(ο:Γ)} = {0},
So h is an embedding map.

22
1 THE ELEMENTARY THEORY OF NEAR-RINGS
1.49 PROPOSITION Let ^Γ be faithful.
(a) If Г is abelian then so is N.
(b) If \j ηεΝ Μ γ,όεΓ : η(γ + ί) = ηγ + ηί then ηεΝ(1 .
Proof: (a) by 1.4B and (b) by a straightforward calculation.
More generally one can prove that, if ^Γ is faithful, every
"identity which holds in Г" (cf. (Gratzer)) also "holds in N".
1.50 PROPOSITION Let ΝΓ be faithful. We assume that N ■= Μ(Γ)
nc(y) -
(by 1.48).
(a) Ω = {о} <-> Nc = {0} <=> ΝεΠ0;
(b) Ω - Γ <"> Νς = Мс(Г) <=> Γ ~Ν Nc.
Proof: (a) If Ω = {о} then \/ γεΓ \j ι^εΝ^
= η (ο) = ο = ό(γ), so ης = 6.
If Nc = {0} then N = Nq£7?0 .
If Νε>30 then Ω = ίο} by 1.19(c) .
(b) If Ω = Γ, take some тгМс(Г). т(о) =:yq.
Then 3nceNc : nc(o) = YQ.
So MycT : m(y) = m(o) = Y0 = nc(°) = пс(у)· hence
nc and Nc = Μς(Γ).
If N. = Mr(r) then the map h: Ν +Γ
с с с
is an
nc-nc(o)
N-i somorphi sm.
Finally,if Г ■
=N N by some N-isomorphism h, take
an arbitrary γεΓ . h(y) =: ncENc. Then h(y) =
" nc = ncnc = nch(y) = h(nc^Y)) = h(nc(°^·
So γ = nc(o) ε Ω and Ώ = Г.
See also Ferrero-Cotti (7) and Scott (15).

1a Fundamental definitions and properties
23
6.) GENERATED OBJECTS
1.51 PROPOSITION
(a) The sets of all ideals (right ideals, left ideals,
N-subgroups, invariant subnear-rings) form inductive
Moore-systems on N.
(b) The sets of all ideals (N-subgroups) of an N-group ,,Γ
form inductive Moore-systems on Γ.
Hence it makes sense to speak about the "ideal (...) generated
by a subset".
1.52 PROPOSITION (Scott (6)) Let R?N with RNsR . Then the
left ideal LR generated by R is an ideal.
Proof: RN ? R «Ξ LR , so Re (LR:N). Since (LR:N) «^ N
by 1.42, LR ? (LR:N) . Therefore LRN ? ι showing
that LR «3 N.
See also 2.16 and 9 .174.
1.53 THEOREM (Beidleman (1))
(a) If N is fg. (0.3) as an ideal (cf. Van der Walt (2),(3))
(e.g. if Νε?!|) then each ideal (right ideal, N-subgroup)
different from N is contained in a maximal one.
(b) If ρ,Γ is f.g. as ideal with N eft then every proper ideal
ideal (N-subgroup) of ,,Γ is contained in a maximal one.
Proof (for ideals I of N). Let N be generated by (say)
x,,...,xk. X:= {I|I<N}. Take some chain I^I^...
of elements of £. I: = !J In 3 N by 1.51(a).
ηεΙΝ
If I = N then all of x1§. ..,χ^εΐ. Hence there is
some seIN with x^,... ,χ^εΙ$ . But then I$ = N,
a contradiction.
So (X,£) fulfills the hypothesis of Zorn's Lemma
(unless N = {0}, a trivial case) and conseguently
contains a maximal element.
If Ν ε7?,, one proceeds as in ring theory.

24 §1 THE ELEMENTARY THEORY OF NEAR-RINGS
b) CONSTRUCTIONS
1.) PRODUCTS, DIRECT SUMS AND SUBDIRECT PRODUCTS
For 1.54 - 1.62 cf. each book on groups, rings or universal
algebra. We cite e.g. from (Gra'tzer).
1.54 DEFINITION Let (N Ί· ) i c ι be a family of near-rings.
X N. with the component-wise defined operations "+"
ΙεΙ 1
and "·" is called the direct product Π N. of the
i ε I
near-ri ngs N. (i εΐ).
1.55 DEFINITION The subnear-ring of Π N. consisting of those
' lei 1
elements with all components - except a finite number
ε IN - equal to zero, is called the (external ) di rect sum
θ N. of the N{'s.
ΙεΙ 1
More generally, every subnear-ring N of Π N. where all
iel '
projection maps ». (ΙεΙ) are surjective (in other words,
\/ i ε Ι \/ η.εΝ. : η. is the i-th component of some element
of N) is called a subdirect product of the N.'s.
The definitions of products, direct sums and subdirect
products of N-groups should be clear now (for direct sums
you need N = NQ). Again we refer to Prehn's papers.
1.56 NOTATION If the Ni (iel) are as above, let N. be given by
N\: = {(...,Ο,η^Ο,,.Οίη^Ν^.

1b Constructions
25
1.57 PROPOSITION
(a) \] ιεΐ: N. - N л N. <l © Ν, Λ TL <1 Π Ν. Λ
1 ] Ί jel 3 jcl J
Λ Ν, Q © Ν. Λ Ν. Q Π Ν, ;
1 Jel J ' jcl J
(b) |N. d Π N. ;
ιεΐ Ί lei
(c) Jel -> © N. Q © Ν. Λ Π Ν. е> Π Ν. .
jeJ J ΐεΐ ] jeJ J ιεΐ 1
1.58 REMARKS (cf. (Gratzer)). If N is a subdirect product of
near-rings N. (ι ε I) then the N.'s are homomorphic
images of N (under the projection maps π·). If Ker π^ =:Ki
we get a family (K.). T of ideals of N with zero inter-
v ι ι ε I
section.
Conversely, if a family of ideals < К.)^ ε τ of some nr. N
with f| К. = {0} is given then N is isomorphic to a
ΐεΐ 1
subdirect product of the near-rings N·: = N/K·.
Of course, 1.56 - 1.58 can be transferred to N-groups in the
obvious way.
1.59 DEFINITION A subdirect product N of near-rings N^ (i ε I)
is called trivial if 3 1 ε I : π- is an isomorphism.
Νε Ύ) is called subdirectly irreducible if N is not
isomorphic to a non-trivial subdirect product of near-
rings .
The same is defined for N-groups.
1.60 THEOREM ((Gratzer), Fain(l)). The following conditions
for a nr. Ν 43 t0} are equivalent:
(a) N is subdirectly irreducible;
(b) If (I ) . is a family of ideals of N with
(Ίΐ = (0} then 3 αεΑ : I = {0};
αεΑ α α
(c) Π I + iO};
{0}4=I<3N
(d) N contains a unique minimal ideal, contained in all
other non-zero ideals.

26
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
Replacing "N" by "Γ" yields an analogous theorem for
N-groups.
1.61 COROLLARY Each simple nr. (N-group) is subdirectly
i rreduci Ы e.
1.62 THEOREM ((Gratzer), p.124).
(a) Each near-ring is isomorphic to a subdirect product
of subdirectly irreducible near-rings.
(b) Each N-group is N-isomorphic to a subdirect product
of subdirectly irreducible N-groups.
The intersection of all non-trivial ideals is also considered
in llartney (3).
2.) NEAR-RINGS OF QUOTIENTS
1.63 DEFINITION Let N be a nr. and S a subsemiqroup of (N,·)·
A near-ring Ns is called a near-ring of rinht (left)
quotients ofNw.r.t. S if
(a) Ns eTlj
(b) N^N$ (by h, say)
(c) V seS: h(s) is invertible in (N<-.·)
(d) V qeNs 3 seS 3 ηεΝ : q = h(n)h(s)_1 (q = h(s)_1h(n)),
Of course there arise the questions about existence and
uniqueness of such near-rings of quotients. We will settle these
questions after the following
1.64 DEFINITION N is said to fulfill the right (left) Ore
condition (Ore (1)) w.r.t. a given subsemiqroup S of
(N,·) if
\/ (s,n) ε $χΝ 3 (Sj,n.) ε S xN : nSj = snj (s^n = n^s)

1b Constructions
27
1.65 THEOREM (Graves-Ma lone (1)). Let S be a subsemigroup of
(Ν,·)· N has a nr. of right quotients w.r.t. S <->
<-> (a) S + 0
(b)V seS : s is cancellable (on both sides)
(c) N satisfies the left Ore condition w.r.t. S.
Proof. =>: Assume that N has a nr. N<- of left quotients
w.r.t. S, and let h be as in 1.63.
(a): By 1.63(d)
(b):\/ seS \j m,ncH : ms = ns => h(m)h(s)
—> h(m) = h(n) => m = η .
Similarly, sm = sn=> m = n.
,-1
h(n)h(s)-
so by
(c): Take ηεΝ, scS. q: = h(s) xh(n) ε Ν$
1.63(d)
3nlEN 3slES : h(s)"!h(n) = h(ni)h(Sl)-1 ;
Therefore hfnsj) = h(snj) , whence ns ^ = snj
<—: Similar to rinq theory (cf. (N. Jacobson), p. 262):
Define an equivalence relation ъ on N*S by
(n ,s )^(n' ,s' ): <=> Bn^N 3 SjcS : (ss^ = s'n1 and
ns ^
η
п'П|). Let - be the equivalence class of (n,s) and
NxS/^ =: N<-. If -,—reNs, we might follow a suggestion of
H.J.Weinert to net "common denominators": - = —Д, —r =
s s s ι s
η 'η ,
sn.
η η .
with ( η . , s , ) ε Ν χ S fulfillino s'n,
ss. 1 1 - 1
We then are able to define with these notations:
ss , ε S
ns.+n η .
ss.
and
n_ η '
s 'I-1-
nn,
s s,
<S fulfills nSj^n's. ε S.
+ and · are shown to be well-defined and (N5,+,·)
turns out to be a nr. with identity e = -| (s any
element of 3). If teS, the map h:N ■* Ns: η -+ -^
is a mononorphi sn and e'jery h(s)=-z- eri(S) has the
inverse Дг. Every — can be written as h(n)h(s)
S t S \ I \ I

28
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
1.66 THEOREM If N has nr!s of left quotients N$,N^ w.r.t. S
then
Ns - Η' .
(We may therefore speak about "the nr. of left quotients
w.r.t. S".)
Proof. N fulfills the right Ore condition by 1.65.
Let h,h' be as in 1.63(b). Define F: Ns -* ГЦ
by h(n)h(s)_1 - h'(n)h'(s)-1. F is well defined:
if h(n)h(s)_1 = h(m)h(t)-1 then Bft^n^eSxN :tn2 =
= st. and nt. = mn.. So h'(n) = h ' (m) h ' (n1 )h ' (t)~ '
and h'tnp = h ' (t)"1h ' (s)h ' (t1 ). Therefore we get
h'(n) = h ' (m)h ' (t) h'(s), which allows us to conclude
that h1 (n)h' (s)"1 = h'(m)h ' (t)"1.
Clearly, F is an isomorphism.
1.67 REMARKS As Maxson (1) and Graves-Mai one (1) pointed out,
1.65 does not hold for near-rings of left quotients,
because addition in N s (as in 1.65) is not necessarily
wel1-defιned.
Tewari (1) even showed that there exist near-rinqs having
nr'.s of left (right) quotients but no nr!s of right (left)
quotients.
1.68 DEFINITION If S = {seN|s cancellable} then N$ (if it
exists) is called the right (left) quotient near-ring
of N.
In the section on near-integral domains (§9b2)) we will
consider the case that Νς is a near-field.
1.69 REMARK In Chan-Chew (1) a characterization of right
quotient near-rings by means of "semi-N-homomorphism"
is given. See also Holcombe (6), Oswald (9), Seth (1),
Seth-Tewari (2), Shafi (1 ).

1b Constructions
29
3.) FREE NEAR-RINGS AND N-GROUPS
For this number, we again use Gra'tzer's terminology and results.
Let V*be any variety of near-rings (e.g. all near-rings, all
abelian near-rings, all near-rings with unity or all distributive
near-ri ngs).
Let X be any non-empty set.
1.70 DEFINITION A nr.
F ε V is called a f ree-near-ri rui in_V"_
over X if 3 f:X-Fv V Νε V \/ g:X+N 33 heHom(Fv,N):h°f=g
— X л
f
(in diagram notation:
///
->FV
)
/\i
From (Gra'tzer) we deduce
1.71 THEOREM In this case, f is injective while X can be
regarded as a subset of F„ and generates F„. F„ is (up to
isomorphism) uniguely determined by V" and |X| and has
the form indicated in (Gra'tzer), p. 163. One therefore
is able to speak about "the free nr. in V" determined
by some cardinal a".
After several glasses of wine one would describe Ρχ
loosely as the "set of all sums and products of elements
of Χυ{0} (and possibly more 0-ary symbols such as I)
where one can calculate according to the laws which hold
in V".
1.72 DEFINITION If V = Yl, we simply speak about the free nr.
on X. A near-ring is called free if it is free over some
set X.
1.73 REMARK In the same way we define free N-groups in some
variety of N-groups, free N-groups over some cardinal
number and free N-groups. 1.71 can be transferred to
N-groups by making the obvious alterations.

30 §1 THE ELEMENTARY THEORY OF NEAR-RINGS
1.74 EXAMPLES If Νε Ύ)ι then ^N is free over {1}.
It is harder to be a free nr. (N-group) than a free ring
(module). This fact is rewarded by
1.75 THEOREM If F,F' ε Ύ) (Νφ'Νφ'εΝ^* are free over sets X,X'
then F - F' (Φ -N Φ') <=> |X| = |X'| .
Proof. <" is settled by 1.71.
—>: Ύ) (m^[) contain finite structures with more than
one element (fornr'.s e.g. the field TU, for
N-groups e.g. (2»,+) with nO = nl = 0 for all
ηεΝ). Now apply (Gr'a'tzer), p. 197.
1.76 REMARK Note that the theorem above also holds e.g. for a
free nr. (N-group) in the variety of abelian nr!s (N-
groups) ("free abelian nr'.s (N-groups)").
1.77 REMARK Let N<t> be free over X. The usual characterization
(in the case of unitary (ring-) modules) of X as a base
("linearly i ndependent generating set") does not carry over
to the case of N-groups directly: N-groups do not have
to be unitary, the lack of commutativity in ,,Φ causes
"linear combinations" (defined as usual) to be influenced
by the order of the summands (as Maxson (1) pointed out,
one has to define linear combinations in terms of ordered
sets of elements of мФ). and - most troublesome of all -
N
Φ usually consists of more then the set of all linear
combinations, since in general η(η,γ, + ...+ nkYk) is no
linear combination any more.
Anyhow, generalizing the concept of linear independence
gives something like a base: let W (ηεΙΝ„ ) be the set
э => П v 0 '
of all n-ary words over some set X in a variety V of
N-groups and (for ^Γε V") w the induced function Γη ■+ I
Define in W„ wvw': <—> \/Δεΐ7": w, = w! .
η ν * Δ Δ

1b Constructions
31
1.78 DEFINITION A subset В of .Γε V is called independent if
\/ neINQ \/w = w(x1 xn)eWn Ϋ^ι·· · · ·βηεΒ* Bi+Sj for
i+j : (WpCPj Sn) = о -> w -γό).
1.79 REMARK Let RM be a unitary ring-module with \/reR: rM =
r {o} ~> r = 0 (otherwise «M would have no linearly
independent subset at all).
Then each subset of RM is linearly independent iff it is
independent in the sense of 1.78.
1.80 DEFINITION Β € ΝΓ is called a base for ^Γ if
(a) В generates ,,Γ
(b) В is independent.
As usual, the following questions arise:
(a) Which N-groups have a base ?
(b) Are different bases equipotent ?
1.81 THEOREM В ч ,,Γ is a base for ,,Γ iff the inclusion map
t: В ■+ Г can be extended to an N-i somorphi sm Φ-*Γ,
where Φ is the free N-group on B.
Proof. ->: Let В be a base for „г.
Consider the diagram
Since Φ is free on B, there is exactly one
ίιεΗοπι,,(Φ,Γ) with h°f = ι . We have to show that
h is an N-isomorphism:
(a) Β?ίι(Φ) л В generates Г -> ίι(Φ) = Γ
(b) Let φεΦ be in Ker h. Represent φ by some word
w(f($1),...,f(Sfi)) over f(8) (S^fij for 1+J)-Now
о = И(ф) - h(w(f(B1),....f(3n))) =
- wr(h(f($j) h(f(Bn))) = wr(i(B1),...,i(Bn)) -
«r(8, 3,
is indeoendent. со
= о -

32
§1 THE ELEMENTARY THEORY OF NEAR-RINGS
<=: Suppose that ι extends to an N-isomorphism
η:Φ-»Τ. By the construction of ,,Φ , f(B) is independent
and generates ,,Φ. So f(B) is a base for
В = г(В) = h(f(B)) is a base for п(Ф) = Г.
Ν4
Hence
From this theorem we immediately deduce
1.82 THEOREM
(a) ΝΓ has a base <=> ,,Γ is free.
(b) ,,Γ has a base В <=-> each map f from В to some N-group
N
Δ can uniquely be extended to an N-homomorphism
ΝΓ * ΝΔ
And from 1.75 we get
1.83 THEOREM Let ,,Γ bf> a non-zero N-group possessing a base.
Then all bases are equipotent.
1.84 EXAMPLE If N is in У1 λ then ^N has a base (namely {1})
and all other bases consist of one single element.
1.85 REMARK See more on free products etc. in Fro'hlich's
paper (4), in Meldrum (2),(3) and in Rao (1).
Cf. also Frohlich (4) and Maxson (1) for a characterization
of a base in terms of free products.
See Zeamer (1) for an "arithmetic" in free near-rings. Free
sums (products) are studied in Prehn (3) and Rao (1). See also
Meldrum (13), Banaschews ki -Mel son (1) and John (1).

1c Embeddings
33
с) EMBEDDINGS
1. ) EMBEDDINGS IN М(Г)
The reader might be wondering if all near-rings are near-rings
of functions on some group Г. This is true, although near-rings
are also considered under totally different aspects.
The main result is
1.86 THEOREM tf Νε>7 3 Γε^ : Nc>M(r).
Proof.(Heatherly-Malone (1)). Let Γ be any group properly
contai ning (N, + ).
For ηεΝ, define f : Γ ■+ Γ .As one can
fηγ γεΝ
γ Ιη υΉ
easily see, V η,η'εΝ : fn+fn. * fn+ni л
л fn°f , = fnni. Thus the map h : N - M(r) is
n * fn
a homomorphism.
If h(n) = h(n') then f = f , . In particular,
V γεΓ\Ν : η = fn(Y) = ^,(υ) - η'.
This implies that h is in fact a monomorphism and
an embedding map, as desired.
1.87 REMARK There are several proofs for 1.86. See e.g.
Berman-Si1verman (3), Nbbauer (8), Heatherly-Malone (1).
While Nbbauer embeds in Μ(Γ) with Г: = (M((N, + )), + ) ,
Heatherly-Malone suggest Г: = (N,+) β (Ζ2·+)·
1.86 and its proof have many interesting corollaries. Some
are in

34
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
1.88 COROLLARIES
(a) If N is abelian there is an abelian group Γ with
Μς,Μ(Γ) .
(b) If N is finite there is some finite Γ with Nc+M(r).
(c) V Νε>70 3 Γε<$ : Ν <=»Μ0(Γ) .
(d) V НсПс 3Γε^ : N<*Mc(r).
(e) If ΝεΤ), embeds in Μ(Γ) by φ and ηεΝ is invertible
then φ(η) is bijective.
(f) Every near-field is isomorphic to a nf. F of functions
on a group Γ, where all ίεί* are bijective.
1.89 THEOREM Each N-group can be embedded into some faithful
N-group. So each nr. N has some faithful N-qroup.
Proof. If ,,Γε,,^ , take some group Γ' ?Γ with Ne»M(r").
Evidently, Г' is an N-group in the natural way and
moreover a faithful one.
1.90 REMARK Heatherly (1) showed that N = Nrf =->
"> 3Γε^ :NcyE(r) (1.15). See Meldrum (13) for an example
of a faithful, simple N=N -group Γ (with Ν ε У\.) which is
not un i tary.
It is sometimes desirable to look for an embedding of N into
М(Г) with a "smaller" Г as above. Recall that in 1.86 and 1.87
e.g. М(Г) is embedded into the much bigger Μ(Μ(Γ)©'Ζ_)!
For doing this, we generalize a concept due to Menger:
1.91 DEFINITION В ? N is called a base (of equality) if
V η,η'εΝ: ( VЬгВ : nb = n'b) -> η = n'.
1.92 REMARK Clearly В forms a base iff (0:B) = (0), so it
would not be necessary to use a special name. But we do
it, because it is a very suggestive one.
1.93 EXAMPLES In М(Г) the set Μ (r) (a group isomorphic to Γ)
forms a base. In MCont(r) (1.4(a)) it suffices to take
a dense subset of Γ.

1c Embeddings
35
This motivates the interest in the case that the constants N
form a base. This can be achieved by force:
1.94 PROPOSITION Let π be the natural epimorphism N+N/.-.. )'
Then π(Ν ) forms a base for ττ(Ν) . c
Proof. If η = π(η) and η, = π(η,) are επ(Ν) then
( ν^ςεπ(Ν(.) : nn"c = H^) ->( \/ nccNc : ηη(;-η1η(;ε(0 : Nc ) )=>
-> (VnceNc: 0 = (nryn^n,. = nnc-ninc = (n-n^nj =>
-> (π-π1)ε(0:Ν(;) => η = rij .
1.95 EXAMPLES
(a) In Μ(Γ), the constants form a base.
(b) In 7 [χ] , the constants 1 do not form a base.
In fact, xp-x =f" о (zero polynomial), but
VcteZ : (xp-x)(a) = ap-a = 0. Therefore χΡ-χε(0:Ζ ).
(0:2 ) consists of all polynomials whose correspondinq
polynomial function is the zero map.
The following solves the problem stated after 1.90.
1.96 THEOREM If Β <Ν Ν , the following conditions are equivalent:
(a) B is a base (of equality);
(b) В is a faithful N-group;
(c) Ν«*Μ(Β).
Proof. 1.48 and 1.92.
1.97 COROLLARY If Nc is a base then N can be considered as a
near-ring of functions on N. In view of 1.95(a), this
is "the natural representation of N".
1.98 DEFINITION Let Γ,Δ be groups. feM(r) is called kernel-
free if \j γεΓ : (f(y) = ο -> γ = ο). Put М(Г)«^к Μ(δ) if
there is some h: Μ(Γ) >+ Μ(Δ) such that h sends kernel-
free elements of Μ(Γ) into kernel-free ones of Μ(Δ),
andM(r)=. Μ(Δ) if М(Г)е>кМ(Л) by an isomorphism h.

36
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
1.99 THEOREM (Heatherly-Malsne (1)). Let Γ,Δ be groups.
Then Ге>д <-> Μ (Г)с% Μη(Δ) <-> М(Г)с>М(Д).
о к о
Proof, (a) Let Γ<*Δ by h. If Г = {о} , the result is
obvious. Assume that Г + {о} and take some
arbitrary, but fixed γεΓ, γ =f о .
If feMQ(r) , define ίγεΜ0(Δ) by
f : Δ - Δ
Ύ
6 ^ rh(f(h_1(5))) δείιη h
L h(f(Y)) 6£Im h
If f is kernel-free, the same applies to f and
the map f*f embeds Μ0(Γ) into MQ(A).
(b) If Μ (Γ) с» Μ„(Δ) by (say) g then take some
о k о
fixed 5εΔ* (Δ = ίο} is again trivial).
h : Μ°(Γ) -ν Μ°(Δ) (Notation as in 1.4(a)) is a
f« * fg(fa)(5)
group homomorphism. If a + 0, f is kernel-free,
a
h is moreover injective:
h(f ) = о "> h(f )[s) =o, so δ=ο (a
contract α
diction) or α = ο , whence * = б.
Hence М°(г)е»М°(л) (as groups). But M°(r) and
Γ are isomorphic groups, and the same applies to
Μ°(Δ) and Δ .
So Гс»Л .
(c) If Γ<^Δ then proceedinq as in (a) one sees
that М(г)«чМ(л) .
(d) If Μ(γ)«*Μ(δ)then Mc(r)c,Mc(δ) by restriction.
МС(Г) = Γ and Мс(л) = д implies that Гс»Л .
1.100 COROLLARY (Beidleman (5)) Г - Δ <-> MQ(r) - М0(д) <->
<=> М(Г) - Μ(Δ). к
с
Since each group г can be embedded into some M(S) = S (S a
suitable set) by Cayley's theorem, we get from 1.99

1c Embeddings
37
1.101 COROLLARY (Nb'bauer (8)). For all ΝεΎ) there is some set
S with Nc>M(M(S)). (More precisely, for every nr. N
there is a set S and a near-ring N' in M(M(S)) such
that N - N'.
2.) MORE BEDS.
Since M(r) contains an identity (id ) we pet from 1.86
1.102 COROLLARY Every (finite, abelian, zerosymmetric) near-
ring can be embedded into a (finite, abelian, zerosymmetriс)
near-ring with identity.
1.103 REMARK Despite this analogy to ring theory the
embedding is totally different from the one in ring
theory. Moreover, N is not always embedded as an ideal.
1.104 PROPOSITION Each ring (ring with unity, field) can be
embedded into a non-ring (non-ring with identity, non-
field).
Proof, (a) For rings (rings with identity) it follows from
1.86. See Clay (3) for another proof.
(b) Let F be a field (Maxson (7)). Take F(x) and
define for f = ^eF(x) d(f): = deqfj-degfg . For
a.beF , a + 0 put θ :F(x)+F(x) . Clearly θ is 1-1.
У ■* ay+b
Define for f.geF(x) f*fig: = {° H/f\ = °
θ l(9d(f)°g).f f + О
Then it is easy to see that F.(x): = (F(x),+,»„)
is a near field and a field iff θ = id .
i:F ■+ F„(x) is the desired embedding map.
1.105 COROLLARY (Maxson (7)). Each commutative ring R without
zero divisors -f 0 can be embedded into a non-field.

38
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
Proof. R can be embedded into an integral domain,
which can be embedded into some field.
Now apply 1.104.
1.106 REMARK See more on embeddings in the chapter on nf.'s and
near integral domains. See also Beidleman (10), Plotkin (1),
(2) and Prehn (1).
d) SOME AXIOMATIC CONSIDERATIONS
In this section we compile some results on the axiomatics of
near-rings: conditions for N to be a ring or to be abelian,
cancellable and invertible elements and a brief survey of
structures which are closely related to near-rings.
1.) MISCELLANEOUS RESULTS
1.107 PROPOSITION Let N be a nr.
(a) N abelian л N commutative <—> N is a commutative ring;
(b) N abelian л N distributive <=> N is a ring;
о
(c) Ν = Ν Λ Ν distributive -> N is a ring (Taussky (1)).
Proof, (a) and (b) are obvious.
(c): \/ η,η'εΝ 3 a,b,c,deN: η = ab л η' = cd.
Computing (a+c)(d+b) in two different ways yields
ad+cd+ab+cb = ad+ab+cd+cb, so cd+ab = ab+cd,
therefore n+n' = n'+n. Now apply (b).
From 1.107(c) and the fact that there exist non-abelian
distributive near-rings (1.15) we deduce
1.108 COROLLARY Not every distributive nr. can be embedded into
a distributive nr. with identity.

1d Some axiomatic considerations
39
1.109 PROPOSITION Each of the following conditions imply a
near-ring N with identity to be abelian:
(a) V ηεΝ : n(-l) = -n;
(b) (Ligh (6)) N finite Λ \/ηεΝ : η(-1) = η -> η = 0.
(c) (В.Η. Neumann (1)) ( \/ηεΝ 3 ΗεΝ : η = h + h) л
л (\/ ηεΝ : η(-1) = η => η = 0).
Proof, (a) V η,η'εΝ : η+η' = (-η)(-1)+(-η')(-1) =
= (-η-η')(-1) = -(-η-η' ) = η'+η.
(b) Define α: Ν -»■ Ν . Clearly αεΑυί(Ν, + ) and
2 η - η("1)
α = id. α(η) = η implies η = 0. So by a theorem
of group theory (e.g.(W.R. Scott),p. 357), N is
abeli an.
(c) α (as above) is again a fixed-point-free
automorphism of order 2. From group theory (B.H.
Neumann (1), p. 206) we know that N is abelian.
1.110 REMARK McQuarrie (2) showed that 1.109(b) does not hold
in the infinite case.
We now consider cancellable elements.
1.111 PROPOSITION Let N be a nr.
(a) (Maxson (1)) ηεΝ is right cancellable <=>> η is not
a right zero divisor;
(b) (Maxson (1)) ηεΝ is left cancellable "~± η is not a
left zero divisor;
(c) (Timm (3)) If Νε7) then the left cancellation law
implies the right one.
Proof, (a) is shown as it is done for rings.
(b) If η is left cancellable and if nn' = 0 = nO
then n' = 0. To see the "<+"-part, consider near-
rings of the type Μ?(Γ) (as introduced in 1.4(a)).

40
§1 THE ELEMENTARY THEORY OF NEAR-RINGS
(c) If n'n = n"n, η + 0, then (n'-n")n = 0 =
= (n'-n")0, so the left cancellation law implies
n' = n" .
1.112 REMARKS Heatherly (1) proved that a finite nr. N has
either only right zero divisors or a right identity.
Ligh (1) showed that if the right identity is unique
then N is a nf.
Moreover, Ligh (1) proved that in a finite non-abelian
near-ring without non-trivial zero divisors each element
has a unique square-root.
Ramakotaiah-Reddy (1) showed that if N is generated by a
left zero divisor then N£2?..
An application of the embedding theorem 1.86 is
1.113 PROPOSITION (Heatherly (1)) Let ΝεΤ^ be finite.
If ηεΝ has a one-sided inverse then this inverse is
two-sided.
Proof. By 1.88(b) there is a finite group Γ with
h
N С*М(Г). If ηεΝ has a one-sided inverse then
h(n) has the same. Since г is finite, h(n) is
a 1-1-map and therefore invertible.
1.114 REMARK If N,N' are nr's and h is a homomorphism (an
automorphism) from N to N' then h/N and h/N are
also nr.-homomorphisms (nr.-automorphisms).
Conversely one might ask whether each pair h : N ■* N'
and hc: Nc -»· N' of homomorphi sms (automorphisms) can
be "mated" together to give a nr.-homomorphism
(automorphism) from N to N'. As Malone (1), (4) pointed out,
this is not the case in general:
h: N ► N' is a near-ring homomorphism iff
Vnc * ho<no> + hc(nc>
\/n0eN0 \jnQzHc : hc (П(. ) + ho(no) = ho(n0)+hc(nc) and
VneN VmeNQ : h(nm) = h(n)h (m).

1d Some axiomatic considerations
41
2.) RELATED STRUCTURES
1.115 SEMINEAR-RINGS
A set S together with two binary operations "+" and "·"
is called a semi near-ri ng if (S, + ) and (S,·) are
semigroups and \/s,s',s"eS : (s+s')s" = ss" + s's" .
EXAMPLES: The sets of all mappings on an (additively
written) semigroup with pointwise addition and composition,
e.g. М(Ш) .
REFERENCES: Pilz (5), Van Hoorn, Van Hoorn-Van Pootselaar,
Van Rootselar, Weinert (14).
1.116 NEAR-ALGEBRAS
A vector space A over a (skew-) field F together with an
additional binary operation "·" is called a near-algebra
over F if (A,+,·) is a near-ring and
\/ a.beA V XcF : (Xa) -b = X(a-b).
EXAMPLES:Take the sets of all mappings of a vector space
r.V into itself with pointwise defined addition, composition
and forming λ-folds.
REFERENCES: H.D. Brown, Marin, Timm (8), Williams (2),
Yamamuro (1) - (4).
ATTENTION Cf. Holcombe (5) for an essentially different
definition of a near-algebra.
1.117 COMPOSITION RINGS (TRI-QPERATIONAL ALGEBRAS)
A set R together with 3 binary operations "+", "·", "°"
is called a composition ring (tri-operational algebra,
TOA) if (R,+,·) is a ring, (R.+.°) a near-ring and
if V r,r',r"eR: (r.r')°r" = (r»r")·(rOr").
EXAMPLES: The sets of all mappings of a ring into itself
with pointwise addition, multiplication and composition.

42
§ 1 THE ELEMENTARY THEORY OF NEAR-RINGS
REFERENCES: Adler, Burke, Clay-Doi (2), Heller,
Hannos, Menger, Milgram, Nobauer (1) - (9),
Penner (1), Pilz (1),(3), Riedl, Steinegger,
Stueben, Suvak.
1.118 A GENERAL PROCEDURE
Take a universal algebra A = (Α,Ω), form the set M(A)
of all self-maps of A and define the operations of Ω
pointwise on M(A). Adding the binary operation "°"
of composition yields a new algebra M(A) = (M(A), Ωο{ο})
EXAMPLES:
naked set
semigroup
group
module
vector space
ring
near-ri ng
1i near algebra
Ω-group
M(A)
semi group
semi near-ring
near-ring
( - )
near-a1gebra
composition ring
( - )
( - )
"Ω-composition-oroup"
( - ): there exists no special name.
REFERENCES: Berma n-S i 1 ve rman (3), Hule, Lausch (2),
Lausch-Nbbauer, Mlitz, Nobauer (8),( 10) , (11) , Pilz (4),
Poli n, Stefanescu (1) .
Algebras of functions in more than one variable are studied
e.g. in Menger (3), Nobauer (2) and Stueben (1).
Such a lot of interesting structures ! Since one might be
attempted to start looking at them more thoroughly we switch
back to near-rings very quickly.

43
§2 IDEAL THEORY
In this paragraph we develop an ideal theory similar to that
one for rings.
After defining sums and (internal) direct sums of ideals
(of N and ,,Γ) we note that, unlike the ring case, internal
and external direct sums of ideals are not necessarily
isomorphic. We call an internal direct sum with this property
a "distributive sum", and prove that for N = N each direct
sum is distributive.
Also, we consider the lattices of ideals (and left ideals in N ).
Of course, these lattices are complete modular ones. If N = N
and no non-zero homomorphic image of N is a ring then the
lattice of left ideals is even distributive. Chain conditions
play an important role throughout this monograph. We prove
for example that N has some chain condition iff a direct
summand I of N and N/I have the same one.
If N has the DCC on ideals then it is a finite direct sum of
indecomposable near-rings. N is called completely reducible if
it is the direct sum of simple ideals. This is the case iff
every ideal is a direct summand, and then each ideal of N has
the same properties. N is a finite direct sum of simple ideals
iff N is completely reducible and has DCC and ACC on ideals
or (equivalently) iff N is completely reducible and has one
of the chain conditions on ideals or (again equivalently)
iff N is completely reducible and finitely generated which
is in term equivalent to the existence of finitely many
maximal ideals with zero intersection. Any two such
decompositions are isomorphic.
Finally, we develop the theory of (semi-) prime and nil(potent)
ideals which runs fairly parallel to ring theory: every near-
ring has minimal prime ideals; the intersection of prime ideals
is semiprime; if I d N then N is nil(potent) iff I and N/I
are nil(potent); if N = N has DCC on left ideals then N is

44
§2 IDEAL THEORY
a prime near-ring iff N has a unique minimal ideal which is
not nilpotent.
Many results carry over to N-groups with Hcfl' .
a )
SUMS
1. ) SUMS AND DIRECT SUMS
2.1 - 2.11 are formulated for ideals of near-rings; but all
(except 2.6(b)) can be transferred to N-groups with ΝεΠ by
making the usual changes. The proofs in these considerations
run parallel to group or ring theory and are therefore omitted.
Also, it is pointed out that these results follow from the
general theory of "Ω-groups" (see (Kurosh) or (Higgins))
(note that nr'.s are Ω-groups, and N-groups are Ω-groups if
ΝεΤ) ) . See also Prehn (1 ).
2.1 THEOREM Let (I|()keK be a family of ideals of a nr.
N.
Then the following sets are equal:
(a) The set of all finite sums of elements of the K's;
(b) The set of all finite sums of elements of different
(c) The sum of the normal subgroups (Iu>+);
(d) The subgroup of (N, + ) generated by |J I. ;
keK
(e) The normal subgroup of (N,+) generated by
(f) The ideal of N generated by (J I■,
keK K
keK K
2.2 DEFINITION The set (a) - (f) above is called the sum of
the ideals It (keK) and denoted by £ I.
keK
(for К = {1,2,...} also by Ιχ+Ι2+...)-
From 2.1 (d) - (f) we readily deduce

2a Sums
45
2.3 COROLLARY
(a) The sum of ideals of N is again an ideal of N.
(b) Forming sums of ideals is an associative and commutative
operation.
Certain sums are of particular importance:
2.4 DEFINITION Again let (Ik)keK be ideals of N. Their sum
У I. is called an (internal ) di rect sum if each element
keK K
of I I. has a unique representation as a finite sum of
keK K
elements of different U's·
In this case we write for the sum У/ Ik (or Ι^+Ι2+...
as in 2.2). keI
2.5 PROPOSITION For each family Пк)кеК of ideals of N the
following conditions are equivalent:
(a) The sum of the Ik's is direct.
(b) The sum of the normal subgroups Пк» + ) ^s direct.
(c) V kEK : Ikn ( I lt) = {0}.
Я+к
2.6 PROPOSITION Let \'I.
keK
i f j . Then
(a) a + b = b + a
(b) a1(a + b) = a'a
(c) ab = aO
[d) If N = NQ then ab = 0,
be direct, a ,a ' ε I · , b ,b ' ε I ■ ,
2.7 EXAMP
LE In the notation of 1.56, У Л. = @ Ν,
ΐεΐ Ί ιεΐ Ί
2.8 THEOREM ("First isomorphism theorem") (cf. Prehn (1)),
If 1,0 3 N then
I л J ^ J л I+J/
I
J/
I л J
2.9 REMARK If the reader should have the same difficulties
as the author in remembering this formula he might note
the alternating appearance of I and J in the isomorphism
statement.

46
§2 IDEAL THEORY
2.10 DEFINITION I «3 N is called a direct summand (of N) if
3 J3N: N = I+J.
J is then called a direct complement of I in N.
2.11 PROPOSITION I 3 N is a direct summand <=> \/αεΑυί I :
: α can be extended to an epimorphism N -* I.
The following result will be used frequently.
2.12 THEOREM If I 3 N is a direct summand then each ideal
of I is an ideal of N.
2.13 COROLLARY If, as in 1.56, N = ® N, and J. <i N{ then
2.14 REMARK In general, the (group-theoretic) sum of two N-sub-
groups is not an N-subaroup any more. But:
2.15 PROPOSITION (Fain (1)).
If Δ <Ν Γ and Ε 3Ν Γ then Δ + Ε <Ν Γ .
Proof. \j δεΔ \/ηεΕ \/ηεΝ : η(ό+η) = η(δ+η)-ηδ+ηδ ε Ε + Δ =
= Δ + Ε.
2.16 COROLLARY (Mli tz (2)). If iy then the N-subgroup
of ,,Γ generated by Δ is given by Δ+Ω.
Proof: 2.15 and 1.35(b).
With this equipment we can consider the relation of ideals of
N and of some homomorphic image N' more closely than in 1.30.
2.17 PROPOSITION Let h: N-& H' be an epimorphism with
Ker h =:K. Let A,A' be ideals (left ideals, N-subgroups)
of Ν,Ν' , respectively. Then
(a) h(h_1(A')) = A'.
(b) h_1(h(A)) = A+K > A.
The same applies to ideals or N-subgroups of N-groups.

2a Sums
47
Proof, (for ideals A,A' of near-rings N,N').
Let ηεΝ, η'εΝ' .
(a) n'ehih'^A')) <-> 3ηεη-1(Α'): n' = h(n) <-> η'εΑ'.
(b) neh_1(h(A)) <=> h(n)eh(A) <=> ЗагА: h(n-a) =
= h(n)-h(a) = 0 <=>ЗагА: п-агК <=> ηεΑ+Κ .
2.18 PROPOSITION R <ir N -> R = Rr*(No + Nc) = R(iN0+RaNc = RQ + RC·
Proof. \/ rcR 33 η,,εΝ 33 nreNr : r = пл + пг ·
——«^—^^— 0 0 L. I* ΟΙ*
R :3 N => n„ = η 0 = (n„+n„)0 = ^R , so η ,^R, too.
г с с v о с' о
The rest is trivial.
2.19 REMARKS 2 18 does not hold for left ideals L of N. All
£εί have the form I = η +n with n0£N0 and nc£Nc t
but in general %il- and ncH:
Consider N = Z[r.] and L : = {Ea ,· x1 | Ia^2Z = {0 ,±2 ,±4 ,...}}.
L is a left ideal of N (even a maximal one - see So (1)),
but £: = χ + ΐεΐ decomposes as I = η +n with nQ = x£L .
2.20 THEOREM Under forming sums and intersections, the ideals
of Ν (,,Γ with ΝεΤ? ) form a complete modular lattice.
Proof, follows from (Kurosh), p. 143.
2.21 REMARK These lattices are not necessarily distributive.
But cf. the following considerations and 2.18 (and also
Scott (3)).
2.22 PROPOSITION (Scott (4)) If А,В ^ Г and Α,Β <Ν Γ
then \/ ηεΝ \/ αεΑ \j βεΒ : η(α+Β) = na+nB(mod АлВ).
Proof, η(α+Β)-ηΒ-ηα ε Α+Α = A
So η(α+β) Ξ na+nB(mod A).
Similarly, η(α+β) ξ nS+na(mod Β) ξ na+nS(mod В),
and the result follows.
One can suspect that 2.22 will be particularly important
for А а В = {о}: see 2.29.

48
§2 IDEAL THEORY
2.23 PROPOSITION (Wielandt (2)). If Νε??0 and Α,Β,Δ ^ Γ
then
Γ': = (Α+Μ0(Β+Δ)/(ΑαΒ)+Δ
is commutative and \j ηεΝ \j γ,,γρεΓ' : η(γ,+γ~) = ηγ,+ηγ~
Proof. (Betsch (5)). Ε: = (ΑηΒ)+Δ ; Η: = (Α+Δ)λ(Β + Δ) .
Let η,,η2εΗ anc* ηε^ ·
Then 3 αεΑ 3 ΒεΒ: n1 ξ a(mod Ε) л п2 ξ g(mod E).
Now α+Β ξ B+a(mod Ал В) and η(α+6) ξ na+nS(mod AnB]
by 2.22.
Since Ал8 ? Ε we get
Hj+n2 Ξ α+β ξ β+α Ξ n2+nj(mod E) and
"(Πι+γιο) = η(α + β) ξ ηα+ηβ ξ nn.+nn2(mod Ε), and the
proposition is proved.
2.24 COROLLARY (Betsch (6)). With the assumptions and notations
of 2.23, 17: = Ν/,0;Γ,. is a ring.
Proof. Γ" can be considered as a faithful TT-group in the
obvious way. Now the result follows from 1.49.
2.25 COROLLARY (Betsch (6)). If Νε^ and Νεϊ^ and if no
non-zero homomorphic image of N is a ring then the
lattice of left ideals of N is distributive.
Proof. Let L1,L2,L3 be left ideals of N. Consider
the N-group Γ : = (L1 + L3) о (L2 + L3y{L ^ ^ +^ .
If Г + {о} then (ο:Γ) + Ν, for ΝεΤ^. From
2.24 we know that Ν//ο·Π is a ring ^ ^' a
contradiction. So Γ = {о} and the lattice of left
ideals is distributive.
Finally, lattice theory provides us with two more laws for the
ideal lattice of a nr. N or ,,Γ (Ней), Let I,J,К be ideals.
Modular law: If K=I then In(J+K) = (InJ) + K.
Cancellation law: If I^J then InK = JnK, IuK = JuK implies I=J,

2a Sums
49
2.) DISTRIBUTIVE SUMS
2.26 DEFINITION
(a) A direct sum Γ I =:I of ideals I of Ν (αεΑ)
αεΑ
is called di stri buti ve: <=>
αεΑ α βεΑ Β αεΑ α βεΑ ρ αεΑ α
(b) A direct sum У*Л = :Δ of ideals Δ„ (αεΑ) of
4 ' ϋ. α α
αεΑ
,,Γ is called distributi ve: <=>
<-> V У δ εΔ VneN : η( Τ δ ) = У ηδ„ .
αεΑ αεΑ αεΑ
(Note that the sums involved are actually finite ones;
all summands should come from different ideals.)
2.27 EXAMPLES If N = Щ N then N is the distributive sum
αεΑ α
of the ideals TTa (1.56).
The same applies to N-groups. Moreover:
2.2B PROPOSITION Let (I ) , be a family of ideals of N whose
v α'αεΑ J
sum is direct. Then У" I ~ ffl I <=> У* I is distri-
αεΑ αεΑ αεΑ
b u t i ν e.
The analogous result holds for N-groups with N = N .
Proof, obvious.
2.29 PROPOSITION (Heatherly (2)). Let (Δ ) д be a family
of ideals of МГ with У Δ = У*Л =:Δ. Then
Ν Λ· α L. α
αεΑ αεΑ
\/ ηεΝ„ \/ Σδ εΔ : η(Σδ ) = Σηδ .
* ο ν α * α' α
Conversely, if ,,Γ is faithful and if for neN
\/Σδ εΔ : η(Σδ ) = Σηδ then ηεΝ„ .
α ν or α ο
Proof. The first assertion follows from 2.22 and by
induction. See also 2.6(b).
If for ηεΝ and all Σδ εΔ η(Σδ ) = Σηδ then
α χ α' α
η(ο+ο) = ηο+ηο, hence no = ο. So
\j γεΓ : (ηθ)γ = η(0γ) = no = ο = 0γ and consequently
nO = 0.

50
§2 IDEAL THEORY
From 2.29 we get the following satisfactory result (recall
that for N =f N there is no chance at all that always
У*Л —μ ® Δ , for the Δ 's are not necessarily N-groups)
αεΑ αεΑ
2.30 THEOREM (Betsch (3)). Each direct sum of ideals in
if Ne)j0 also in „г) is distributive.
Proof. The statement for ..Γ is clear from 2.29.
If ['I =:I and У i , У 11 εΐ then
αεΆ α *U α β^Α β
( Σ 1α)( Σ φ « I i ( Σ ig) ·
αεΑ α βεΑ β αεΑ α βεΑ 5
Now 1 ( [)!)» Ι 11' =ii' by 2.6 (b).
(and ,
βεΑ
ΒεΑ
b) CHAIN CONDITIONS
2.31 REMARKS By 1.51, the ideals form an inductive Moore-
system. It makes sense to speak about things like
"the ideals fulfill the OCC" etc.
By 0.10, if the ideals fulfill the ACC then each ideal
is f.g. .
2.32 CONVENTION If the set of ideals fulfills the OCC we say
that "N fulfills the DCC for ideals" or more briefly that
"N has the DCCT'.To simplify statements, the phrase
"Let N have the DCCI" will be abbreviated by "DCCI".
Similar conventions apply to right ideals (DCCR), left
ideals (DCCL) and N-subaroups (DCCN).
Of course, the same is done for the ACC.
2.33 REMARK Clearly the DCCN implies the DCCI if N = NQ; in N,
DCCR or DCCL imply the DCCI. If N = NQ then the DCCN
implies the DCCL.
The same holds for the ACC.

2b Chain conditions
51
2.34 EXAMPLES
(a) (Beidleman (1)). Let a group Γ contain only finitely
many normal subgroups but an infinite chain Γ =
= Δ^ =&2 =» . . . of subgroups (such groups are known
to exist). N: = {feM(r)| V i ε IN : ί(Δ1)^Δ1>.
Then it is immediate that ,,Γ has the DCCI but not
the DCCN (since all Δ1 <Ν Γ).
(b) Each ring satisfying the ACCI but not the DCCI (Z, for
instance) or conversely is of course an example of a
nr. with the same properties.
2.35 THEOREM
(a) If I «3 N and N has the DCCI (DCCN, DCCL) then the
same appli es to N/I.
(b) If I ^ N and I is a direct summand then N has the
DCCI (DCCN, DCCL) <=> I and N/I have the DCCI (DCCN,
DCCL).
(c) If Δ Щ, Γ (ΝεΤ? ) is a direct summand then Γ has the
DCCI (DCCN) iff Δ and Γ/Δ have this property.
Proof, (for ideals of N and the DCCI)
(a) Let Jj^Jp?... be a descending chain of ideals
of N/I. If J^: = π"1(31-) (ΐεΙΝ) then Jj?^?...
by 1.30.
So 3 ηεΙΝ \t k>n : J*k = Jp. Since \J i ε IN : π (J i) =
= irU"1^)} = Ji by 2.17(a), Jk = Tn for all k>n.
(b) =>: It remains to show that I has also the DCC.
But this follows from the fact that each ideal of I
i s an ideal of N.
<-: Let I and N/I have the DCC and let JjHvJg?...
be a chain of ideals of N. The chains ϋ,ο I?j„r»l=>. . .
and ir(J,+ 1 )?π (Jp + I)?. . . get constant after some
ηεΙΝ . Therefore V k>n : J.nl = J-,λΙ λ π( Jfc + I) =
■ *(Jn+i) ·

52
§2 IDEAL THEORY
Since -rr"1 {π{ 0η.+ 1)) = J1 +1 + 1 = J^+I, Jk+I = J +1
for all k>n. Now tf xeJ : xej +1 * J. +1 ,
-i-i η η к
so 3 yeJn 3 ι'εΐ : χ = y + i.
Therefore x-yelo J. = J л I ? J„ and so xej„ .
КПП Π
This shows that \/ k>n: Jk = J .
(c) The proof is similar to the one of (b).
2.36 REMARK Lausch (4) showed that if ΝεΤ?1 has the DCCN
and eeN has some e'eN with e'e = 1 then ее' = 1.
The "Jordan-Holder-theory" carries over to near-rings and
N-groups with Νε??0 (but we only formulate it for near-rings).
The proofs are nearly word for word the same as in group or
ring theory and hence omitted. This omission is again justified
by the fact that all of 2.37 - 2.41 is a special case of the
Jordan-Holder-theory of Ω-groups (see e.g. (Kurosh), IV, §2).
2.37 DEFINITION A finite sequence
О 1 с
■Nn = (0>
(*)
of subnear-rings Ni of N is called a normal sequence of
N <-> \j ιε{1,...,η} : Ni <| N^ .
In the special case that all Ni 3 N we call the normal
sequence (*) an invariant sequence.
η is called the length of the sequence (*) and the near-
rings N. .. (ϊε{1,...,n}) are called the factors
л —
of (*).
Another normal (invariant) sequence
Ν = Μ =·Μ.=*Μ,=
oiz
.мт - ίο}
(**)
is called a refi nement of (*) if
\/ ΐε{0,...,η} 3 je{0 m} : N1 = M. .
(*) and (**) are called isomorphic if η = m and the
factors of (*) and (**) are (after a possibly necessa
re-ordering) isomorphic.
ry

2c Decomposition theorems
53
(»») is called a proper refinement of (*) if (*) is not
a refinement of (**).
A normal (invariant) sequence (*) is called a compos ition
sequence (principal sequence) if (*) has no proper
refinement.
2.38 PROPOSITION (*) is a composition (principal) sequence <=>
<-> all factors are simple.
2.39 COROLLARY A sequence isomorphic to a composition (principal)
sequence is itself a composition (principal) sequence
We now state the famous Jordan-Hb'Ider-theorem:
2.40 COROLLARY Let N have a composition (principal) sequence.
Then each normal (invariant) sequence can be refined
to a composition (principal) sequence and all these
sequences are isomorphic.
2.41 THEOREM N has a principal sequence <—> the ideals of N
fulfill both chain conditions.
See also Kaarli (2), (4), (6) and Oswald (8),(10).
c) DECOMPOSITION THEOREMS
2.42 DEFINITION Ν (ΝΓ) is called decomposable if it is the
direct sum of non-trivial ideals (or, equivalentlу, if
it has a non-trivial direct summand), otherwise
indecomposable.
2.43 EXAMPLES Clearly each simple nr. (N-group) is indecomposable.
The ring Ж is indecomposable, but not simple.
Between the concepts of simplicity, indecomposabi1ity
and minimality of an ideal (which is at the same time
supposed to be an N-subgroup in the case of N-groups)
there are the following relations:

54
§2 IDEAL THEORY
minimal <=- simple => indecomposable
If the ideal in question is even a direct summand, we get
minimal <=> simple —> indecomposable.
2.44 REMARK The next considerations concern merely N-groups
with ΝεΤΤ. The reason is obvious: in qeneral the ideals
of N-groups are not necessarily N-groups again. But
we have to speak about "simple ideals" etc. . Cf. also
Roth (1) and the remarks preceding 2.1.
2.45 THEOREM Let Ν (^Γ with ΝεΎ?0) have the DCCI. Then
N (Nr) is the finite direct sum of indecomposable ideals.
Proof (for near-rings). If N is not indecomposable then
there are non-trivial ideals ΙιΊ? Wltn
N = I ji-12 .
If I,,I2 are indecomposable, we are through. If not,
I, or I~ decompose again properly, et cetera. By
the DCCI, these decompositions stop after finitely
many steps thereby proving that N is the direct sum
of finitely many indecomposable ideals.
The corresponding assertion in nuclear physics is much harder
to prove!
2.46 DEFINITION Ν (Nr with Νε7)0) is called completely
reducible if Ν (,,Γ) is the direct sum of simple ideals.
1:3 N (&<fJr) is completely reducible if Ι (Δ) is
completely reducible when considered as a near-ring (N-group).
2.47 REMARK Another usual name is "semisimple". However, "semi-
simple" will have another meaning in §5. More on that can
be found in Oswald (2),(3) and (5).

2c Decomposition theorems
55
2.48 THEOREM (Roth (1), Beidleman (1)). If Hcfl, the following
conditions are equivalent:
(a) Every ideal of N is the sum of simple ideals.
(b) N is the sum of simple ideals.
(c) N is the direct sum of simple ideals.
(d) N is completely reducible.
(e) Each ideal of N is a direct summand.
(f) \/ I<IN : I and N/I are completely reducible.
(g) N is the sum of minimal ideals.
The analogous theorem holds for N-groups with Νε*Μ' .
Proof.(for near-rings).
(a) => (b): trivial.
(b) =-> (c) : If N = У I , define Л : =
aeA a
= {B?A| [ I s J*1й}- Л + 0· By Zorn's Lemma,
βεΒ ΰ βεΒ Β
,4-contains a maximal element (w.r.t. ?) Ε.
Τ I0 = :N'. V αεΑ: (Ι ηΝ' - Ι ν Ι ΛΜ' = {0}) .
вёв β α α
Ι λΝ' = {0} is a contradiction to the maximality
of B. So V αεΑ : I =N' and hence N = N' = 7*1
(c) —> (d): by definition.
(c) =-> (e): If I <l N, consider an ideal J maximal
(Zorn!) with the property that Jnl = {0}. N':= I+J.
If N + N', 3 J0<1N : JQ simple л J fN'AJo+{0}. Then
JonN' = {0},
so
J+J =»J,
0
Also, (J+J )M = {0},
since χ = J+Joe(J+J0 )r\l implies that j
N'aJq = {0}. This contradicts the
= x-je(I+J)njQ .
maximali ty of J.
Therefore I+J = N and I is a direct summand.
(e) -> (a): If I «3 N, denote by Τ the sum of all
simple ideals of I. Assume that 7 =f I.
Τ Й i ^ Ν Λ I is direct summand => Τ <| Ν . Hence
Τ is itself a direct summand and there is some
J <3 N with T+J = N.

56
§2 IDEAL THEORY
Consequently each simple ideal of Τ is a simple
ideal of N. T+(JnI) = I, since each ί ε I has the
form i = T+j with ΤεΤ and jeJ; because of
T^I we know that jel. We now show that Jnl
contains a simple non-zero ideal of N and arrive
at a contradiction. By assumption, Jnl -j* {0}.
If Jnl is f.g. then there exists a maximal ideal
I* in Jnl, and each direct complement (existence
as before) of I'
ι η
Jnl is a simple non-zero ideal
of Jnl and of N.
If Jr\I is not f.g., take any fg. ideal F 4= (0}
of Jnl. Then F =f Jnl. F contains a maximal ideal
Μ <l F. As before, each direct complement of Μ in F
is a non-zero simple ideal of F and of N contained
in Jnl.
(c) =-> (f): Since (c) => (a), every I < N is the sum
(and by (b) => (c) the direct sum) of simple ideals,
implying that I is completely reducible.
If I <l N, take some J ^ N (again, J is completely
reducible) with I+J = N. But then N/I - J by 2.8
and N/I is completely reducible.
(f) -> (d): trivial (take I = N).
(a) -> (g): trivial.
(g) =■> (e): as in (c) => (e).
2.49 COROLLARY The direct sum of completely reducible near-rings
(N-groups with ΝεΤ? ) is again completely reducible.
Near-rings (N-groups) which decompose into finitely many
simple ideals are especially important.
The following theorem will be used frequently throughout this
book.
Much more on this subject can be found in Blackett (1), Chao (1),
Hartney (2), Oswald (3) , (4 ) , (5 ) , (10 ) , Natarajan (1), Ramakotaiah
(3) and Scott (7).

2c Decomposition theorems
57
2.50 THEOREM (Beidleman (1), Betsch (3)). Let N be a nr. .
Equivalent are:
(a) N is the sum of finitely many simple ideals.
(b) N is the direct sum of finitely many simple ideals.
(c) N is completely reducible and has the DCCI and the
ACCI.
(d) N is completely reducible and has the ACCI.
(e) N is completely reducible and has the DCCI.
(f) N is completely reducible and every ideal of N is f.g..
(g) There exist maximal ideals I,,...,I of N with
zero intersection, but all J : = Π IL + {0}.
Г k+r k
(in this case, N = [J- and J j, ·..,J n are simple).
r= 1
(h) There exist maximal ideals Ij In with
r=l r
The usual changes yield analogous results for N-groups
with ΝεΤ70
Oswald (2))
with ΝεΤ7 (remark also the additional results in
Proof, (a) <=> (b): as in 2.48.
η
(b) => (c): If N = I Ik (all I. simple) then
k=l
N is clearly completely reducible. Moreover,
* · .
n - i1+...+in = 4+... + V1
Ij =» {0} is a
principal series, so N fulfills both chain conditions
by 2.41.
(c) -> (d) and (c) —> (e) are trivial.
(d) <-> (f): by 0.10.
(d) -> (b) and (e) -> (b): If N = I'I , the
aeA
ACC (OCC) forces A to be finite.

§2 IDEAL THEORY
(b) -> (g): If N = Jj+...ί·^ (Ji simple ideals).
cause of N/Ik =- Jk, all
defi ne I,
r+k
Be
S I„.
I. are maximal ideals. If χ ε II I. , χ = J , +.. . + j
(j.eJ.j) and if 3 ke{ 1,. . . ,n}: Jk + 0 then xs|:Ik ,
η
a contradiction. So Π 11, = (0).
k-1 K
Since Π Iu = J„ + {0}, we are throuqh.
k+r r
(g)=> (h): trivial.
(h) => (b): Let I,,..., I be minimal w.r.t. the
property that their intersection = {0}. Then each
J„: · (1Il + {0}. Since V reil,...,n}: J φ I ,
r k+r K r r
but Jrnlr = {0}, we have N = Jr+Ir· Hence
Jr ™ N/I and J is simple.
Let for re{l,...,n} Кр: = I ^r\ ... η 1 r .
We claim that N = Ji+...+J +K and prove this by
induction on r.
If r = 1 then Kr = Ij and J^^ = N.
Assume that it is shown for r (< n). We show
the assertion for r+1.
Since *r+i+Kr = N (ЬУ maximality),
"/>r.r,»'*VI~.aVwKr ■'■/",♦,·
Since N/, is simple, the same applies to
/ r+1
К /„ and К ., is a maximal ideal in К .
r/Kr+1 r+1 r
Jr+i° Kr+i = < 0 1к)^(ГП i4) = П ik - to}.
r+1 r+1 k+r+1 K £-1 l k-1 K
Also Jp+1 - Kr , but Jr+1 i Kr+1
Hence Kr+i+Jr+i = Kr and N = Jj+-·-+Jr+Kr =
- Ji;---;Jr;Jr+i;Kr+l ·
η
But К = {0}, so N = У J,, .
n k-1 K

2c Decomposition theorems
59
2.51 REMARKS The proof of (h) => (b) in 2.50 could also be done
by using subdirect products and "words generating prime
ideals" similar to (McCoy), p. 59. Cf. also (Higgins), §9.
At a first glance one might assume that"f.g'.' implies
already"completely reducible" This is not the case: take
the zero-nr. N on the dihedral group D„ on 8 elements.
Then normal subgroups and ideals coincide. But D„ is known
to have G 3 0g and Η <l G, but Η £| Dg . By 2.48(e)
and 2.12 N cannot be completely reducible.
2.52 COROLLARIES
(a) If N fulfills one (and hence all) of the conditions in
2.50 and if I <l N then the same applies to I (use
2.48(f), 2.48(e) and 2.35(b)) .
(b) If N has the DCCI and is a subdirect product of simple
near-rings Ν; (ι ε I) then 3JeI, J finite:
N = @ N, (apply 2.50(h) and 1.58).
Again, corresponding statements hold for N-groups with N = N.
2.53 DEFINITION Two decompositions of N : N = £'I = ['J.
αεΑ α βεΒ Β
are called isomorphic if |A| = |Bl and the I 's
and J„'s are - up to order - isomorphic.
P
The Krul1-Schmidt-Theorem reads as
2.54 THEOREM (Roth (1)). If Ν (ΝΓ with Νε7?0) fulfills one
(and hence all) of the conditions of 2.50 then any two
decompositions of Ν (,,Γ) into simple ideals are
i somorphi с.
Proof (for r.r.'s) If N = I. + ... + I = J. + ...+J
ч ' 1 η 1 m
(Ik, J. simple) then N=>I,+. . .+1 ,=>. . .=>I ,={o} and
N=»J, + . . .+J ,=». . .=»J.=>{o} are two invariant sequences
with simple factors
1-1 ' k=l
Ik 2: ir and

60
§2 IDEAL THEORY
JjL - J (2<r<n and 2<s<m)
By 2.4D these sequences and therefore these
decompositions are isomorphic.
Compare the following result with §9 of (Higgins).
2.55 THEOREM
η
(a) If N « l'lr (all Ir simple) and if I <| N then
r = 1
there is a subset S of {1 n} with
seS s
(b) If I,J 3 N are such that N/I and N/J are completely
reducible then Ν/. , is completely reducible, too,
all of whose simple summands being isomorphic to one
of the simple components of N/I or N/J.
Again, the corresponding theorem holds for N-groups with
N - N0.
Proof, (a) Let Kr: = I+Ij+...+Ir (l<rsn), KQ: = I.
Then Kn = N. V red....,η-Π: «r <l Ν Λ
Λ K„r\ I ,:s1I .. . Thus we have either К .. = K„
Г Г+1 Г+1 Г+1 г
or Kr + 1 - Kr;ir+r
Hence 3 Te{1 η}: Ν = 1+ Vl¥, and so by 2.8
teT z
' teT τ r L I teT z s"
S: = {1 n}\T .
(b) Let K: = I+J. Then K/I <i N/I and 3 M<N: N/I =
= (K/l$M/l), whence K+M = N and KnM = I.
So MnJ = MnKnJ = In J and M+J?I+J = K,
M+J^M+K = N, hence M+J = N. Consequently

2d Prime ideals
61
N'lnJ = M/InJ + J/IaJ
"/inJ = "/«ftj - M + J/J - N/J and
J/
In J
K/I 3 N/I are completely reducible, so
N/¥ , is completely reducible by 2.49.
1 Λ J
The rest follows from (a) and the first line on this
page.
Ferrero-Cotti showed that N = 11 + 12 , Ν φ Nc , I,2/»^ I22
implies that all ideals of N are given by {0}, Ip I2 and N.
d) PRIME IDEALS
1.) PRODUCTS OF SUBSETS,
2.56 NOTATION If S,T «ξ N then ST: = {st | s£SAteT},
For ηεΙΝ , the definition of Sn is then clear.
2.57 PROPOSITION (Maxson (1)).
(a) M R.S.T 5 N: (RS)T = R(ST).
(b) If h: N - Л then \/ S,T 5 N: h(ST) = h(S)h(T)
and \/ S",T η IT: h-1(ST) ? h"1 (^)h'1 (T).
(c) V I <| N \/S,T Ε Ν: (S + I)(T + I) = ST + I.
Proof, (a) and (b) are immediate.
(c) follows from (b) for π:Ν * N/I.
2.58 REMARK Note that ST has no particular structure in
general. Even if S,T are ideals, ST is not even
a subsemigroup of (N,+) except in some very special
cases.

62
§2 IDEAL THEORY
2.) PRIME IDEALS
2.59 DEFINITION Ρ 3 N is called prime if \/ I,J<N: IJ=P ->
-> IsP ν JeP.
2.60 NOTATION For SsN, let (S) be the ideal generated by S.
({n}) =: (n) .
2.61 PROPOSITION (Van der Walt (1)). Let Ρ be an ideal of N.
Equivalent are
(a) P is a prime ideal.
(b) V I,J <i N: (IJ) = P=->IsPvJeP.
(c) \j i ,jeN: i t Ρ л j { Ρ -> (i)(j) $ P.
(d) V I,J <) N: I =. Ρ л J =» Ρ -> IJ^P.
(e) V I,J <) N: I $ Ρ л J $ Ρ *=> IJ «f: P.
Proof, (a) <=> (b) <-> (e) is trivial.
(a) -> (c): If (i)(j)«=P then (1)eP or (j) = P,
so i ε Ρ V jeP.
(c) => (d): If I=PaJ=>P, take iεΙ\Ρ and jeAP.
Then (i)(j)<JEP, so IJ«fP.
(d) => (e): If I«|PaJ$P, take ιεΙ\Ρ and jeJ\P.
Then (i)+P=>P and (j)+P=«P. Then ((i )+P) ((j )+P)*P .
So 3 1'ε(1) 3 J'e(j) 3 Ρ,ρ'εΡ: (i'+p)(j'+p')(P.
Therefore i'(j'+p')-i'j'+i'j'+p(j'+p')£Р. But since
i '(J'+P1 )"i 'J'eP and p(j'+p')eP, i 'j 4p· hence
IJ«fP.
2.62 PROPOSITION Let (P ) . be a family of prime ideals,
totally ordered by inclusion. Then Π Ρ =:Ρ is a prime
ideal, too. αεΑ *
Proof. We may assume that A is ordered such that for
α,βεΑ α<β =■> Ρ = P„.
= Ο. ϋ

2d Prime ideals
63
Of course, Ρ is an ideal. Let I,J be ideals of N.
IJ ς Π Ρ -> \/ αεΑ: IJ 5 Ρ . If 3 αεΑ: I «f Ρ ''
л α α τ α
αεΑ
then J «ξ Ρα. V β>α: J s Ρ If 3 γ<α: J «| Ργ
then 1 ? Ρ , so Ι = Ρ , a contradiction.
So V αεΑ: J s Ρ and J 5 Π Ρ .
α «α
αεΑ
2.63 PROPOSITION (Maxson (1)). If I «3 Ν is a direct summand
and Ρ ^N is prime then Рл I is a prime ideal in I.
Proof. If JjJ2 ? ΡλΙ (J1,J2 <l I) then JjJ2 = Ρ and
J^.Jo^ N, so J. ϊ Ρ or J2 ? Ρ and therefore
J1 s Рл I or J2 5 Pnl.
2.64 PROPOSITION If I «3 N and Ι ε Ρ «3 N and if
π: Ν ■+ Ν/Ι =: Ν" is the canonical epimorphism as usual then:
Ρ is prime <=> π(Ρ) is prime.
Proof. ->: If Ί1^2^π(Ρ) (3"^ <) ft), let J^- ir"1^)
(ΙείΙ,Ζ}). By 2.57, J^ = π" l (TTj )π_1 (U"2 ) «=
Ειγ'^Τ,Τ^Ειγ^ΜΡ)) = P + I « P.
So J.sP
-1,
ν J^P, hence Tj = π(π (Jj)) = π(01)Ειτ(Ρ)
or J2stt(P).
<-: If JjJ^P then π^Μ^) = ir( J ^2 )«Ξπ (Ρ ).
So π(ϋ1)?π(Ρ) or ir (J2 )*π (Ρ ). This shows that
either J^Jj + I = ir"1 (ir( d^) )stt_1 (π(Ρ)) - P + I = Ρ
or J2«iP.
2.65 DEFINITION Call N a prime near-ring if {0} is a prime
ideal.
linaldi (1) studied nr.'s whose proper ideals are all prime.
If N eU is not simple and has this proj
nost two minimal and two maximal ideals,
If N eU is not simple and has this property then N has at

64
§2 IDEAL THEORY
2.66 EXAMPLES
(a) Every integral near-ring is of course a prime near-
ring (for I«J ϋ {0} and I 4= {0}, J + {0} would
guarantee the existence of some ιεΐ*, jeJ* (see p. 1)
with ij = 0).
(b) N is a prime ideal of N, so {0} is a prime ring.
More generally:
2.67 PROPOSITION If I <3 N, I is a prime ideal iff N/I is
a prime ring.
Proof. Take Ρ = I in 2.64.
2.68 EXAMPLE If Ne1?c then each normal subgroup of (N, + )
is a prime i deal.
2.69 COROLLARY Each constant near-ring is a prime near-ring.
2.70 PROPOSITION N simple -=> N is prime or N is a zero-near-
ring. The proof is trivial. More generally:
2.71 PROPOSITION If I <1 N is a maximal ideal then I is either
prime or N ? I.
Proof. N/I is simple. By 2.70, N/I is either prime
(implying that I is a prime ideal) or N/I is a
2
zero-nr, which causes N s I.
2.72 COROLLARY If I < Νε^ is maximal then I is prime.
2.73 REMARK If I is prime, I is not necessarily maximal (not
even for finite dgnr's.: see Beidleman (8), Laxton (4)
and Laxton-Machi η (1 )).
2.74 DEFINITION An ideal minimal in the set of all prime ideals
containing some given ideal I is called a minimal prime
ideal of I.

2d Prime ideals
65
Applying 2.66(b) and Zorn's lemma on ({P<N|Р=1лР prime},?)
we get
2.75 PROPOSITION For each ideal I there exists a minimal prime
ideal of I.
2.76 DEFINITION A minimal prime ideal of {0} is called a
minimal prime ideal (in N).
2.77 COROLLARY
(a) Each prime ideal contains a minimal prime ideal.
(b) N has a minimal prime ideal.
Proof. Take I = {0} in 2.7E.
As in ring theory (cf. e. g. (McCoy)), the complements of prime
ideals deserve some interest.
2.78 DEFINITION Μ 5 N is called an m-system if
V a.beM 3 aje(a) 3 bje(b) : a^eM.
2.79 EXAMPLES
(a) 0 and N are trivial examples of m-systems.
(b) V ηεΝ: {η,η ,η ,...} is an m-system.
2.61(c) gi ves us
2.80 COROLLARY If Ρ <3 Ν, Pisa prime ideal iff N\P is
an m-system.
2.81 PROPOSITION (Van der Walt (1), Ramakotaiah (3)).
Let Hi Ν be a non-void m-system in N and I an ideal of
N with I л Μ = 0.
Then I is contained in a prime ideal Ρ $ N with PnM = 0.

66
§2 IDEAL THEORY
Proof. I: = {J<N: J =? I л J л Μ = 0 ). Ι ε I. By Zorn's Lemma,
I contains a maximal element P. Ρ is an ideal =f N.
Ρ is in fact a prime ideal:
If J.=aP л J2=>p then take some j,eJj^ м and
J2eJ2 л М.
{J* χ) {J 2' " J1J2' and 3 J i ε < j j) 3 J 2 ε (j 2) : jjj'2cM.
So (J1J2)oM + 0, (J1J2)^P and JjJ2 f P.
3.) SEMIPRIME IDEALS
2.82 DEFINITION S 3 N is semiprime: <=> У I <l N: I2eS => IsS.
Evidently, each prime ideal is semiprime.
Similar to 2.61 we get
2.83 PROPOSITION For an ideal S of N the following conditions
are equivalent:
(a) S is semiprime.
(b) V ΙϋΝ: (IZ)^S -> I*S.
(c) V ηεΝ: (n)Z?S => n£S.
(d) У I<N: I=»S => IZ^S.
(e) У ΙϋΝ: lis =■> I2«£S.
2.84 PROPOSITION If (S ) . is a family of semiprime ideals
v α'αεΑ ν
then || S is again semiprime.
αεΑ α
Proof. If I «3 N and I2 E П S then \/ αεΑ: IZsS ,
αεΑ
so \/ αεΑ: IsS , hence Ι ε Π S ·
α αεΑ α
As in 2.63 and 2.64 we get
2.85 PROPOSITION Let I 3 N be a direct summand and S <d N
be semiprime then Sol is semiprime in I.

2d Prime ideals
67
2.86 PROPOSITION I<N л I?S<N. Then S is semiprime iff
tt(S)en/I is semiprime.
2.87 DEFINITION N is called a semiprime near-ring if {0} is
a semi prime ideal .
2.67, and 2.74 - 2.77 can again be transferred to semiprime
near-ri ngs (ideals) .
2.B8 DEFINITION (Maxson (1)). S ε Ν is called an sp-system
if \j scS 3 si»s2e(s) : s,s2eS.
2.89 PROPOSITION (Maxson (1)).
(a) Each m-system is an sp-system.
(b) \/ S ^ N: S is semiprime <=> N\S is an sp-system.
The proof is trivial.
2.90 PROPOSITION (cf. 2.81). Let S be a non-void sp-system
in N. Let I be an ideal of N with In S = 0. Then I is
contained in a semiprime ideal + N.
Now we study some relations between prime and semiprime ideals.
Since each prime ideal is semiprime we get at once from 2.84
2.91 PROPOSITION Any intersection of prime ideals is a semiprime
ideal .
2.92 PROPOSITION (Maxson (1)). Let S be an sp-system and seS.
Then there is some m-system Μ with seMES.
Proof. scS => 3 s,,s2e(s) : s^SgeS =■> 3 si«s^(s^s2) :
: sJsieS.
Continuing this process, one gets a sequence
■ ι (к) (к)
S, ^1^2' ^1^2* ···' ^ι ^2 ' *""
with \/ keIN : s}k)s^k)eS and (s)?(s ^^(s^)?. . . .

68
§2 IDEAL THEORY
Take M:= {s, s^, s^s^. ...)· We show that Μ is
a desired m-system.
If s[k)s£k). sj^s^cM (w.l.o.g. Л<к) then
(s{k>s<k>) * (s«*>s<£>) · Take 5[*+1,Ц* + 1> ε
e(s(Ms(M} s (5<кЦк>); then 5<*+1Ц*+1>еМ.
2.93 DEFINITION If I <! N, call £»( I): = fiP the
Ρ prime id.
Ρ э I
prime radical of 1. Gojan (1) calls it the Baer-radical .
Df course, ^(I) is a semiprime ideal (by 2.91) containing I.
2.94 PROPOSITION η ε JP( I) => 3 keIN : nkel.
2 3
Proof. M:= {n,n ,n ,...} is an m-system (2.79(b)).
If UM = 0 then by 2.81 there is some prime ideal
Ρ ? I with Ρ η Μ = 0, a contradiction to nejp(l).
Hence ΙλΜ* 0 and 3 keIN: η εΐ.
2.95 THEOREM (Gojan (1), Groenewald (1), V.S.Rao(D) Let IjN.
(a) I is semiprime iff f{I) = I.
(b) If I is semiprime then I is the intersection of all
prime ideals containing I.
(c) У (I ) is the intersection of all semiprime ideals
containing I.
(d) A semiprime near-ring is a subdirect product of prime nr's
Proof. Since (a )=-> ( b )=> (c ) and by 1.58 and 2.91, we only have
to show => in (a). Suppose ρ ε J*( I )\I . Since N\I is an
sp-system, 2.92 provides us with an m-system Μ such that
a ε Μ SN4I . But MflIS(NM)nI = 0 contradicts aef(I).
For more and some other related material see Beidleman (7).
Ferrero-Cotti (7), Gojan (1), Oswald (5),(8) and Ramakotaiah-
P.ao (5), Santhakumari (2). Semiprimary near-rings were
considered in a series of papers by Kaarli. See in particular
Kaarli (7) and 9.260.

2e Nil and nilpotent
69
e) NIL AND NILPOTENT
2.96 DEFINITION
(a) ηεΝ is called nilpotent if 3 keIN: η = 0.
(b) S e N is called nilpotent if 3 kelN : Sk = {0}
(c) S e N is called ni1 if all seS are nilpotent.
2.97 REMARKS
(a) S ε Ν nilpotent =-> S nil. (In 3.40 we will see that
if ΝεΤ? has the DCCN then"ni1 "and"ni1potent"coincide
for N-subgroups.)
(b) S s Τ ? Ν Λ Τ nil (potent) -> S nil(potent).
2.9B EXAMPLES
(a) In Z.[x], 2x is nilpotent.
(b) If ηεΝ is nilpotent then η = 0.
2.99 COROLLARY If I <} N is nil then I s Nq.
Proof. By 2.18, I = I0+Ic. so by 2.97(b) I = N л-I
is nil, hence by 2.98(b) I = {o} and I = I s N .
2.100 THEOREM (Ramakotaiah (3)). I g N. N is nil(potent) <=>
<=> I and N/I are nil(potent).
Proof (for nilpotence)
->: by 2.97(b), I is nilpotent.
If 3 keIN : Nk = {0} then (N/I)k = Nk/I = {I}.
к к
<-: 3 k.eIN : (N/I) l = {I}, so N !sl and
k,
3 k-εΙΝ : Ι ά = {0}.-
к к
Therefore (Ν ) 2 = {0} and N is nilpotent.
The proof for "nil" is similar.

70
§2 IDEAL THEORY
2.101 PROPOSITION (Ramakotaiah (1)). Let Ι (αεΑ) be ideals
of N.
(a) ( VaeA: I nilpotent л A finite) => J I is nilpotent,
aeA
(b) ( VaeA: I nil) => У I is nil .
a ' iu a
αεΑ
Proof. Let I,J be nil(potent) ideals. I+J/T = J/T ,.
By 2.100 j/taj and ЬУ assumption I are ni 1 (potent) ,
Harnessing 2.100 again, I+J is nil(potent).
By induction we get (a) and (b) for a finite A.
In (b), let 1ε I I : i = 1 +...+1 (say).
αεΑ α
α1
Then ιεΐ +...+Ι and i is again nilpotent.
al ak
Scott (11) even showed that the sum of a nilpotent N-subgroup
and a nilpotent ideal is nilpotent if N = N.
2.102 PROPOSITION (Polin (2)). Let N be isomorphic to a sub-
direct product of near-rings Ν (αεΑ) without non-zero
nil(potent) N-subgroups, left ideals or ideals.
Then N has the same property.
Proof (for N-subgroups of N). Let И <. N be nil(potent)
Let τ, : N -» N be the usual epimorphisms (1.58).
Then all π (Μ) are nil(potent) in Ν , hence = {0},
so Μ = Π Ker π = {0} and therefore Μ = {0}.
αεΑ α
(If N£7)o then 3 αεΑ: Ν $YIQ. So neither in N nor
in N there are nil(potent) N-subgroups and the
proposition is meaningless in this case).
The following proposition will be useful later on.
2.103 PROPOSITION (Polin (2)). I,J <i Ν Λ I nil(potent).
Then I+J/, is nil(potent) in N/J.
Proof: by 2.8 and 2.100.

2e Nil and nilpotent
71
There are several connections between nil(potent) and (semi)prime
i deals:
2.104 PROPOSITION (Maxson (1)). If I 3 N. Then N/I has no
nilpotent ideals iff I is semi prime.
Proof.
>: Assume that N/I has no nilpotent ideals and
2
π(ϋ)2 = tt(J2) = {1}
that J 3 N, J = I. Then
(zero ideal of N/I ). So tt(J) = {1} and J
k = Ρ then
J14) ? Ρ by right distributivity. So
J <= Ρ by induction. By 2.95(b), this also holds
if I is only semiprime. By 2.103, N/I has no
nilpotent ideals.
I.
o*: If Ρ is prime and JSN with J
(Jk"1)J ' ' lk
2.105 THEOREM
(a) (Polin (2)). If I <| N then 09(1) contains all
ni1 potent i deals of N .
(Ь)з»((о>:
is nil.
Proof, (a) φ{1) is semiprime. So NL,,, has no nilpotent
ideals. Assume that J is a nilpotent ideal of N.
By 2.103, J+PCIJ/y/j) is nilpotent in N/^.j.
and therefore zero. Hence J s P(I),
(b) follows from 2.94.
2.106 THEOREM (Laxton (4)). If Νε7?ο has DCCI then N is a prime
near-ring <*=> N has a smallest ideal I under all non-zero
ideals and I is not nilpotent.
Proof. =■>: If N is prime, {0} is a prime ideal. Let I
be a minimal ideal (existence guaranteed by the
DCCI). I is not nilpotent by 2.104.
If J is another minimal ideal then {0} + IJ = I,
so (IJ) = I. Similarly, (IJ) = J, so I = J.
If К is another non-zero ideal, К contains a minimal
ideal of N, so К contains I. Hence I is the smallest
of all non-zero ideals.

72
§2 IDEAL THEORY
<": Conversely, let I be the unique minimal ideal
and suppose that I is not nilpotent.
If Jx, J2 <) N, JjJg = {0}, but Jj 4= {0} and
Jo + {0} then J,,J2 contain a minimal ideal of N
by the DCCI and this minimal ideal = I.
2
Hence I s Jj A I ? J2, so Ι ξ J^ = {0} and
I is nilpotent, a contradiction.
From 1.60 we get the following
2.107 COROLLARY If ΝεΤ^ is prime and has the DCCI then N
is subdirectly irreducible.
2.108 REMARKS See Oswald (2) for a discussion of "strictly
(semi-)prime" near-rinqs and Holcombe (1) for "0-, 1-
and 2-(semi -)prime ideals" and theit connection to v-
primitive ideals (4.2 (c), cf. 4.34). Thereby I <1 N
is called 0-(l-,2-) prime if for all ideals (left ideals,
M-subgroups) Λ,Β of N: AB = I *=>A s I vB Ε I , and
similar for 0-(l-,2-) semiprime ideals. So 0-(semi-)prime
ideals are just our (semi-)prime ideals. See also
Ramakotaiah-Rao (2).
The correspondence between "nil" and "nilpotent" is
further discussed in 3.40 and 5.48 below, as well as in
Kaarli (4), Oswald (6) and Scott (16).
Local nilpotency is studied in Gringlaz (1). See also
Beidleman (7) and Gojan (1).

PART II
STRUCTURE THEORY
§3 ELEMENTS OF THE STRUCTURE THEORY
§4 PRIMITIVE NEAR-RINGS
§5 RADICAL THEORY

74
§3 ELEMENTS OF THE STRUCTURE THEORY
Irreducible (ring-) modules RM (i.e. simple ones with RM 4= CO})
play an important role in ring theory. They have e.g. the
property that ty meM: Rm = (ο) ν Rm = M. However, simple
N-groups ,,Γ with ΝΓ + ίο} do not enjoy this property. It
might also come to mind to use N-simplicity or N -simplicity
(both equivalent to simplicity in the ring case). So we define
3 types (type 0,1 and 2) of N-groups, all coinciding with
irreducibi1ity in the case of modules, with type 2 implying
type 1 and this in turn type 0. Monogenic N-groups (3 γεΝ:
Νγ = Γ) are particularly important. For instance, we prove
that every monogenic N-subgroup of N contains a right identity
1f N=N has the DCCN.
о
We then study the effect on the type of ,,Γ of changing N into
N/I , NQ or Nc.
In c), modular left ideals are introduced in the same way as
for rings. Many theorems of ring theory carry over to near-rings:
each modular left ideal is contained in a maximal one, modular
left ideals are exactly the annihilators of generators of
monogenic N-groups, the intersection of two maximal modular
left ideals is modular, etc. It is advisable to call a modular
left ideal L v-modular if N/L is an N-group of type v.
We prove e.g. that if I is a direct summand of N=N then every
v-modular left ideal in I is the "trace" of one in N.
We also introduce quasiregularity for abusing it to show that
"nil" and "nilpotent" coincide for Μ <Ν Ν if N is a zero-
symmetric near-ring with DCCN.
If Νε?10 has a right identity e and if N is the finite direct
sum of left ideals then decomposing e into Ее. yields
"orthogonal" idempotents e·. Another method (due to S.D. Scott) to
get orthogonal idempotents is presented and central idempotents
are discussed.

За Types of N-groups
75
Finally, we consider zero-symmetric near-rings N with minimum
condition on N-subgroups and show e.g. that every "minimal
non-ni1 potent" N-subgroup (left ideal) contains a riqht
identity (a non-zero idempotent, respectively), and that every
minimal ideal is a finite direct sum of N-isomorphic minimal
left ideals.
a) TYPES OF N-GROUPS
3.1 DEFINITION
(a) ΝΓ is mono gem" с: <=> ] γεΓ: Νγ = Γ.
(In this case we say that ,,Γ is "monogenic by γ"
and γ is called a generator for ΝΓ.)
(b) ,,Γ is strongly monogenic: <·=> ..Г is monogenic and
\j γεΓ: (Νγ = (ο) ν Νγ = Γ).
3.2 REMARK Observe that a strongly monogenic N-group ,,Γ
has Ω = {ο} or Ω
3.3 EXAMPLES
(a) Each ,,Γ with Ω = Γ is strongly monogenic.
(b) м/г\Г and M ,„,r are strongly monogenic.
See also the examples 3.8, 3.9 below.
Now we list some properties of monogenic N-groups which are
useful for the sequel .

76
§3 ELEMENTS OF THE STRUCTURE THEORY
3.4 PROPOSITION Let ^Γ be monogenic (by yQ). Then
(a) L <!, N => ίγ0 <!„ Г.
(b) If e is a left identity of N then \/ γεΓ: ey = y.
(c) If e is a left identity of N and if ,,Γ is faithful
then e is a two-sided identity.
(d) If Г is N -simple (Г can be considered as an N -group!)
then either ΝΓ = {o} or ..Г is strongly monogenic.
<e> ΝΓ Ι Ν/{ο:Ύο)
(f) ,,Γ is simple <=> (°:y0) is a maximal left ideal
or = N.
(g) ,,Γ is N-simple <·=> there is no N-subgroup strictly
between (0:Y0) and N <-> (°:Υ0)+Ν- is a maximal
N-subgroup or = N.
(h) Γ is N-simple <-> (°:γ0) is a maximal N-subgroup
or = N.
(i) (Betsch (6)) If Nr is faithful, Νε^, and if
3 Lj,L2 at N with L1+(o:y0) = L2+(o:yQ) = Ν ,
but L,n L~ e (ο:γ ), then N is a ring.
Proof, (a) \/ γεΓ 3 η εΝ: γ = η γ . So
V £γ εΐγ \/ ηεΝ \/ γεΓ: η(γ+£γ )-ηγ =
= η(η γ +£γ )-ηη γ = (η(η +£)-ηη )γ„ ε Lv .
ν γ'ο Ό γ ο Υ γ' ο Ό
In the same way one shows that Ly is a normal
subgroup of Γ.
(b) As in (a), ey = εηγγ0 = ηγγ0 = γ.
(c) V γεΓ \/ ηεΝ: ο = ηγ-ηεγ = (η-ηβ)γ, so η = ne.
(d) \/ γεΓ: Νγ <,, Γ implies (γ = ο) Ω to be = {о}
о
or Ω = Г. So each Νγ equals either {o} or Γ.
(e) Consider the N-epimorphi sm h: N ■+ Γ and
η - ηγο
apply the homomorphism theorem.

За Types of N-groups
77
(f) - (h) follow from (e), 2.16 and the "second
isomorphism theorem".
(1) By (e), Г \ "/<р:чо) =
- (Li + <°^o>b(Lz+(o:Y0))/(LiAL2)+(0:Yo) -:r'.
(θ:Γ·) = (о:Г) = {о}. So by 2.24, N is a ring.
3.5 DEFINITION A monogenic N-group Г with Г + {о} is said
to be of
type 0: <=> ,,Γ is simple
type 1: <=■> ,,Γ is simple and strongly monogenic
type 2: <=> Г is N -simple.
The definition of "type 2" cries for
3.6 REMARK Of course it seems more natural to define "type 2"
by "N-simple" (see e.g. Fain (1)). But N-simplicity says
very little about Г. For instance, every non-zero sub-
near-ring N of M(r) with N?Mc(r) has fJr of'type 2"then.
So one can get nearly everything except nice structure
theorems. Moreover, we would not get
3.7 PROPOSITION
(a) ΝΓ of type 2 -> ^Г of type 1 -> ^Г of type 0.
(b) If ,,Γ is of type 1 or 2 then Ω = {ο} or Ω = Г.
(c) If ,,Γ is a unitary N = N-group then ,,Γ is of
type 1 <=> ,,Γ is of type 2. In this case,
\j γεΓ* : Νγ = Γ (see also 3.19(a)!).
Proof. (a):by 3.4(d) and 1.34.
(b) follows from (a) and 3.2.
(c) ->: Let ΝΓ be of type 1. If Δ <Ν Γ then
о
V όεΔ: N6 = {о} or = Г. Hence Δ = {о} or Δ = Γ,
si nee each δεΝδ.
<-: by (a).

78
§3 ELEMENTSOF THE STRUCTURE THEORY
3.8 EXAMPLES If Γ
define (Betsch (3))
= {feMo(r)|f(2)e{0,2}}
= {feMn(r)|f(2) = 0}
= ifeMn(r)|f(3)
0}
Then
Γ is of type 0, but not of type 1
'r is of type 1, but not of type 2
Ί
N
Γ is of type 2
(where ηγ is defined as in 1.18(c)).
This can be seen by simple calculations.
3.9 EXAMPLES If Ν = Νς then
ΝΓ is of type 0 <=> j,r is of type 1 <=> Γ is a simple
group with Ω = Γ,
,,Γ is of type 2 <=■> Γ is a cyclic group of prime order.
This holds since N-kernels (N = {0}-subgroups) in Г
coincide with normal subgroups (subgroups, respectively)
°f Γ; 3 ΥΛεΓ: Я, = Г results in Ω = No = Ny = Γ.
0 ' о о
3.10 PROPOSITION (Betsch (3)). Let fJr be of type 0 (with
generator γ) and let L Sj, N be a minimal left ideal
with L «f (ο:γ). Then L =N Γ.
Proof. By 3.4(a), ίγ ^ Γ. By Lf(0:y), LY + {o}.
Since ΝΓ is simple, Ly = Γ.
h: L -» Γ ε HomN(L,r) and Ker h = Lft(o:y) = {o}
(since L is minimal).
3.11 COROLLARY Let N = У L.eTL, where I is some index set and
.L τ ι о
ι εΐ
all
Li are minimal left ideals of N. Let ν be ε{0,1,2}.
(a) Each N-group of type ν is N-isomorphic to some L..
(b) I finite »> there are only finitely many classes of
non-N-isomorphiс N-groups of type v.

За Types of N-groups
79
Proof, (a) Let ,,Γ be of type ν and generated by γ.
Since Γ = Νγ + (ο), 3 ι'εΐ: Ц i (ο:γ).
Now apply 3.10.
(b) Follows from (a).
3.12 LEMMA (bcott (4)) Let N have the DCCN and let Μ <Ν N
be monogenic (by m ) and 3 πι,εΜ: (0:m,) = {0}.
Then Μ contains a right identity and (0;то)м = ί0>.
Proof. Let all annihilators be taken in M.
(a) Since (Oim^ = {0}, the map h: Μ + Mm1 is
an N-i somorphi sm; moreover, Mm, = M. If Mm, «= Μ
2
then applying h we get Mm, «= Mm,, and so on,
contradicting the DCCN. So Mm^ = Μ and
3 eeM: em, = m,. But then \/ meM: mem^ = mm,,
so Μ meM: те-те(0:т ) = {o}.
Therefore e is a right identity in M.
(b) Mm = Μ => Ηπι,εΜ: m,m„ = e. If mm, = 0 then
v ' о с с о i
also me = mm^m = 0m = 0, so (0:тр) ? (0:e) = {0},
So (0:m2) = i0} and - as in the beginning of (a)-
M = Mm2.
Suppose that пце(0:т ). 3 ιτι^εΜ: т, = m.m~. So
П1л = m.e = m.m0m = m,m_ = 0
4 4 4 с о 3 0
This shows that (OimJ = {0},
го* = Голб = m.m0m = m,m_ = 0 and m, = т«т0 = 0.
4 4 4г!о 3o 3 4^
3.13 THEOREM (Scott (5)). If N = NQ has the DCCN and
Μ <N N is monogenic (by m ) then Μ contains a right
identity and (0:т0)м = ί°Ь
Proof. Again, all annihilators are to be taken in M. In
view of lemma 3.12 we "only" have to show that
3 rn^M: (0:m1) = {0}.
Suppose that 3 Μ' έ., Ν 3 m'eM': Μ' monogenic by m^
and (0:m') =f {0}. W.l.o.g. we may assume that M' is
minimal for containing such an m'.
Let mleM' be such that (0:m^) is minimal in
{(0:т')|М'т' = M'} (so also M'm^ = M'). Therefore

80
§3 ELEMENTS OF THE STRUCTURE THEORY
3 m'eM': m'm' = m\ and (0:m2) 9 (0:mj) as in 3.12(b).
If mi generates M' then minimality of (0:mi)
forces (0:m2) = (0:pi[).\/ hi'eM':m'm^m| = m'm^,
so V m'eM': m'mi-m'e(0:mi) = (0:mi) and hence
\/ m'eM': (т'т2)т2 = m'm2 showing that mi is a
right identity in M'mi
M'
and so (0:m£)M, ={0}.
Consider
By 3.12(b), (0:mQ) = {0}
If mo does not generate M' then M'mi < Μ
the sequence M1 > M'm2 > M'(m2)2 £ ... .
3 keIN : M'(m£)k = M'(m2)k+1 = ... . Thus
(M'(m£)k)(m£)k+1 = Μ'(m^)k and since m£: =
= (m£)k+1 ε M'(m2)k, m3 generates M'(m2)k.
By the minimality of M', (0:m^)Λ Μ'(m2)k = {0}.
Again using the minimality of M' we see that each
generator m^ of M'(m2)k=M'm' has (O'.^jnM'iiij
= {0}.
We shall show that г
statement.
(a) mimi generates M'mi, for m\ = mimi and
M'm^ = M' imply that (M'm3)(m|m3) =
= M'm|m^ = M'm3.
= mlmi violates this
(b) Observe that (0:m3) + {01. f°r otherwise
M' -N M'm3 < M'mi < M'. Take some non-zero
mge(0:m3). 3 mieM': m^mj = mi, since ml
generates M'.
Now 0 = mimi = mimJmi = nigmimJmi.
Hence т£т3г(0:т|т3)л M'mi, but m^mi + 0
since nigmimJ = nigm| » mi ^ 0.
So we arrive at a contradiction and the proof is
complete.
N-groups of type 0 over a semiprimary (see 9.260)
near-ring N are studied in Kaarli (2), (4) and (6). Holcombe-Walker
(1) study N-groups ΝΓ of type 3 (i.e. Nr is of type 2 with(V π ε Ν :
ηγ =ηγ')=> γ = γ'. The sum of all left ideals L of N = N , where ,,L
is of type 1, is called socle of N (see e.g. Ramakotaiah (3)).

ЗЬ Change of the near-ring
81
b) CHANGE OF THE NEAR-RING
Up to now we had an unjust situation: a near-ring keeps an
harem of N-groups, but not conversely. Now we let an N-group
„Г change into Ν/ΙΓ (for some I 3 Ν), Ν Γ, Ν Γ. These
oc
changes will be an important tool in later considerations.
3.14 PROPOSITION (Betsch (3)). Let I be an ideal of Ν, Γ a
group and νε{0 ,1,2}.
(a) If Γ is an N-group with I s (ο:Γ) then
(η+Ι)γ: = ηγ
makes Γ into an N/I-group м/тг·
If Nr is of type v, so is Ν/ΙΓ.
If ,,Γ is faithful, the same applies to ц/1г-
(b) If Г is an N/I-group then
ηγ: = (η+Ι)γ
makes Γ into an N-group ,,Γ with Ι ε (ο:Γ)ν·
If м/тГ is °f tpye υ, so is ,,Γ.
If
Ν/ΙΓ is faithful then I = (ο:Γ)Ν·
The proof is a collection of straightforward arguments and
therefore omitted.
Observe that (N/I)Q = ίη0+ΙΙη0εΝ0>·
Each N-group Γ can be viewed as an N-group ,, Γ and as an
N-group м Г in an obvious way (by restriction). In 3.4(d)
c \
we already mentioned this fact. We now study the relation
between N Γ, N Γ and ^Γ:
о с

82
§3 ELEMENTS OF THE STRUCTURE THEORY
3.15 PROPOSITION Let Γ be an N-group and Δ a subset of Γ.
(a) ,,Γ is faithful iff N Γ and N Γ are faithful.
о с
(b) Δ <)Ν Γ <-> Δ <!Ν Γ
(с) Δ <Ν Γ <"> Δ <Μ Γ Λ Ω = Δ
"Ν
Ν.
Proof, (a) If ,,Γ is faithful, the same trivially applies
to N Γ and N Γ. Conversely, let ηΓ be = {o}.
о с
Then (with η = n+n as in 1-13) V γεΓ: nnY+nro =
о' с
noY+ncY = nY = °-
Taking γ = о yields ηςο = ο. So Μ γεΓ: η γ = ο
and η = 0. But no = ο gives V γεΓ: η γ = ο,
hence η„ = ο. Therefore η = η +η„ = 0.
с ос
(b) -> is trivial. If Δ 3Ν Γ then
о
i/ 6εΔ V γεΓ \/ ηεΝ: η(δ+γ)-ηγ = nQ (δ+γ)+η(. (δ+γ) -
"nc°"noY = η0(δ+γ)+ηε0"ηο0"ηογεΔ·
(с) is even more trivial
The relation between ,,Γ and N Г is particularly important.
3.16 COROLLARY Let Nr с ^.
(a) Nr is simple <-> N г is simple.
N
N.
(b) N Г is monogenic by γ -> ΝΓ is monogenic by γ.
N
(с) N Г is strongly monogenic =■> ..Г is stronnly mono
N
genie or {o} =j= Ω =j= Г.
(d) Γ is N -simple =-> Г is N-s1mple.
3.17 EXAMPLES If Ν = Μς(Ζ4) then 24 is N-simple but not
N -simple (since {0,2} is an NQ = {0}-subgroup).
So N-simplicity does not imply N -simplicity.
Plugging all together yields

ЗЬ Change of the near-ring
83
3.18 THEOREM Let ΝΓ be an N-group and νε{0,1,2}.
(a) ΝΓ is of type ν »> м Г is of type \> or Ν Γ = {о}
Ν
Ν
(b) Ν Γ is of type υ (for ν = 1 assume that in
о
ΝΓ Ω = {ο} or Ω = Γ) -> ,,Γ is of type v.
Proof, (a) Anyhow, ^r is simple, therefore also ,. Γ
by 3.16(a).
Let ,,Γ be monogenic by γ. Then Ν γ <L Γ by 3.4(a)
Hence Nqy = {o} or Nqy = Γ.
If Ν γ = Γ, Ν Γ is monogenic, too.
о
If Ν „γ = {о} then Γ = Νγ = Ν γ+Ω = Ω implies
ο ο
that \/ γεΓ: Νγ = Γ.
Again by 3.4(a), \/ γεΓ: Ν γ = {ο} or = Γ.
So either Ν Γ is monogenic or Ν г = {о},
о
If ΝΓ is of type 1 then Ω = {ο} or Ω = Г.
If Ω = {о} then V ΥεΓ: Νγ = Ν γ+Ω = Ν γ and
и Γ is again of type 1.
о
If Ω « Γ then each γεΓ generates ,,Γ so
(again by 3.4(a)) V γεΓ: Νογ = {о} or NqY = г.
So N Γ is either of type 1 or NQr = {o}.
о
The assertion for ν = 2 is trivial,
(b) By 3.16.
3.19 REMARKS
(a) 3.18(a) and (b) show that 3.7(c) holds for arbitrary
near-rings!
(b) Information about the behaviour of „Γ with Μ <Ν Ν
can be found in Mlitz (3).

84
3 ELEMENTSOF THE STRUCTURE THEORY
c) MODULARITY
3.20 DEFINITION L 4 N is called modular: <=>
<»> 3 eeN V ηεΝ : η-ηβεί.
In this case we also say that L is "modular by e"
and that e is a "right identity modulo L" (since
V ηεΝ: ne ξ η (mod L)).
3.21 REMARKS
(a) If L1,L2 ά Ν with Ц e L2 and Lj is modular by e
then L2 is modular by e, too.
(b) {0} is modular iff N contains a right identity.
(c) Every normal subgroup of (N ,+) is a modular left
ideal of N (by any element of N ).
(d) If L is modular by e in Νε1?0 then eel iff L = N.
3.22 PROPOSITION (Betsch (3)). Each modular left ideal L+Ne7)0
is contained in a maximal one (which is modular, too).
Proof. Let L be modular by e. Apply Zorn's Lemma to the
set of all left ideals I ? L with e £ I and
use 3.21(a).
Proposition 3.22 is not always true if N -f N : see 3.21(c).
3.23 PROPOSITION (Betsch (3)). L <3 N is modular <=■>
<"> 3 ΝΓεΝ# 3 ΥεΓ: цГ mon°9enic by γ Λ L = (ο:γ).
Proof. =>: Let L be modular by e. Then ,,Γ: = N/L is
monogenic by γ: = e+L, since N(e+L) = {ηε+ί|ηεΝ} =
= ίη+ί|ηεΝ} = N/L = Г. Moreover, ηε(ο:γ) = (L:e+L)<=>
<=> n(e+L) = L <=> netl <—> ηεί.
<=-: Let ,,Γ be monogenic by γ. Then 3 ezH: ey = γ.
But then \/ neN:ney = ηγ, so V ηεΝ: η-ηβε(ο.-γ) = L
and L is shown to be modular by e.

3c Modularity
85
Applying 3.4(e) we get
3.24 COROLLARY L ^ N is modular => L ? (L:N).
Proof. Take some (by γ) monogenic N-group Γ with L = (ο:γ)
Then L = (ο:γ) ? (ο:Γ) = (o:N/L) = (L:N).
3.22 - 3.24 are similar to the ring case ((Jacobson), pp. 5-6).
Looking at (L:N) more closely gives for future use (cf.
Ramakotaiah (1)):
3.25 PROPOSITION Let L be modular by e. Then (L:N) = (L:Ne)
and this is the greatest ideal of N contained in L.
Proof. (L:N) ίΞ (L:Ne) is clear. If n£(L:Ne) then
\J η'εΝ : nn'eeL. But nn'-nn'eeL, hence
V η'εΝ: ηη'εί. So ne(L:N) and (L:N) = (L:Ne).
By 1.42, (L:N) is a left ideal and it is easy to
see that it is even an ideal of Ν , (L:N) s L
holds by 3.24.
If I 3 N with I s L then trivially I s (L:N).
3.26 THEOREM (cf. (Kertesz), p. 122). If N = Lj+L2, where
Li,L2 are modular left ideals, then L,nL2 is again
modular.
Proof. Let L,,L~ be modular by e,,e„, respectively.
Decompose e,,e2:
ν/ΠΘΓΘ "< ι ("niCLt j 19* 99 ?'
e1 = Ии + Л12
6p я 919 9
We claim that L^L- is modular by *·?1 + )ί12 =:e*
If ηεΝ then n-ne = п-п(Л?,+Л,~) = n-ni.p + nfc.»-
-п(Л2,+Л.2) = n-ne,+ne,-n(-il, ^e, )+ηί.,2-η(Л~j + i.,2)
But η-ηθ,εί^, nej-n(-Л,i+e,)εί, and
пЛ« 2"η()ί21+)!Ί2^ε^1"
Therefore V ηεΝ: η-ηεεί,.
Similarly, \j ηεΝ: η-ηεεί2, and we are through.

86
§3 ELEMENTSOF THE STRUCTURE THEORY
3.27 COROLLARY
(a) If L is a modular and Μ a maximal modular left ideal
then L л М is modular.
(b) A finite intersection of maximal modular left ideals
is modular.
(c) If N is a direct sum of two modular left ideals then
N contains a right identity.
(d) (Betsch (3)). If N contains a finite family of maximal
modular left ideals with zero intersection then N
contains a right identity.
3.28 DEFINITION Let ν be ε{0,1,2}. A left ideal L of N is
called v-modu1ar if L is modular and N/L is an N-group
(via n(n'+L):= nn'+L) of type v.
Let X (N) be the set of all v-modular left ideals of N.
3.29 REMARK So a 0-modular left ideal is just a modular maximal
one and a 2-modular left ideal L is a modular maximal left
ideal with no N -subgroup strictly between L and N.
(Beidleman calls these left ideals "strictly maximal".)
v-modular left ideals turn out to be very useful in determining
radicals of related near-rinqs.
3.30 PROPOSITION Let (Ν η- ) i j be near-rings and N their direct
product. Let L^ be a left ideal of H. for some icl.
f N. i + j
Denote И Μ. with Μ.: = i J by Γ. <|, N.
Jel J J [ I. i = j Ί г
Then for νε{0,1>2}, Li is v-modular in N. iff Γ. is
v-modular in N.
Proof, (a) If Li is v-modular in Ni then N (l^./L.)
is of type v. By ni ((. . . ,n ! ,. . . )+Ц ): =1
= (. . . ,Ο,η.ηί ,0,. . . )+Γ^ , Ν/Γ1· becomes an N.-group
and clearly N/Ci =N Nj/Li. So N/t\ is an

3c Modularity
87
N.-group of type \>.
If
{0}· (notation as in
the statement), Ni - П/J., so 3.14(b) shows that
Ν/Γ. is an N-group of type ν (and the multiplication
is the same as in 3.28). Hence Γ^ is v-modular in N.
(b) If Г. is v-modular in N then Ν/Γ- is an
N-group of type v. Similar to (a), N/Ii =N N./L^
where (...,n·,...)(n!+L.) : = n^ni+L·. The annihilator
of N^/L-j in N contains J. (as in (a)), so by
3.14(a) N-j/L, is an N.-group of type ν in the sense
of 3.28. Therefore L^ is v-modular in N..
From 2.28, 2.30 and 3.30 we get
3.31 COROLLARY If I«N is a direct summand in N and if L ε "Ϊ (I)
then there is some Ur/^N) with L = L/iN.
There exist examples (see e.g. no. N) 1) in the appendix (nr.'s
of low order) such that 3.31 does not necessarily hold for v=1 if
I is not a direct summand. See also Ex. 6.32 in Mel drum (13).
Is, in 3.30, every LcY (N) given by some L^ L = f(x,y)|xsy mod 2}
is a counterexample in the constant near-ring on 1x7. But:
3.32 PROPOSITION Let N be the direct sum (or product) of the nr.'s
ΝΊ· (i ε I ) and L < N. For i ε I let L i : = {1 . εΝ· | ( . . . , 0 ,1 . , 0 ,. . .
el]. If ©N. ^L ε У (N) then L i с £ (Ni ) for some i ε I
(provided that N = NQ if v = 1 ) .
Proof. It is easy to show that each L - is a left ideal of N ,■ .
If Lis modular by (... ,e ..... ) then L. is modular by e ■ .
If all L.=N. then ΦΝ^ L. So suppose now that L. =(- Ni
for s ome i ε I.
(0) If v=0, assume that L· is not a maximal left ideal
in N.. Then there is some left ideal Li strictly
between I. and N . . Now if I.] :={(.. ,0 ,1 ! ,0 ,. . ) | 1 I εΙ\ }
then L+U is a left ideal if N properly containing
Take n^N^Ll . Then
,0,ni ,0,
L , whence L + L. ■
= (..,ii5..)+(
all j^i and n.=l-+lleL.+LI=L!, a contradiction
So L- is a 0-modular left ideal in N..
,0,1 ! ,0,. . ) ε L + L. '■ . Hence 1, = 0 for

88
§3 ELEMENTS OF THE STRUCTURE THEORY
2) If v = 2, we proceed similarly. L is 0-modular, too, so
L. is a maximal left ideal. If it were not strictly
maximal then there is some bigger (N. ) -subgroup L ! .
Define L! as in (0) and apply 2.15.
1) Now let v=1 and N=NQ. We must show that Ni/Li is
strictly monogenic. Suppose that η,εΝ, fulfills
Li<Ni(ni + Li)<Ni/Li. If n: = (. . ,0,n. ,0,.. ) then
N(n+L)=0+L or N(n+L)=N/L. In the first case, Nn =
= (..,0,N.n.j,0,..) (since N = NQ) and this is in L,
whence N.n-sL·, hence N · (n · +L . ) =0 + L · , a contradiction.
In the second case, take some η!εΝ. with η! + L. not in
N.j(n.+L-) and let η ':=(.. ,0 ,n ! ,0 ,..). There is some
n" = (. . ,n'.',. . ) with n'+L=n"n + L. In the i-th component
we get η!-η!·'η.εΙ-., whence n! + L · ε N,(n.+L^), again a
contradiction. Hence L. is 1-modular in N..
.LARY If N =0Ni (and N=NQ for v= 1) and L εϊ^Ν)
^ ' ■ -- ■ ■ з.зо) iS in <£V(N), too
3.33 C0R0LLAR, .. ,. ^...
then Γ. (as defined in
3.34 THEOREM (Kaarli (4))
5<fl N, SN«=S and if L-/2(S), L£So,
and S/L is an N-grouo of type 2.
If S
then L is an i deal of „S a
Proof. If ηεΝ, Ι ε|_ and s ε s, we have to sho.v that
n(l+s)-ns ει; w . 1 .
can assume that
Case I: S
_ , о . g . we
.qs^L. If NQs ^L then L + N
ηεΝ
о '
„---. . a <e s eS> , s ' eS ,
о oo
0--t-- ^'"о" "" " °о""о'
s'=l'+n's. Now s s'=s Μ'+n's) = s„(1'+n's )-з n's+n'seL+S s=
о о ο ν о ' оч o'oo.o о
=L. Hence S S=L . Now s -s eeL for some esS. Hence S sL,
о oo о
a contradiction. Therefore S ss L always implies N
о J v с
But for s£S, s (l+s)e|_, hence η (l+s)eL, whent
n(l+s)-ns£L, as desired
Case II : S s^L.
о
ssL.
ice
n(l+s)-nseL, as desired.
Case II : S^^L. Then SQs + L = S. Let Y(s): = (L:s)N . Then
Y(s)n S = Y(l+s)nS holds for all UL. Also, No/Y(s)^S/L
by h:N ^S/L, n^ns+L. From this isomorphism we know that
Y(s) is maximal in „N So if Y(s)fY(l+s) we get that
Y(s ) + Y(l + s) = NQ. This shows that SQSosY(s), hence
y(s; + t(, i + s; = in . ims snows
SoSQssL. Since SQs+L=S, we
the contradiction L=S. Sin
Now each ηεΝ can be writt
о
we get that
"ο"ο-·ч"'' e
erive S SsL and from thi
ce SQ^Y(s), we get Y(s) + Sc=N_
и - и О
Now each ηεΝ can be written as n=y+I with уεΥ(s)=Υ(1+s)
and s eS . Hence for 1cl we finally get what we want:
oo ·
n(l+s)-ns = y(l+s)+?o(l+s)-s"os-ys ε L .

3d Quasiregularity
89
d) QUASIREGULARITY
3.35 NOTATION For ζεΝ, denote the left ideal generated by the
set {n-nz|neN} by L.
(Note that for L = Ν, ζ = 0, NQ has still one single
meani ng.)
3.36 DEFINITION
(a) ζεΝ is called quasi regul ar ( = : q_r) if ζείζ<
(b) SsN is called quasi regul ar ( = : q£): <=> \j scS: s is qr.
3.37 REMARKS
(a) If ΝεΤ)0, ζεΝ is qr <-> Lz = N.
(b) L is modular (by z).
(c) Beidleman (1) calls (for a near-ring ΝεΎΙ with
identity 1 ) ζεΝ quasiregular if 3 ΥεΝ: y(l-z) = 1.
In this case, ζ is also quasiregular in the sense
of 3.36.
3.3B PROPOSITION (Ramakotaiah (1)). Let N be cf)0.
(a) ζεΝ nilpotent =■> ζ is qr.
(b) Each nil subset of N is qr.
(c) If L <l N is modular by e then e is not qr.
(d) If e is a non-zero idempotent then e is not qr.
Proof, (a) If zn = 0, consider any χεΝ. Then
χ-χζεί , χζ-χζ ει χζ " -χζ ει .
Hence χ-χζηεί , so χεί and L = Ν.
(b) Follows from (a).
(c) \/ ηεΝ: n-neeL. If e is qr. then L = Le = N.
(d) Assume that the idempotent e is + 0. Consider
о
the N-endomorphism h · N * N . h (e) = e = e +0
e χ * xe

90
§3 ELEMENTS OF THE STRUCTURE THEORY
shows that h„ + o.
e 2
\j χεΝ: h (x-xe) = xe-xe = 0,
and e cannot be quasirenular.
so
Ker h„ + N
e '
3.39 PROPOSITION Each nil ideal I of a near-rinq N is
quasi reqular.
Proof. Proceeding as in 3.38(a) one sees that
V 1εΙ, Hr
{χ-χΟ|χεΝ} ξ Li
If ΐ ε I then by 2.99 ieNQ,
so
i ε L · and i is qr.
3.40 THEOREM (Ramakotai ah (1)). ΝεΤ?0> DCCN, Μ <Ν N. Then
Μ is qr <=> Μ is nil potent <=> Μ is nil.
Proof. Let Μ be qr. For keIN , let
Μ
(к)
be the
N-subgroup of N generated by Μ (2.56). We get
a chain Μ ? M^2) э M^3^ ξ? ... . By the DCCN,
3 kcIN : M(k) = M(k + 1) = ... =:P.
If Ρ = {0}, we are through.
If not, observe that p' ' = Ρ =f {0}. so
Pe{K| Κ <Ν Ν Λ K?P л PK + {0}} =: 3° . So У + 0.
The DCCN assures the existence of a minimal element
K0 in Ψ. Since PK0 + {0}, 3 VV PkQ + {0}.
Pk <t, N, Pk =K =P, P(Pk ) 4= {0} (since otherwise
,(2)
p - (0:kQ), so Ρ = Ρ
These three assertions qualify Pk
Ξ (0:k ), a contradiction)
to be ε ?°
Since Pk„ = Κ .
о о
Рк„ = К„.
о о
Therefore 3 ρεΡ: pk
So \/ ηεΝ: (n-np)k = nk -npk = 0.
Hence V ηεΝ: n-npe(0:k ) + N, so L + N and p is
not quasiregular. The rest follows from 3.38(b).
3.41 REMARKS Kaarli (4) showed that in 3.40 the DCC for
monogenic N-subgroups suffices. If NfN then Μ £Ν Ν, Μ qr.
implies Μ nilpotent. 8ut for N = Nc with DCCN, nil does not
imply nilpotent. If q is qr, the left ideal L generated by
q is not necessarily qr, (take e.g. N=Z3[x], q = 1, L = (0:1)
and 1 £ L). Ramakotaiah (3) showed that if L « N = N and
q ε L is qr. in L then q is qr. in N.
See also Oswald (6) and Ramakotaiah-Santhakumari (1).

Зе Idempotents
91
е) IDEMPOTENTS
3.42 DEFINITION A set Ε of idempotents is called orthogonal if
Ve.feE: e+f=>e-f=0.
The standard method to get orthogonal idempotents is to
decompose a right identity:
3.43 THEOREM (Beidleman (1),(6)). If Ν ε 7)Q contains a right
к к
identity e, if N = J' Li (Ц. 4 N) and if e = I ei
i = l
i = l
(e.eL·), then e,,...,ek are orthogonal idempotents
and each e· is a right identity in l- which generates
Li - Nei·
к
Proof. If e = e, + ...+e., then W ηεΝ: η = ne = η £ e. =
к i = 1
I nei (by 2.30). If ηεΐ^, the uniqueness of
the representation yields η = ne., so e. is a
right identity for L·. In particular, e. is
idempotent, while for i =f j one gets ече< = 0
by taking η = e. above.
Finally, Ц = Ne.j, since each ^еЦ can be
written as I. = i.e..
If one has a right identity in 3.43, but no direct decomposition
it is sometimes still possible to get orthogonal idempotents:
3.44 THEOREM (Scott (5)). Νε7\,, DCCN, Μ <N N, L χ ,L2 <|fc N,
L,,L2 - M. If Μ contains a right identity e and if both
L,,Lo are minimal for the property that L.+L» = Μ then
there exist orthogonal idempotents 6ιε4· β2εΙ_2 with
(a) ej+e2 = e (mod L^n L2)
(b) (OiejjriM = L2 and (0:е2)лМ = Lj.

92
§3 ELEMENTS OF THE STRUCTURE THEORY
Proof. If e = Л^^
= *1{i1+*2).
(fcjeL,, Л„е1-2) then
*1 = V
But ЛЛЛ^ЛЛ-Л^-Л^Ц, so * χ (A l + Jt2 ^ Ξ
о
i Jtr-+Я. j й. 2 (mod Lj)
2
and ί.Ί (A1+«.p)-A,-Jt1Ji.pELp, so А.(г.+Л?) =
1^1
1 Ί*2
= л.л2+Л, (mod L2).
о
From 2.22 we conclude that «.j ξ л,Л2+Л^ (mod LjnL2),
hence ^i"*·! = Л.Л^Цл L2> so \/ melM : Л.^Л^ (mod L,nL2)
Similarly, \j meIN : ЛтнЛ2 (mod LjoLj).
Now let i be ε{1,2}.
All ЦЛ? <N N. By DCC 3 kelN: LJ^ = Ч^*1 = ···
Therefore (ЦЛк)Лк + 1 = 1..Лк and Лк + 1 generates
L^k . Moreover, Л^еЦЛ1?. We can apply 3.13 and
get
(0:Лк+1)п L.£k = {o}
and L-Л. has a right identity e· wiU
e Лк+1 - Лк+1
(b) By 1.13, Μ = Ме^Оге^л Μ = Lj + fOrej)* M.
k+1 k+1
From e^*+1 = Л*Г1 we get
(Oie^nMe (0:Лк+1)лМ? (LjnL^^lnM.
By the remarks at the beginning of the proof,
e ξ Лк+1+Лк+1 (mod Цп L2).
Thus Μ mefLjrtL^l^lflM: m = me = тЛк+1+тЛк+1
k + 1
= тЛ2 l (mod LjrtL2) (since me = т(Л1+Л2)).
But
,k+l
k + 1,
2 εί2, so meL2, hence (l^n LgrtJ*1)?!^
and by the minimality of L? we get (Oie^n Μ = l«.
By symmetry, (0:е2)лМ = L,.

Зе Idempotents
93
к + 1 к+1
(a) ej + e2 = (ej+e^e ξ (е1+ег)Л1 +(ej+e2)fc2 (mod LjnLg)
к+1 .к+1 а.
= Л, and
(as above). Because of e^i - *-i
ce (L,nL2:tj + 1
,к+1А„к+2 _
e2lj+1cLjnL2 (since (Lj л L2: i.^ + 1) = L2) we get
ei+e2 ξ Л?'* + г.2т'" ξ Л.+Л- = е (mod Lin LJ.
Finally, since e, eL, = (0:e2)nM, eie? = 0 anci ЬУ
symmetry, e2el = °- So el,e2 are 0rtn°90na1
i dempotents.
3.45 REMARK See Lausch (5) for applying sets Ε of orthogonal
idempotents to get a decomposition of N into "blocks"
"spanned" by some partitions of Ε (similar to (Artin-
Nesbitt-Thrall)). See also Deskins (2), Williams (1).
See Fain (1) (Th. 6.4) and Lyons (3),(4) for more
decompositions induced by orthogonal idempotents.
3.46 DEFINITION An idempotent eeU is called central if it is
in the center of (N,·), i.e. if \/ ηεΝ: en = ne.
3.47 PROPOSITION (Betsch (3)). Let e be a central idempotent
with Ne <| N. Then N is the direct sum of the ideals
Ne and (0:Ne) = (Ore).
Proof. Clearly Ne (by assumption) and (0:Ne) (by 1.43(b))
are ideals. By 1.13, N = Ne+(0:e) and
Nen(0:e) = {0}. But (0:e) = (0:Ne), since е is
central.
3.48 PROPOSITION (Fain (1)). Let Ε be a set of orthogonal
central idempotents and le? any sum of distinct elements
of E. Then
(a) E - Nd.
(b) Ее. is idempotent.
(c) V neN: nle . -Enei ε(0 : Ε ) <l N.
(d) (ΟεΕ ν |E| > 2) -> N = N.

94
§3 ELEMENTS OF THE STRUCTURE THEORY
Proof, (a) is trivial.
(b) (Ее.)2 = Ze^Ze^ = EEe^ = Ze.e^ = Eei .
(c) V eeE: (nEei-Enei)e = 0.
(0:E) = Π (0:e) 3. N (by 1.43(a)). Moreover,
eeE *
(0:E)N ^ (0:E) since
V ηεΝ \j me(0:E) Μ eeEr (mn)e = men = On = 0.
(d) If ΟεΕ then clearly N = NQ.
If |E| > 2, let e + f be in E. Then for all
ηεΝ, nO = nef = efn = On = 0.
3.49 REMARK A ring is called biregular if each principal ideal
is generated by an idempotent. In (3), Betsch defined a
near-ring to be biregular if there exists some set Ε of
central idempotents with
(a) \/ eeE : Ne 3 N.
(b) V ηεΝ 3 eeE: Me = (n) (principal ideal generated
by n).
(c) V e.feE: e+f = f+e.
(d) \/ e.feE: efeE Λ e + f-efeE.
Ramakotaiah (1) showed that each commutative biregular
near-ring is isomorphic to a subdirect product of fields
and hence a biregular ring.
More information can be obtained in Courville (1), Courville-
Heatherly (1), Miron-Stefanescu (1) and Ramakotaiah (3).

3f More on minimality
95
f) MORE ON MINIMALITY
We conclude this paragraph with some results concerning
minimality of non-ni1 potent N-subgroups and left ideals of N.
As we will see, considering minimality does not imply that
the results can be reached by minimal efforts.
However, we first reap the fruits of previous sections.
3.50 DEFINITION Μ <Ν Ν (L ^ N) is called a minimal non-nil
potent N-subgroup (left ideal) if it is minimal in the
set of all N-subgroups (left ideals) of N which are not
ni1 potent.
Clearly if L ^. N = N and L is a minimal non-ni1 potent
N-subgroup then L is a minimal non-ni1 potent left ideal.
3.51 THEOREM Νε?70, DCCN.
(a) (Scott (5)). Μ <Ν Ν is a minimal non-ni1 potent
N-subgrcup =·> Μ contains a right identity e with
Ne = Me = Μ (see also Beidleman (6)).
(b) If L ^j, N is a minimal non-ni 1 potent left ideal
then L contains a non-zero idempotent.
(c) (Beidleman (1)). If L £. N is a minimal non-nil-
potent N-subgroup then Lisa direct summand of NN.
2
Proof, (a) If meM is not nilpotent then m eMm^M is
not nilpotent, so Mm = Μ and by 3.13 Μ contains
a right identity e. By the minimality of M, Ne
(not nilpotent!) = Μ = Me.
(b) By the minimum condition in N, L contains a
minimal non-ni1 potent N-subgrouD Μ. Μ has a right
identity e by (a) and so L has a non-zero idempotent
(c) L contains a right identity e by (a) with Le = L,
By 1.13, N=L+(0:e) and L is a direct summand of ^N.

96
§3 ELEMENTSOF THE STRUCTURE THEORY
3.52 COROLLARY (Blackett (2)). ΗεΎΙ0, DCCN, N without non-zero
nilpotent N-subgroups. Then each minimal N-subgroup Μ is
generated by an idempotent e which is a right identity of M.
We now turn to minimal ideals.
3.53 PROPOSITION (Scott (4)). Let I be a minimal ideal, Μ <, Ν,
Μ nilpotent, Μ = I. Then IM = {0}.
Proof. Let Mk be = {Oh k>2, Mk_1 + {0}. Then
Mk-1-M = {0}, so Mk_1 is contained in the ideal
k- 1
(0:M). Hence the ideal J generated by Μ is
contained in (0:M). Since {0} 4 J -1» J = I and
IM = {0}.
See also Kaarli (2) and Scott (16).
3.54 THEOREM (Scott (6)). Νε7?0, DCCN, I a minimal ideal.
Then I is a finite direct sum of N-isomorphic minimal
left ideals of N (and therefore completely reducible
when considered as Л).
Proof. We need 3 lemmata and keep the assumptions of the
theorem.
Lemma 1. Let Nr be faithful and Δ be a minimal
ideal of ΝΓ. Let {0} + L ε (Δ:Γ) be a
left ideal such that \j γεΓ: Νγ = Γ ν Ly = {о}.
Then Lisa finite direct sum of N-isomorphic
minimal left ideals of N.
Proof, (a) Since ^r is faithful and L 4- {0},
3 Υ^Γ: L^oiy^, so (ory^L = (o:Yj)nL«=L.
If (o:yl)l + {0} then 3 γ2εΓ: (o:Yl)L »
ϊ»(ο:γ1)ίΛ (o=Y2)l·
Proceeding in this manner, by the DCCN we
eventually obtain elements γ,,γ~ ,...,γ εΓ
η
with П (o.-Yj), - ίθ>·
i = l 1 L

3f More on minimality
97
Thus (ο:{γ1,...,γη))|_ = ί) (o:Y.)L = (0).
Anyhow, we get a non-empty subset Σ of
{γ,,.,.,γ } of minimal order for the
property that (ο:Σ), = {0}.
Set Ζ =: {oj,... ,ak).
(b) Define 1^:= L if к = 1 and
Ц: = (οιΐΜσ^^ If к>1 (1е{1 к}).
Δ are N-iso-
We now show that h^: L.
morphi sms.
I ■* la.
к = 1: Then {0} + L = Lj and (6:a1)L={0).
Thus Loj = LjOj 4= i°J and Ha ^ = Г.
Since ίΕ(Δ:Γ), LjO^A. By 3.4(a),
Since Δ is minimal, L,°i = Δ and
h, is surjective.
Also, Ker hj = (ο·.σ^). = {0}; hence
h, is an N-isomorphism.
к > 1: Suppose that 3 je{l,...,k}: L-σ· =
= {o}. Then L,5(o:a·)» so
J J
LjE(o'.E). = {0}, a contradiction to
the minimali ty of Σ.
Hence all L.a. ·)■ {o} and (as above)
Цо^ = Δ. Also, Ker h· = (ο:σ^)π L· =
= (о:о.|)л (о:Д{а())л L = (o:E)L =
= {0}, and again h. is an N-iso-
morphism.
(c) Let i be ε{1,. . . ,k}.
L. is a minimal left ideal of N:
ВУ 3.4(e), Μ/(0.σ } \ Νσ^ = Г. By (b),
Ца.,· = Δ i s minimal .
Thus Lj + (o:o,· )/,„. ч is a minimal ideal
1 1 \OtO г )
of the N-group Ν/. \· Since
Цп(о!0^) = {ο} (by (b)), 2.8 gives

§3 ELEMENTS OF THE STRUCTURE THEORY
Li "N Li+^0:CTi'/(o:a.) ' s0 Li is a min1'mal
left ideal of N.
(d) Since all Ц =,. Δ, the L.'s are N-
i somorphi с.
(e) We show that L = Lj + ...+Ц. We may
assume tha t k>l.
If JleL, V ιε{1 к}: ta.cb. By (b),
L.oi = Δ, so 3 *ieLi: tCTi = liai-
Set Л' := l^. . .+lk.
If i + J. ί-.εί^(ο·.σ1·), so l^a^ = o.
So for all ie{l,...,k} £'σ· = ί-.-σ·.
Therefore (Л-Л')Е = {о}, so 4-1' ε(ο :Ζ )r> L =
= (0 }. Hence I = I' = l^+. ..+{.,.
(f) The sum in (e) is direct:
If I = Л1 + ...+«,|< = μι + ...+μ|< (^,-,μ,-εί.)
Then \/ ιείΐ,. . . ,Η: la. = u^ = Я·^·
Thus \/ i ε {1,. . . , к}: ί,-μ.ε(ο:Γ)η L and
the proof is complete.
Lemma 2. If IN + {0} and Μ <Ν Ν is minimal for
IM + {0} then
(a) M contains a right identity.
(b) 1л М is minimal amongst all non-zero ideals
of the nr. Μ which are also N-subqroups.
(c) Ι π Μ is the sum of minimal ideals of ,,Μ.
Proof, (a) We shall show that ^M is monogenic.
If V πιεΜ: Mm + Μ, ΙΜΜ = {0} since
\/ ηιεΜ: I(Mm) = {0} by the minimality of M.
Denote the ideal generated by IM by J.
Since IM is contained in the ideal (0:M),
Js(0:M) and JM = {0}.
But IM + {0}, IM=I and so J = I and we
arrive at the contradiction IM = {0}.

3f More on minimality
99
So Μ is monogenic and the result (a) follows
from 3.13.
(b) Since IM + {0} and IMeI л Μ, Ι α Μ is
a non-zero ideal of M.
Let {0} + К 3 Μ be such that Κ <N N and
ΚεΙλΜ. Since KMeKsI, Ks(K:M)nI and
(К:М)л1 + {0}, so (K:M)n 1 = 1 and
I=(K:M). It follows that IM«=(K:M)M«=K.
By (a), M contains a right identity e, so
(InM)e = InM. Thus In M«(InM)M*IM=K
and (b) fol1ows.
(c) Since Ι η Μ + 10}, there exists a
minimal ideal W of NM in Ir\M.
Take some meM.
If Mm = Μ then Wm is either = {0} or
a minimal ideal of ,,Μ (since Wm is
an N-endomorphic image of W).
If Mm -f Μ then IMm = {0}. But Μ contains
a right identity e, so We = W and thus
Wm = Wem^IMm = {0}.
Hence \/ meM: Wm
M.
ideal of
Set L:=
N
I Wm? I.
иеМ
{0} or Wm is a minimal
L <м М.
W
WeS L and L + {0}.
So L is the sum of minimal ideals of ,,Μ.
Of course, L ^ M. Also \j meM: Lm =
= J Wmm^L, so L <l M, and L is a non-
meM
zero ideal of Μ contained in
L = Ι η Μ and (с) is shown.
Ιλ Μ. By (b),
Although the reader might be gasping for breath, we
need a third Lemma, which will be used in the proof
of 4.47.

100
§3 ELEMENTS OF THE STRUCTURE THEORY
Lemma 3. If IN + t°} there exists an N-group
such that
Δ % Γ
and a minimal ideal
(a) (o:r)n I = {0}.
(b) I + (o:r) = (Δ:Γ).
(c) \/ γεΓ: Ιγ = {ο} ν Νγ = Γ.
Proof. Let Μ, W, e be as in lemma 2. By part
(c) of this lemma, In Μ is the sum of
minimal ideals of ,,Μ. By the fact that
WsinM, we conclude from 2.48(e) that W
is a direct summand.
So 3 U ^ ΙΛΜ: In Μ = W + U.
Define Γ:= M/U and Δ:= ΙηΜ/
So Δ is a minimal ideal of
We now prove the lemma.
(а) (о:Г)п I = {0} or = I. If
then Is(o:r). So ΙΓ = {o} and
But (1лМ)е = ΙηΜ, so I n № (I л M)M?IMsU .
Thus Δ = {о}, a contradiction.
U
Г.
w;u/u 2:
N
W.
(о:Г)М =1
IMsU.
(b) Since IMElnM, Ι?(Δ:Γ) and
Ι+(ο:Γ)=Ι+(Δ:Γ) = (Δ:Γ).
(c) If γεΓ, 3 meM: γ = m+U.
If Nm«=M, the minimality of Μ gives us
I(Nm) = {o}. So ΙΝ«=(ο:γ). If (IN)^ is
the left ideal generated by IN then
(ΙΝ)^(ο:γ). By 1.52, (IN) ^ is the ideal
generated by IN and therefore equals I.
So Ιγ = {о}, completing the proof.
Tired, but happy we are ready for the
Proof of the theorem. Suppose that IN = {0} and L is a
minimal left ideal of N contained in I. So LN = {0}
and L 3 N. Thus L = I and the theorem holds.

3f More on minimality
101
If IN + {0}, let Γ,Δ be as in Lemma 3. If
Ν/ίο·η =:Ν'' Ν'Γ is faitnful and has Δ as a
mi nimal ideal .
Set I·:- 1 + <о:Г)/(о:Г)ш
By lemma 3(c), \/ γεΓ: (Ν'γ = Γ ν Γγ = ίο}).
By lemma 3(a), Ι ~Ν Ι/{0} =Ν I'. Thus by
lemma 1, I X I* is the direct sum of minimal
N-isomorphic left ideals.
The proof is now complete.
Note that if ΝεΤ) is simple and has the DCCN then „N is
completely reducible and 2.50 is applicable. Cf. 4.46 and 4.47.
3.55 COROLLARY (Scott (4),(6)). Νε7?0, DCCN, I a non-nil-
potent minimal ideal of N; 0(N):= sum of all nilpotent
left ideals. Then Q(N)0 I = {0} and Q(N) is nilpotent.
Proof. See Scott (1) or (4) for the proof that Q(N) is nil-
potent. By 3.54 and 2.48, l)(N)n I is a direct summand
of NI. Let L ul I be such that I = Q(N)nI+L. By 2.22
V i el V qeQ(N)n I V «-ει: i (q+A) ξ iq+U (mod Q(N)nlnL).
Since (Q(N)nI)nL = {0}, i (q + A) = iq+U. Hence
I2 = I(Q(N)r,I + L) s I ( Q ( Ν ) η L) + IL. But by 3.53,
I(Q(N)r»I) = {0}, since Q(N) is nilpotent. So I2=ILsI
and the left ideal generated by I , the ideal generated
о
by I (by 1.52) and I (by minimality) coincide.
So I is contained in the left ideal generated by ILSL,
I = L and Q(N) л I = {0}.
3.56 REMARK There also exist results concerning near-rings with
ascending chain conditions. For "Goldie-type" ones, see
Oswald (2). For more results, consult Scott (1), Kaarli (2),
(4), Di Sieno-Di Stefano (4), Ramakotaiah-Santhakumari (1)
and Zand (1).

102
§4 PRIMITIVE NEAR-RINGS
This paragraph presents a discussion of the "building stones,
near-rings are made of", the so-called "primitive near-rings".
Similar to ring theory, the "atoms" are not the simple near-
rings as one might expect at a first glance. There is, however,
an important connection (4.47). The idea to consider primitive
near-rings comes from the bible ("You will recognize them by
their fruits"): given a near-ring N, we look at all of its
fruits (= N-groups) and ask, whether there are faithful and
"enough simple" ones among them. If this is the case, we call
N "primitive on this N-group". Since "enough simple" is not
precise we fix its meaning in wanting N-groups of type \>.
The resulting concept is that of "v-primitivity".
We get the hierarchy 2-primitivity<?> 1-primitivity<J> 0-pri-
mitivity, discuss conditions, which force some of these concepts
to coincide and make a lot of work towards a density theorem
which is comparable to the celebrated one in ring theory due
to N. Jacobson. We really get one for 2-primitive near-rinqs
with identity (4.52). Adding a chain condition, we arrive at
a Wedderburn-Artin-1ike structure theorem (4.60). Before that,
we get "better and better" density-like structure theorems
for 0-, 1- and 2-primitive near-rings. It comes out that many
theorems on v-primitive near-rings can be derived from zero-
symmetric v-primitive near-rings where they are much easier
to obtain since these ones behave more like rings. However,
many proofs concerning even zero-symmetric near-rings differ
totally from the comparable ones in ring theory.
Anyhow, the "building stones" mentioned above (2-primitive
near-rings with identity) are shown to be dense in HomD(r,r)
or Maff(r) (if NQ is a ring) or in MQ (6y(r) °r
MG (б}(Г'+Мс(Г) (if No is a П0П-Г1"П9)» where GQ is the
fixed-point-free automorphism group AutN (Γ). In particular,

4a General
103
if GQ = {id}, the latter two ones are Μ0(Γ) and М(Г).
Finally, the density property is seen to be a kind of an
interpolation property and a "purely interpolation-theoretic"
result will be obtained. Recall again (p.l) that Г*=Г\{о}, and
so on.
a ) GENERAL
1.) DEFINITIONS AND ELEMENTARY RESULTS
4.1 CONVENTION In all what follows, ν will be any number
ε{0,1,2} unless otherwise specified.
4.2 DEFINITION
(a) N is called v-primitive on ,,Γ: <=■> ΝΓ is faithful anrf
of type v.
(b) N is v-primi ti ve: <=> 3 мГем^: N Ί*s v-primitive on „Г.
(c) I £ Η is called a v-primitive ideal of N: <=-> N /1 is
v-primi ti ve.
4.3 PROPOSITION Let I be an ideal of N. Then the following
conditions are equivalent:
(a) I is v-primitive.
(b) 3 NrcNty: I = (ο:Γ) Λ ^Γ is of type v.
(c) 3 L Зг Ν: I = (L:N) л L is v-modular.
Proof. (a) =■> (b): I is v-primitive => N/I is v-primitive
on some „,,Γ -> ,,Γ (as in 3.14(b)) is of type ν and
I = (ο:Γ).
(b) -> (c): Let Г be = Νγ + {ο}. (ο:γ) =:L. Then
L is modular. By 3.4(e), Ν/. -^ Γ, so L is v-modular.
Finally, I = (ο:Γ) = (o:N/L) = (L:N).
(c) => (a): Take N/L =:Г. Then ^Г is of type ν and
(as above) I = (L:N) = (ο:Γ) .

104
§4 PRIMITIVE NEAR-RINGS
4.4 COROLLARY The following conditions are equivalent:
(a) N is v-primitive.
(b) {0} is a v-primitive ideal.
(c) 3 L <lt N: L v-modular л (L:N) = {0}.
4.5 REMARKS
(a) Observe that (c) in 4.3 and 4.4 give "intrinsic"
characterizations of primitivity - that will be
extremely helpful, for it enables one to recognize
primitivity "within N".
(b) If N is v-primitive on Г then N 4- {0}, Г + {о} and
if I <l N is a v-primitive ideal then I + N.
(c) 2-primitivity implies 1-primitivity and this in turn
implies 0-primitivity (always on the same group).
(d) The near-rings Nv of 3.8 are examples of v-primitive
near-ri ngs. (on Ζή).
(e) If U is v-primitive on Г then N «► M(r) (1.48).
(f) See §5 of Betsch (3) for a discussion of the spaces
of v-primitive ideals (v = 1,2) of ΝεΊ?0.
4.6 PROPOSITION Let N contain either a left or a right
identity e. Then
(a) Every v-primitive ideal I of N is modular.
(b) If e is a left identity of N then N is 1-primitive
iff N is 2-primitive (and in this case e is a two-
sided identity).
Proof, (a) If e is a left identity in N then (because
N/I is v-primitive on some N/jr) e+I 1S an identity
of N/I by 3.4(c). So \j ηεΝ: en ξ ne = η (mod I).
If e is a right identity, the assertion is trivial.
(b) Let N be 1-primitive on ΝΓ. By
3.4(c), e is a two-sided identity for N. By 3.4(b),
ΜΓ is unitary. Now apply 3.7(c) and 3.19(a).

4a General
105
4.7 PROPOSITION Let N be simple and ,,Γ be of type v. Then
N is v-primitive on Г.
Proof. (о:Г) 3 Ν, so (о:Г) = {0} (for (о:Г) = N gives
the contradiction ΝΓ = {o}).
4.8 PROPOSITION (Betsch (3)). Let the ring N be v-primitive
on Γ. Then N is a primitive ring on the N-module Γ
((Ν. Jacobson) , p. 4) .
Proof. If Γ = Νγ then Γ =N ΝΛ . and (Γ. + ) is
abelian. If η^+(ο:γ), η2+(ο:γ)εΝ/, ν then
ty neN: n(n^+(o:γ)+η2+(ο:γ)) = nnj+(o:γ)+ηη2+(ο:γ) =
= η(η1+(ο:γ))+η(η2+(ο:γ)).
Hence \/ γ,,γ2εΓ \j ηεΝ: η(γ.+γ2) = ηγ,+ηγ2> and
„Γ is a (ring-) module.
Each N-submodule of ..Γ is an ideal, so = {o} or
= Γ. Finally, ΝΓ 4= to} by assumption, so „Γ is
irreducible and N is primitive on Γ.
4.9 COROLLARY (Ramakotaiah (1)). If N is commutative and
v-primitive then N is a field.
Proof. By 4.4(c), 3 Let (N): (L:N) = {0}. L <J Ν, since
N is commutative. By 3.25, (L:N) is the greatest
ideal in L, so L = {0} and N contains a riqht
identity. By 1.107(c), N is a ring, hence a primitive
ring by 4.8 and by (N. Jacobson), p.7 a field.
In (3), Ramakotaiah shows that if I < N = NQ and Ι ε £ (N) then I is
a O-primitive ideal. Near-rings N with a faithful, simple, non-
trivial N-groun are called s-primi ti ve and are studied in Hartney
(4), Meldrum (7),(13). See also Beidleman (7), (8), (9). Holcombe-
Walker (1) study 3-primi ti ve near-rings 14, which means that N
has a faithful N-group of tvpe 3 (see the last lines of p. 80).

106
§4 PRIMITIVE NEAR-RINGS
2.) THE CENTRALIZER
4.10 DEFINITION
(a) EndN(T) = HomN(r,r) =: CN(r) =: С is called the
centralizer of ^Г (cf. Kaarli (2), Ramakotaiah (3)),
(b) AutM(r) =: GM(r)
>}
G; AutN (Г) =: GQ
о
(c) G
o. . (Gu{6]
if όεΟ;
otherwise
1i kewi se G.
4.11 REMARKS
(a) (C,°) is a monoid, ((!,») and (G0»°) a>"e groups,
(G°,o) and (Go>0) ("groups with zero") are monoids
(b) όεΟ <=■> Ω = {ο} .
(c) If ΝΓ is faithful then N e» Mc /Γ\(Γ) ^ Μβ(Γ).
4.12 NOTATION If ηεΝ, fp: Г - г ; FN(r):= ifJneN}
Υ -*■ ηγ
F.
4.13 PROPOSITION (Mlitz (3)).
(a) If J is monogenic then С,.(Г) =
= {πιεΜ(Γ)| VfcF: m°f = f°m} =: Мр(Г).
(b) \/ heCN(T): h/Ω = id.
(c) If Ω = Г then CN(T) = {id}.
Proof, (a) If ηεΟΝ(Γ) and fntFN(T) then
V γεΓ: (h°fn)(y) = η(ηγ) = nh(Y) = (fneh)(Y); so
heMF(r).
Conversely, let f be εΜρ(Γ). If γ,,γ^εΓ = Νγ and
ηεΝ then 3 n^.ngeN: γ, = η,γ Λ γ~ = η~γ. Then
f(nY,) = (f0fn)(Y,) = (fn°f)(Yi) - nf(Yl) and

4a General
1
f(Yl+Y2> = Π^Υ + η,,γ) = (f°fni + n2)(Y) =
= (fn +n °f)(Y) * (nj + n^ffY) = η^(γ)+η2ί(γ)
= ftn^J+fingr) = f(Y1)+f(Y2).
Hence feCN(T).
(b) у ηεΟΝ(Γ) \j ηοεΩ: h(no) = nh(o) = no.
(c) follows from (b).
*. 14 NOTATION θ
30(ΝΓ): = {γεΓ|Νγ = No = Ω}.
)Χ(ΝΓ): = {γεΓ|Νγ = Γ}.
4.15 REMARKS (Betsch (6)).
(a) οεθ0> so θ0 + <Ц.
(b) θ, + 0 <=> ,,Γ is monogenic.
(c) 9Qn 6j - 0 <=> Ω + Г.
(d) ,,Γ is strongly monogenic =-> Γ =
(e) ΜΓ is uni tary => θ
Ί·
Ν
Ω (for γεθ0 =-> Νγ = Ω =>>
—> γ = ΙγεΩ and ω = ηοεΩ => Να; = Νηο^Νο = Ω =*> ωεθ
(f) G(6 ) = θ Λ G(θ, ) = θ,, so G induces permutation
groups on 6Q and (if θ^ + 0) on θ^.
The next proposition is a "Schur-type lemma".
t.16 PROPOSITION (Betsch (6), Mlitz (3)).
(a) ,,Γ is simple Λ Ω = {ο} »> С = {ό}υΜοη,,(Γ) and
(hcC Λ] γεθ^ h(γ)εθ1) => ίίεβ.
(b) ΝΓ is N-simple => С = Ερι"Ν(Γ,Ω)« Ερι'Ν(Γ,Γ) (if N
ΕρίΝ(Γ,Ω) = {6}!).
(c) ΝΓ is NQ-simple =-> С = G°.

108
§4 PRIMITIVE NEAR-RINGS
Proof, (a) follows from the fact that \/ hcC: Ker h ^N Γ,
so either Ker h = {0} (then ηεΜοη.,(Γ)) or
Ker h = Γ (then h = 6). We may assume that Γ -f to},
If heC л 3 γεθ^ h(r)eej then h + 6, so
πεΜοηΝ(Γ). Now h(r) = h(Ny) = Nh(y) = Γ.
(b) V heC: Im h s^ Γ, so either Im h = Ω or
Im h = Γ.
(c) follows from (b).
We are mainly interested in the case that С = G , in which
every non-zero N-endomorphism of Г is an N-automorphism.
4.17 PROPOSITION (Betsch (6)).
(a) G is fixed-point-free (1.4(b)) on Θ,.
(b) If ΝΓ is simple then С = G° <=■> \/ Δ <Μ Γ: Γ i
N
N
=N
Δ.
Proof, (a) Assume that for geG and γεθ, g(y) = γ.
\j δεΓ 3 ηεΝ: δ = ηγ. Then g(6) = g(ny) = ng(r) =
= ηγ = δ. So g = i d.
(b) —>: Assume that h is an N-i somorphi sm Γ ■+ Δ <., Г.
Then hεC = G° = (δ}υ AutJr), a contradiction.
<-: If hεC, h + δ then Ker h + Γ, so Ker h = {0}.
Therefore h is a monomorphism and Γ =■ Im h. So
Im h = Γ, and heAutN(r).
4.18 COROLLARY (Betsch (6)). If
N1
is of type 1 or if
N1
is simple and finite then С = G .
Proof. If Nr is of type 1 then ΝΓ is simple. Assume
N
N1
that h is an N-i somorphi sm Γ ■+ Δ <N Г. Represent Г
Νδ
Νη(γ) = η(Νγ) =
as Γ = Νγ and call η(γ) =: 6.
= h(T) = Δ.
If δεθ^ then Νδ = Γ, so Γ = Δ, a contradiction.
If δεθ0 then Νδ = Ω = ίο}, so Δ = {ο} and
therefore Γ = {ο}, which again is a contradiction.

4a General
109
Now apply 4.15(d) and 4.17(b).
If ,,Γ is simple and finite, apply 4.17(b).
4.19 NOTATION For γ,δεΝΓ we define
γ -v 6: <=■> (ο:γ)Ν = (° = δ)Ν ;
о о
γ % δ: <=> GQ(Y) = Go(6).
4.20 REMARKS (Betsch (5)).
(a) ■>»,■>» are equivalence relations in Γ.
(b) The equivalence classes of л, are exactly the orbits
of GQ on Г.
(c) \/ γ,δεΓ: γ^δ -> γ л, δ (for γ^δ -> 3 9ε^0: g(y) = δ =->
-> (ο··ύ)μ = (ο·.Γ)(γ))Ν = (ο:δ)Ν => γ^δ).
The reason for defining ■>,,л, via N instead of N stems
from 4.13(c): in the frequent case that Ω = Γ, ^ would
otherwise be the all-relation in any case.
4.21 PROPOSITION (Betsch (6)). If ΝΓ is unitary and N = NQ
then л, and ^ coincide on θ ι .
Proof. If γ^δ (γ,δεθ,) then for all η^,η,,εΝ
η. γ = η~γ =·> η.-η~ε(ο:γ) = (ο:δ) —> η,5 = η^δ.
Therefore h: Γ * г is well defined, h turns out
ηγ ■+ ηδ
to be an N-automorphism, so hεG.
Now h(y) = h(ly) = 1δ = δ, hence γ <ν δ.
In Kaarli U) it is shown that if ΝΓ is of type 0, I 9 N and
N1
jГ г* (0) then Θ1 (ΝΓ) = Θ1 (jT:

110
§4 PRIMITIVE NEAR-RINGS
3.) INDEPENDENCE AND DENSITY
An appropriate frame for our next considerations is given by
4.22 DEFINITION (Mlitz (9)). Let Μ be an arbitrary set and
•f(M) the set of all finite subsets of M. A map
r: ^ (Μ) -»■ IN is called a rank map i f
(a) r(0) = 0
(b) \j Fef(H) \j mcM: r(F „{m}) = r(F)+0 with σε{0,1}
(c) V Fcf(M) у m.ncM: [r(F0{m}) = r(F„{n}) = r(F) =>
-> r(Fu {m,n}) = r(F)] .
F is then called r-independent if r(F) = |F|.
4.23 REMARK With respect to r-independence, Steinitz's theorem
is fulfilled (see A. Kertesz, "On independent sets of
elements in algebra", Acta Sci. Math. (Szeqed) 21, 1960,
260 - 269). See also Kaarli (2).
4.24 EXAMPLES
(a) Define r(F): = |F|. Then r is a rank function and
every (finite) subset is r-independent.
(b) Take a vector space Μ over a field K. Set r(F): =
= dim L(F) (linear hull), r is a rank function and
r-independence is just linear independence.
(c) Take an N-group Γ and define for each Φε^(Γ) г(Ф)
as the number of non ^-equivalent generators (i.e.
г(Ф) = Ιφ^Θ,/^Ι). Then r is a rank function and
Φ = {γι,.,.,γ } is r-independent if ΦΞΘ, and
V i+J: Y^Yj ·
This independence is called i-i ndependence.
The same can be done for ^.

4a General
111
In the theory of rings each primitive ring R is isomorphic to
a "dense" subring R of a ring Нотп(Г,Г) for some irreducible
R-module Г and with D = Нот»(Г,Г) (the centralizer) making
_Г into a vector space (see (N. Jacobson), p. 26 - 31). Density
means here (in our notation) that \j selN \/ {γ,,.,.,γ } lin.
indep. in Γ V δρ.-.,δ^Γ 3 retf \j ιε{ 1,. . . ,s} : ?(γ1· ) = 6^ .
(It is clear that only values of independent elements can be
arbitrarily prescribed.)
We are going to prove similar theorems for near-rings.
But before doing so we have to take a look at the density
concept (see also Adler (1) and Ramakotaiah-Rao (1)).
4.25 NOTATION Let Μ be a subset of some Μ(Γ). We introduce
a topology in Μ as in Betsch (7):
If mcM and γεΓ, define S(m,y): = {η)'εΜ|πι'(γ) = m(y)}
and "f: = {S(m,Y)|mcM Λ γεΓ),
4.26 PROPOSITION (Betsch (7),(11)).
(a) ί is the subbase of some topology 7 (the "finite
topology") on M.
(b) N= Μ is dense in Μ w.r.t. У <=>
<"> \/ sclN V ηιεΜ \/ yj γ εΓ 3 ηεΝ \/ i ε{ 1,. . . ,s}:
: n(Yi) = m(Yi).
Proof, straightforward and hence omitted.
In all that follows, "density" means "density with respect to
7 of 4.26".
4.27 REMARKS
(a) If Μ and N are subnear-rings of Μ(Γ) then it is
easy to see that ΝΛ is dense in M„ iff Ν +Μ„(Γ)
J о о о сv '
is dense in MQ+Mc(r). Note that Ν0+Μς(Γ) and
Μ +Μ (Γ) are no near-rings in general (see 4.53(e)),
except in some important special cases. (See 4.54
and 4.60.)

112
§4 PRIMITIVE NEAR-RINGS
(b) If N is dense in Μ then N : = NnH (Γ) is dense
in M0.
(c) Observe that if Η 4= ί i d > is a fixed-point-free
automorphism group of Γ then М„(Г) н Μ (Γ) (since
V теМн(Г) V heH: h(m(o)) = m(o)).
If Η = {id} then ΜΗ(Γ) = М(Г).
(d) We will be mainly interested in near-rings which are
dense in Mr0(r) and Μ 0(г):= МГ„(Г)+М (г)
uo uo о
(4.52 and others).
4.28 THEOREM (Ramakotaiah (2), Betsch (7)). Let Η be a fixed-
point-free group of automorphisms of some group Γ.
(a) V γεΓ* γ δεΓ 3 πιεΜΗ(Γ): (m(y) = 6 л
Л V γ·εΓ\ΗΎ: т(у') = о).
(b) V S£1N \j Yl γ5εΓ* , Ηγ. + HYj for i + j
V 6j 55εΓ 3 тгМн(Г) \/ 1e{l,...,s): m(y. ) = δ,.
(c) If Η + {id}, ΝϊΜΗ(Γ) is dense in ΜΗ(Γ) <=>
<=> V sE]N V γ j γ5εΓ* , Ηχ. 4= HYj for 1 + j
tf 6j &scT 3 ηεΝ \/ ιείΐ s}: n^) = 6j.
(d) If Μ (Γ) s N s ίΤΗ(Γ), Ν is dense in ^Η(Γ) <=>
<"> Μ ScTH \] yl Υ$εΓ, Ηγ. 4= Ηγ, for 1 + j
\j 61,...,65εΓ 3 ηεΝ \/ 1eU,...,s}: η(γ1 ) = «г
Proof. In any case we may assume that Η 4= {id}, for
otherwise MH(r) = Мн(г) = М(Г).
(a) \/ αεΗγ 3] h εΗ: α = h (γ) (since H is fixed-
point-free), г „ (δ) αεΗγ
Define ηιεΜ(Γ) by m(a):= < а . Then
Ι, ο aiHy
clearly πι(γ) = S and тгМн(Г); т is uniquely
determined by the conditions m(y) = 6 л
Λ ( \/γ'εΓ\Ηγ: т(у') = о).

4a General
113
(b) Define maps π).εΜ„(Γ) with π^(γ.) = $i and
\/ Y'iHy.: πι.(γ') = о (as in (a)).
Then m: = m,+...+m will do the job.
(c) ->: By (b) and 4.26(b).
<=: If Η γ, -f HYi for i + J> the result is clear.
If Ηγ. = Ηγ , V πιεΜΗ(Γ) \j ηεΝ: n(Yi) = m^) ->
**> f(Yi) = m(Y,) and the result follows again from
4.26(b).
(d) =>: By 4.27, NQ is dense in (HH(r))0 = ΜΗ(Γ).
If one γ^ (say Yj) = o, take η еМс(Г) to be the
map which is constant = 6^. Take η0εΝ0 with
no^Yi' = δι'~δ1 for 1ε*2 s^' Then n: = no+nc
fulfills V ieil s}: n(Yi) = 6-.
Two or more γ, cannot be zero. If all γ. 4" °· the
result follows from (c).
<=·: If SeIN , Yj γ$εΓ* , Ηγ. + Ηγ for i + j
and 6^ δ εΓ, define Ys + i: = ° and 5S + 1: = °"
Then 3 ηεΝ \j ΐε{1,... ,s + l}: η(γ^ = 5^. Because of
n(o) - ο, ηεΝ and by (c), NQ is dense in Μ„(Γ),
so by 4.27a) N is dense in МН(Г).
4.29 THEOREM (Betsch (7)). Let Γε^, Η < Aut(r) and Τ as in
4.26. Then the following conditions are equivalent:
(a) T is discrete in Мм(Г).
(b) У is discrete in МН(Г).
(c) Η has only finitely many orbits on Γ.
Proof. Again the results trivially hold for Η = {id},
(in this case, Γ is finite). So we assume that
Η + {id}. Then ΜΗ(Γ)ε7?0-
(a) -> (b): Trivial, since МН(Г) н Ν (г).

114 §4 PRIMITIVE NEAR-RINGS
(b) =-> (c): Assume that Η has infinitely many orbits
on Γ. Take πΐεΜΗ(Γ) and a neighbourhood U of m.
s
Then 3 seIN 3 γ. γ.εΓ: U в f] S(m,Yi).
1 s i=l Ί
If Ηγ. = Ηγ. then Sfm.y^ = S(m,Y.). So we will
assume that Ηγ. =f Ηγ. for i 4= J ·
Since Η has infinitely many orbits,
3 Υ$+1εΓ\({ο}υ Ηγχ u ... u Ηγ$).
Then V ϊε{1 s + 1} 3 е^М^Г) \/ je{l,. . . ,s + l}:
e^Y,) =
f Yi Ί' = j
Ι ο 1 + J
Define m,: = m (e,+...+e ) and m~: = mi+es+i·
s
Then m, and m9 are ε Π S(m,Y·) ξ U.
1 L i = l 1
If m{Ys+l' + ° then ml(Ys+l3 = ° + m{Ys+l)* so
m. 4" m■
If m(ys+l) = ° then m2{Ys+l) = Ys+1 + ° = m(Ys+l)'
so m2 + m.
Anyhow, U contains an element =f m 3η<* У cannot be
di screte.
(c) =-> (a): If Η has only finitely many orbits on Γ
then each element of Mu(r) and of ТС„(Г) is
uniquely determined by its effect on finitely many
suitable elements of Γ.
So j i s di screte.

4b O-primitive near-rings
115
b) O-PRIMITIVE NEAR-RINGS
Now we shall prove a "density-like" structure theorem for
O-primitive near-rings. We start with zero-symmetric ones.
We may assume (1.48) that if N is O-primitive on Γ then NSM(T).
Generalizations can be found in Mlitz (4),(8),(12) and Kaarli (6)
4.30 THEOREM (Betsch (6)). Let Νε7?0 be O-primitive on Γ.
If N is a ring then N is a primitive rina on the N-module
Γ and Jacobson's density theorem is applicable.
If N is a non-ring then we get a kind of a density
property:
(D):\/ εεΙΝ \/ Yj Υ,,εΓ, -\--indep.\/ ij 6$εΓ
3 ηεΝ \/ ιε{1,...,ε}: nyi - 6^.
Proof. If N is a ring we only have to apply 4.8.
Now let N be a non-ring. In the terminology of (D),
let s be > 1 and for t£{ 1,.. . ,s-l} let S(t)
be the statement
t
V ke(t + l s}: [\ (o:Yi) $ (о:ук) .
Lemma. \/ tei 1,. . . ,s-l} : S(t).
Proof. By induction on t.
Since for γεθ, (ο:γ) is a maximal left
ideal of N, \/ i,je{l,...,s}: (ο:γ^Ε
s(o:yj) => (ο.-γ^ = (o:yj) -> γ^γ^ => i = j
Particularly: S(l):ty ke{2 s}: (o:yj)^
i(o:Yk).
Now assume S(t), s > 3 and ke{t + 2 ,. . . ,s }
t
Then П (о:у1)ф(о:ук) and (о:yt+1)i(o:yk)
Since (ο:γ.) is maximal,
t
Π (ο:γ1) + (ο:γ|<) = (о :rt + 1) + (o = Yk) = Ν.

116
§4 PRIMITIVE NEAR-RINGS
Since N is not a ring, f) (ο:γ.) η (ο :γ )φ
1 = 1 τ
ί(ο:Υΐ/) ЬУ 3.4(i), which is nothing else
than S(t+1).
Now return to the proof of 4.28 and let γ, γ ,
δ.,.,.,δ be as in (D). Again we use induction on
te{l s}.
If t = 1 then Υιεθ, =■> 3 η,εΝ: η,γ, = δ^.
Now assume that V tc{l s-1} 3 nttN
V ΐείΐ t}: ntYi = 6..
t
By the lemma, L: = f] (° : Y,· )i (° :Y + ^.i ) · hence
i=l 1 ttx
Lyt+1 + ίο). Since LYfc+1 ^ Γ (3.4(a)), LYfc+1 = Γ.
The
refore 3 £εί: Αγ
= 6
t+rVt+i ·
Now we take n* + l: = * + nt and get ^ iε{1, . .. ,t+l}:
: η. ,γ. = δ. , and the proof is complete.
4.31 REMARKS
(a) (D) is no "real" density property since there is no
near-ring in sight in which N is dense (w.r.t the
finite topology). +)
(b) From (D) it follows (Ramakotaiah (2)) that, if seIN
and γ, γβεΓ are -v-i ndependent, Ν/ η ,n.v ^ =Ν
1 Υ5εΓ are -v-i ndependent, N/ pj (ο:γ.)
(Yj.···»Ys}
ίγ,,...,Ύ5}
-,.Г (where for fer
, (nf)(Yi)
= n(f(Yi))).
(c) The content of (D) might be very thin: if e.g.
|θ,| = 1, (D) is trivial. So it is not too surprising
that the converse of 4.28 does not hold :
) (Betsch) : If one changes ¥ of U.25 to У ' : ={S(m ,γ) | πιεΜ,
л γεθ (Γ)} then one gets a "real" density theorem w.r.t.
the resulting coarser topology. See also 9.230.

4b O-primitive near-rings
117
Let N be the non-ring {feMQ(Z4)|f(2)ε{0,2}Af(3) = 3f(l)}.
In N Z4. 6j = {1,3}, КЗ, (О) is fulfilled, but
{0,2} <L TLa , so N I, is not simple and therefore N
is not O-primitive on 2..
(d) (D) is equivalent to the following property:
(D'): у selN \j γχ,. . . ,γ,,εΓ , %-indep. V meM(r)
3 ηεΝ \j ie{l,...,s}: ηγ1· = m^).
Now we turn to arbitrary near-rinns.
4.32 THEOREM
(a) Let N be O-primitive on Γ.
Case 1: HQT -f {o}. Then N0 is O-primitive on Γ,
so 4.30 is applicable (for N Γ), and
NcsMc(r).
Case 2: HQT = {o}. Then N = МС(Г) and Г is a
non-zero simple group.
(b) Conversely, if either N is O-primitive on Г and
Ν έ Μ (Γ) or if Ν = Μ (Γ) where Γ «f {o} is
simple then N is O-primitive on Γ.
Proof, (a) Anyhow, Nc Ε Mc(r)·
If Ν0Γ + {o} then N г is of type 0 by 3.18(a)
о
and N is O-primitive on Г (3.15(a)).
If NQr = {o}, NQ = {0} "by faith", so Ω = Г and
N = Nc = Μ (Γ) by 1.50(b). Since Ncr is simple
iff Г is simple, (a) is shown (observe that Ν Γ ■(■ ίο}!).
(b) Again by 3.18 (this time by (b)), if u Γ
"о
O-primitive then ..Γ is of type 0. Since Ν εΜ (Γ),
N and N (and hence N) act faithfully on Г, so
N is O-primitive on Г.
If Ν = Μ (Γ), Γ =f {о} and simple, the result is
clear.

118
§4 PRIMITIVE NEAR-RINGS
4.33 REMARK (D) would not necessarily mean the same in ,,Γ
and in ,, Г, if % would be defined by y\.6:<=> (ο:γ) =
= (ο:δ) (°n Ν). Cf. 4.19.
4.34 THEOREM (Ramakotaiah (1)). Each O-primitive ideal is a
prime ideal -f N.
Proof. Let I be a O-primitive ideal of N. Let ,,Γ be
of type 0 with generator γ such that I = (ο:Γ)
(4.3).
Assume that 3 Ji.J2 ^ N: ^р^1 л Jl^ л J2^*
For ιε{1,2}, ^Γ = ^Νγ η J.yo ? J.Г. Since
J. i (o.T), J.г = J1-y0 + {o}. By 3.4(a),
JiYo ^N Γ· So JiYo = Γ· Now JlJ2r = Jlr = Γ'
so J.J2 i (ο:Γ) = I, a contradiction.
4.35 REMARK In 5.40 we will see that the converse of 4.34
holds if N = NQ has the DCCN.
4.36 THEOREM (Ramakotaiah (1)). Every maximal modular ideal
I of ΝεΤΙ is a O-primitive one.
Proof. Let I be a modular maximal ideal. By 3.22, I is
contained in a modular maximal left ideal L. Since
(L:N) is the largest ideal of N contained in L
(by 3.25), we get I^(L:N) and (L:N) is modular
by 3.21(a). By the maximality of I, I = (L:N).
By 4.3(c), I is O-primitive, since by 3.29 L is
O-modular.
By the way, if N is O-primitive on Γ and Νγ =: Δ then A is
not necessarily simple (K. Kaarli).
For the rest of this section, we give a description of a class
of O-primitive near-rings which are not 1-primitive. This
discussion is due to Holcombe (5), where the proofs can be
found, too.

4b O-primitive near-rings
119
4.37 NOTATION If ΝΓε|$> let Δ: = ΓΝΘι be the set of "non-
generators". If Δ <Ν Γ, let GA: = AutN ..Q .(A)
(cf. 3.14(a)!) .
4.38 DEFINITION If (Γ, + )ε^, Β5Γ, H<Aut(r), H(B)SB, we call
the triple (Γ,Β,Η) compati Ы e if at least one of the
following conditions is satisfied:
(a) В is no normal subgroup of Г.
(b) 3 γεΙλΒ 3 βεΒ \j ηεΗ : γ + β ή- η(γ).
(c) (3 h'£H 3 γεΛΒ 3 ΒεΒ : γ + β = h'(Y)) Λ
Λ (3 γ'εϊλΒ : h' (γ')-γ'$Β).
4.39 THEOREM Let ΝεΤ^ be O-primitive on Γ, Ν a non-rinn with
identity and DCCL, and let Λ (as in 4.37) be an N-subgroup
of Γ such that ,,Δ is not faithful, but of type 2.
Then N is not 1-primitive on Δ, G (4.10(b)) has finitely
many orbits on 9i, (r,A,G) is compatible and
if Ν//0.Δχ is a non-ring then N={feMf ,ό}(Γ)|f/^M Δ(Δ)},
if Μ/(0.Δ) is a ring then fAifcM^ (Γ) | f /eEnd ItΛ}
(where Δ is a finite dimensional vector space over
the division ring G υ(ό}).
Conversely:
4.40 THEOREM Let Γ be an additive group and Δ be a non-zero
subgroup. Let G be a regular group of automorphisms
of Δ which has only finitely many orbits on Δ. Let Η be
a subgroup of Aut (Γ. + ) such that
(a) (Γ,Δ,Η) is compatible.
(b) each ЬгН is regular on Γ\Δ.
(c) Н has only finitely many orbits on Γ\Δ.
(d) V hεH: h/^GA.
Then Ν = {ίεΜ„ ,_>(Γ)|ί/ΔεΜ .(Δ)} is zerosymmetriс,
G
O-primitive, but not 1-primitive on Γ, has an identity
and the DCCL.

120
§4 PRIMITIVE NEAR-RINGS
If moreover Γ f Δ and Δ is a finite dimensional vector
space over some division ring D and if \/ ηεΗ: h/.εθ
then N = {fcMH гб}(Г)|f/ueEndQ(r)} is also O-primitive,
but not 1-primitive on Γ, Νε770> Νε 7ί, ,and moreover
Ν/
(ο:Δ)
is a π ng,
4.41 REMARK See also Holcombe (4) for the more general case
that Δ is only a finite union of N-subgroups of type 2
with zero intersection.
4.42 REMARK If G = {id} then in the non-ring case of 4.30
we get near-rings of the form N ={fεΜ (Γ)|f(Δ)^Δ}
(see e.g. NQ in 3.8). Cf. Ramakotaiah-Rao (1),(3),(4).
Conversely, if (Γ. + ) is a finite group and Δ a non-
trivial subgroup then N:= {feM (Γ)|f(Δ)ξΔ} is a finite
near-ring with identity, zero-symmetric and 0-, but not
1-primitive on Γ. Δ is just the set of non-generators
and is an N-subgroup such that N/
if
> 2.
(ο:Δ)
is a non-r'inq
c) 1-PRIMITIVE NEAR-RINGS
Now let N be 1-primitive on Γ. Then С = G° (by 4.18),
Г is not N-isomorphic to a proper subgroup (4.17(b)),
Γ = θ0« 9j (by 4.15(d)), Ω = {ο} or Ω = Γ (3.2) and
V L 4t N, L + {0} 3 γεΓ: LY = Γ (by 3.4(a)).
We still assume that N ? M(r).
4.43 THEOREM
(a) Let N be 1-primitive on Γ. Then
Case 1: NQr + ί°} Λ Ω = Γ. Then NQ is 1-primitive
on Γ, Nc = Μς(Γ) and 62 - Г.
If N„ is a ring then N is dense in Μ ГГ(Г)
о э affv '
where Г is a vector space over the division
ring 0: = Hom^ (Г.Г).

4c 1-primitive near-rings
121
If N Is not a ring then (D) of 4.30 is
о
appli cable.
Case 2: HQT 4- {о} л Ω = {о}. Then Ν = NQ is 1-pri-
mitive on Γ and 4.30 holds.
Case 3: Ν Γ = {о}. Then N = N„ = Μ,(Γ) and Γ is
о с сN '
a simple group -f {о}.
(b) Conversely, if a near-ring №H(r) is such that NQ
is 1-primitive on Г with Ν ε{{0}, Μ (Γ)} or if
N = МС(Г) (Г + ίο} and simple) then N is 1-primitive
on Γ.
Proof, (a) If NQr + ίο}, NQ is 1-primitive on Γ by
3.18(a). Since each strongly monogenic N-group has
either Ω = ίο} or Ω = г, the rest follows from
1.50, 3.9, 3.15(a), 4.27(a) and 4.32.
(b) If N is 1-primitive on г and N = {0} or
N,. - М.(Г) then either Ω = {ο} or Ω = Γ (1.50),
с с
so N is 1-primitive on Γ by 3.18(b) and 3.15(a).
If Ν — Μ (г), Г simple and *f {o}, then clearly
N is 1-primitive on Γ.
4.44 REMARK 4.43 is the main reason for defining "strongly
monogenic N-groups Γ" as in 3.1(b) and not by the conditions
"monogenic" and "\J γεΓ: (Νγ = Ω ν Νγ = Γ)", for 4.43 would
not be true in this case:
Take Γ = Z8, NQ: = {feMQ(r)|f(2) = f(6)ε{0 ,2 ,4 ,6 } л
Л f(4)e(0,4}} and Νς: = {ίεΜς(Γ)|f(0)ε{0,2 ,4 ,6}}.
Then one can show that Ν: = Ν +Ν is a subnear-ring of
oc Ό
Μ(Γ) enjoying the following properties:
ΝΓ and t, Г are faithful, simple and monogenic. Moreover,
о
\/ γεΓ: (Νγ = Ω = {0,2,4,6} ν Νγ = Γ). But {ο} + Ω + Γ,
and Ν is not 1-primitive on Γ (it is not even true
о x
that for all γεΓ Ν0γ is either = ίο}, = Ω or = Γ,
since NQ4 = {0,4}).

122
§4 PRIMITIVE NEAR-RINGS
From 4.30 and Γ = θ u 9j we get with a straiqhtforward proof
4.45 THEOREM Let the non-rinq ΝεΤ) be 1-primitive on Γ but
without ^-equivalent generators.
Then N is dense in the near-rinq {feM_(Γ)|f(θ ) = {о}}.
For 1-primitive near-rinqs cfL with DCC we get a whole bunch
of important results (cf. Ramakotaiah (3), Betsch (10)):
4.46 THEOREM (Betsch (3)). Let Nz7lQ be 1-primitive on Г and
endowed with the DCCL. Then
(a) There are only finitely many ^-equivalence classes if
Μ is a non-rinq.
s <
(b) 3 selN : NN = 7'L., L· finitely many pairwise (to Г)
14 i = l 1 1
N-isomorphic left ideals and N-groups of type 1 (so
2.50 is applicable!); if N is a non-rinq then
s = |rA|-i.
(c) All N-groups of type 0 are N-isomorphic and of type 1.
(d) N contains a right identity (not necessarily two-sided).
(e) N is simple.
(f) N is either 2-primitive on Г or there is no N-group
of type 2.
Proof. If N is a ring, (b) - (f) are either well-known
or trivial. So we will assume that N is a non-rinq.
(a) Suppose that there are infinitely many -v-equi val ence
classes with representatives Υ0Ύι»Υ2'··· · We таУ
assume that Ύοεθ0· Then (°:Y0) = N + (°:Yi) for
i > 1, hence γ1,γ2 ,...εθ1. So by (D) of 4.30
(ο:γ )=»(ο:γ^)=»(ο: {γ^,γ2})=». . . which is a
contradiction to the DCCL.
(b) Now let γ ,Ύι,...,γ, be a complete system of
representatives of the ^-equivalence classes with
s
Ύοεθο' Yl" ··'Yse6l· Then .Π (°:γί) = {0} » but

4c 1-primitive near-rings
123
L,··· = Π (ο:γ,) + {0}. By 3.4(f), ί,,.,.,ί are
J 1+J
minimal left ideals.
Now apply 2.50(c) to aet N = [*L.. Since
j = l J
V jcil s): Lj^o-.Yj), Lj -„ Γ by 3.10.
(c) Holds by the proof of (b) and 3.11(a).
(d) By (b) and 3.27(d), N contains a right identity
e. N. of 3.8 shows that e is not necessarily two-
sided.
(e) If I < N, 3 J'eil s): Lj«H. Since Lj is
minimal, L-r.1 = (0). But IL.iU L. = {0}, so
I?(0:Lj) = {0} (for L. ~N Γ), whence I = (0).
(f) By 4.7 or by (c).
Note that 4.46(a) is not valid for rings: If Γ is the vector
2
space IR , considered as an Hom(r,Γ)-modu1e, all (1.x)
(χεΙΚ) are pairwise inequivalent w.r.t. ^, Нот(Г.Г) is a
near-ring which is primitive on Г and has the DCCL.
4.47 COROLLARY ΝεΤ) , DCCN, N contains a left identity; I 3 N,
N + {0}. Then
(a) N Is 1-primitive <=■> N is 2-primitive <=> N is simple.
(b) I is 1-primitive <=> I is 2-primitive <-> I is maximal
Proof, (a) By 3.4(c) and 3.7(c), 1-primitivity and
2-primitivity coincide. In this case, N is simple.
If N is simple then I = N is a minimal ideal and
by 3.4(b) and Lemma 3 in the proof of 3.54 (with
I: = N; then Δ = Γ) Ν has a faithful N-group ^Γ
of type 1, so N is 1-primitive.
(b) follows from (a).
Kaarli (2) showed that if N = NQ is simple and U is a maximal N-
subgroup of N with NU f {0} then N is 1-primitive. Cf. also
Kaarli (4) and Adler (1).

124
§4 PRIMITIVE NEAR-RINGS
d) 2-PRIMITIVE NEAR-RINGS
Again we assume that if N is 2-primitive on Γ then N?M(r),
1.) 2-PRIMITIVE NEAR-RINGS
The structure of 2-primitive near-rings can be described as
follows.
4.48 THEOREM
(a) Let N be 2-primitive on Γ. Then
Case 1: Ν Γ + {ο} Λ Ω = Г. Then NQ is 2-primitive
on Г, Nc = Мс(Г) and 9j = Г.
If N is a ring then N is dense in Μ ,^(r)
(as in 4.43);
if N is a non-ring then (D) of 4.30 is
applicable (for N ).
Case 2: Ν Γ + {о} л Ω = {ο}. Then N = N„ is 2-
o ' о
primitive on Γ and 4.30 is applicable.
Case 3: Ν Γ = {о}. Then N = Μ ίΓ) and Γ is a
о сv '
cyclic group of prime order.
(b) Conversely, if NQ is 2-primitive on Г with
N еН0},Мс(г)} or if N = МС(Г) (Г a cyclic group
of prime order) then N is 2-primitive on Г.
The proof is similar to the one of 4.43 and therefore omitted.
4.49 THEOREM (cf. Fain (1) and Betsch (7)). If N is 2-primitive
on Г and if I 3 N, I + {0}, then I is 2-primitive on Г,
unless I = Μ (Γ) (where Γ is not a cyclic group of prime
order).

4d 2-primitive near-rings
125
Proof, (a) We first show this theorem for NeTL.
Ev dently , .Γ is fai thful.
Assume that Δ ζ, Γ.
If ΙΔ = {0} then consider ΝΔ. If ΝΔ + {0},
3 όεΔ: No + {о}. Therefore Νδ = Γ and ΝΔ = Γ,
But ΙΓ = ΙΝΔ^ΙΔ = {ο}, so Ι = {0}, since jT
is faithful. Hence ΝΔ = ίο}, Δ <., Γ, whence
Δ = ίο}.
If ΙΔ + {ο} then again 3 δεΔ: Ιδ + {ο}. Since
Ν(Ιδ) = (ΝΙ)δΕΐδ, Ιδ <Ν Γ, so Ιδ = Γ.
Consequently Δ = Γ, for ΙδΞΔ.
Therefore I 1s 2-primitive on Г.
(b) Now let N be arbitrary. We may assume that
N + Ν , so case 2 of 4.48 1s excluded.
' 0
If N falls into case 1, N is 2-primitive on Г.
By 2.18 , I0 = InW0 < N0. If
I0 + {0} then I is 2-primitive on Г, hence
I is 2-primitive on Г by 3.18(b).
If IQ = {0} then IiNc = Мс(Г). Since θχ = Γ,
Ιο ^Ν Γ. Ιο = {ο} implies that for all γ = ηοεΓ
and for all ιεΐ ίγ = ino = 0, so I={o}.
Hence Io ■(■ io) and so Ιο = Γ.
Take any m εΜ (Γ); mo =:μ. Because of Ιο = Γ,
3 i ε I : io = y.
Now \/ γεΓ: ίγ = io = μ = mo
my, hence i = m
and we get I = Mc(r). If 3 ΡεΡ
V
I is
2-primitive on Γ; if not, I is not 2-primitive.
If N is in case 3, I is trivially 2-primitive on Γ.
4.50 REMARK 4.49 cannot be transferred to 0- or 1-primitivity,
not even for finite, abelian, zerosymmetric near-rings.
It is easy to show that if N is e.g. O-primitive on Γ and
I ^ N then .Γ is faithful and monogenic. But not
necessarily simple:
Take Γ: = 20, Δ: = {0,2,4,6} and I
■Q·
= (0,4}.
N:
« {fcM {Γ)|ί(Δ)5Δ Λ f(5)=f(l) л f(7)=f(3)}, Ι:- (0:Δ)

126
§4 PRIMITIVE NEAR-RINGS
N is O-primitive on Γ, but Ε ά, Г. Moreover, .Г is.
strictly monogenic and I has a riqht identity. I cannot
even be 0-prim1tive on some other group Γ' =:Ιγ':
Assuming that, take (0:1). and (0:3K in .Γ and put
L: = (° :YO^ I · Then L is a maxitna1 1eft idea1 of l (3·*(Π)
and L + (0:l)j, L 4= (0:3)j (since (0:l)j and (0:3)r
cannot be maximal). Therefore (0:1)r+L = (0:3)r+L = I.
but (0:l)r\ (0:3) = {6}iL, so I would have to be a ring
by 3.4(i), a contradiction.
As Y.S. So pointed out, NN = I + (0:1)n(0:3).
Seemingly there is no "smaller" counterexample than that
above with 4096 elements. See also 5.19(a).
By the way, one can use Zorn's lemma to show that in any
case HN (I f {0}, N v-primitive) has some I-groups
of type v.
4.51 COROLLARY (Fain (1)). Let Ρ be a 2-primitive ideal of
ΝεΤ) . Let I be another ideal
Pisa 2-primitive ideal of I
ΝεΤ) . Let I be another ideal of N containing P. Then
Proof. I/p <l N/p, and N/p is 2-primitive. Since
I + P, I/p + i°} and */p is 2-primitive. Hence
Ρ is 2-primitive in I.
2.) 2-PRIMITIVE NEAR-RINHS WITH IDENTITY
In this case,
Γ*
Also, If N = NQ then
(if ΝεΤ10) or e: = Γ (if ΝφΤ^).
л, = % (by 4.21).
Recall that a 1-primitive near-ring cfl with a left identity
is already a 2-primitive one with identity (4.6(b)).
We are now in a position to get a "real" and fundamental
density theorem.

4d 2-primitive near-rings
127
4.52 THEOREM (Wielandt (1), Laxton (2), Ramakotaiah (2),
Betsch (6) and (7), Kaarli (2),(5), Mlitz (4), Scott (15)).
(a) Let N be 2-primitive on Γ with identity.
Then N is dense in (C : = EndM (Γ) = G^
v О "_
D: = HomN(r,r)):
by 4.18,
"N1
N a ring
N a non-ring
N* N0
(case 1
of 4.48)
Μ
aff
(Γ)
Μ (Γ)
uo
N = N0
(case 2 of
4.48)
HomD(r,r)
Mc (Γ) = МС(Г)
D
Γ a vector
space -f {o}
G f i xed-poi nt-
free on Γ
(b) Conversely, every near-ring which is dense in Maff(r)
or HomD(r,r) (where Γ is a non-zero vector space
over some division ring D) or dense in TTP (Γ) or
Lo
M~ (Γ) (G fixed-point-free on Γ) is 2-primitive
о
on Γ , where С = G w{δ}.
Proof, (a) If N is 2-primitive on Γ and has an identity
then case 3 in 4.48 cannot occur. N is therefore
2-primitive on Γ and has an identity. If N is a
ring, the statement is clear. If N is not a rinq,
note that G is fixed-point-free on Γ, since
V g£G : {γεΓ|9(γ) = γ} < Г.
"о
If Go+{id} then HG (Г) = MGou{6}(r) = Μ (Γ) =
о
= Мг (Г) and the result follows from (D) of 4.30,
Lo
4.28(c) and 4.27(a).
If GQ = {id} then (D) of 4.30 implies that N is dense
in M0(r)· which is trivially dense in (since
equal to) Μς (Γ) = Μ{&}(Γ) = MQ(r).
If Ν + Ν , apply again 4.27(a).

128
§4 PRIMITIVE NEAR-RINGS
(b) Assume now that N is dense in HomD(r,r), where
Г is a non-zero vector space over some skew-field 0.
Then N is a dense subring and therefore a primitive
ring on Г. From this we deduce:
If N is dense in Μ ,ЛГ) then N is dense in
НотЛГ.Г), so Г has no non-trivial N0-subgroups,
and N is 2-primitive on Г. If N is dense in Μ (Γ)
Go
then N = N . If G„ = {id}, Μ „(Γ) = Μ (Γ) and
О О -О О
Go
each dense subnear-ring of that is trivially 2-primitive
on Г. If G + {id} then Μ 0(Γ) = MQ (Г) and
G о
о
4.28(c) shows that ,,Γ cannot contain non-trivial
N-subgroups.
Finally if N is dense in Μ- (Γ) then N is dense
о
in (Mc (Γ)) = Μ- (Γ). As above, Γ cannot contain
о о
a non-trivial N -subgroup (or one can use 3.18(b)).
4.53 REMARKS
(a) It is not true that each 2-primitive near-ring with
identity, N -f N and N a non-ring, is dense in
МС(Г): take Γ finite with {i d}=)=G^Aut (Γ) , G fixed-point free
and N: = ttg(r). Then ,,Γ has Ω = Γ, so
CN(T) = {id} (4.13(c)) and therefore Mc(r) = Μ(Γ).
But N + М(Г), so N cannot be dense in М(Г) by
4.29. This is a late but convincing reason for
introducing this crazy Rr (Г), where one first
Lo
switches down to N (by forming С = End^ (Г) and
о
then back up by adding all of the constants : Mr(r)
would be too big in general.
(b) 4.52(a) does neither hold for 0-primitive near-rings
with identity nor for 2-primitive near-rings without
identity (not even for Η = N and N finite):

4d 2-primitive near-rings
129
If Γ: = 24 and Δ: = {0,2}, N: = {feMQ(Γ)jf{Δ)*Δ}
is O-primitive on Γ with identity, but not dense in
MC (Γ) = Μο(Γ) (4·29!). M: = {feMQ(r)|f(3) = 0} is
о
2-primitive on r, without identity and again not dense
in Mc (Γ) = Μ0(Γ).
(c) All 2-primitive near-rings with identity on Z^»
where N is a non-ring, will be classified in 4.63.
(d) 4.32, 4.43 and 4.48 reduce the theory of primitive
near-rings to those of primitive zero-symmetric near-
rings. We will therefore mainly deal with those ones
iη the sequel .
(e) Recall (4.27(a)) that Ψ,Γ (Γ) is "only a set" in
о
general.
Here is some example: G = {id,-id} (with -id(x): =
= -x) is a fixed-point-free automorphism group on
Γ = IR . C: = {6,id,-id}. Μς( IR ) = {feM( IR ) | f (0) = О Л
Л V xeIR : f(-χ) = -f(x)b
If HC(IR) = :N, take n^ = sin+ίεΝ and n2: =
= id+Tj-εΝ. Consider n: = η, οη~εΜ( IR ).
η = si no( id+^J-si r\{j) is not an odd function, thus
not belonging to M_(IR), whence n^N and N is no
near-ring.
4.54 COROLLARY If N is 2-primitive on Γ with Aut^ (Γ) = {id}
о
then N is dense in either one of the following near-rings
(notation as in 4.52): НотЛГ.Г), Maff(r), MQ(r) or
М(Г) (cf. 4.65).
4.55 THEOREM (Ramakotaiah (2)). Let Νε7)0 be a 2-primitive
non-ring on Γ with an identity. Then any two equivalence
classes w.r.t. *v (except the zero class) are equipotent.

130
§4 PRIMITIVE NEAR-RINGS
Proof. Let Ε be in Γ*/% and ε a fixed element of E.
Consider the map f: G ■+ Ε (with G=Aut,.(r) aqain),
q - 9(ε)
Since G is fixed-point-free (4.52), f is injective.
By definition, f is surjective, so f is a bisection.
3.) 2-PRIMITIVE ZERO-SYMMETRIC NEAR-RINGS WITH IDENTITY AND A
MINIMAL LEFT IDEAL.
N be a nr.
4.56 THEOREM (Betsch (7), cf. Deskins (2)). Let N
with identity which is 2-primitive on г and has a minimal
left ideal L. Then
(a) L =N Г.
(b) 3 e2 = eeL
#.
L = Ne = Le
morphic to
and
!eNe,
:cN(r:
is anti i so-
Proof. (a) Since Lr 4= {o}, 3 γεΓ: Ly 4= {0},
and γεθ,. Now we can apply 3.10.
so
LY = Γ
(b) With γ as above, 3 eeL*
ey = γ.
Therefore
2 2
e γ = ey and e -ecLn(o:y) = {0} (since L is
minimal). Hence e = e and Le =f {0}. Since
l-ie ε (ο:γ) ο L =
Ne = Le = L. By
{0} and Le <N L, Le
'N
N
:r) = cn(d
L. By Le s Ne £ L ,
it can
CN(Ne)
be easily verified that N-isomorphic N-groups have
isomorphic centralizer-semigroups).
For ηεΝ, consider t
Ne ■+ Ne
xe ■+ xene
is wel1 -
defined and eCN(Ne).
Consider next the map h: eNe -* CN(Ne). If ene = eme
ene ■* t
η
then tn = tm, so h is well-defined. Clearly,
h is an antihomomorDhism. If h(ene)
tm and V χεΝ: xene = tn(xe) = tm(xe)
h(eme) then
xeme .
Specializing χ = :e we get ene = eme and h is
shown to be injective.
Finally, \/ ceCN(Ne) 3 ηεΝ: c(e) = ne. Therefore
ene = e c(e) = c(e ) = c(e) = ne and for all χεΝ we
get c(xe) = xc(e) = xene = tn(xe), so с = t .

4d 2-primitive near-rings
131
and h is surjective, hence an antiisomorphism.
4.57 COROLLARY (Betsch (7)). If ΝεΤ^ has an identity and
a minimal left ideal L then all faithful N-groups of
type 2 (if those exist) are N-isomorphic (to L) and N
determines the pair (Г, С^(Г)) uniquely "up to isomorphism"
If e is as in 4.56(b), (eNe,·) is a group with zero and
the group (eNe\{0},·) acts on L = Ne as a fixed-point-
free automorphism group (by right multiplication). Hence e
"brings back" some information on Г out of NsM(r). Cf. 9.227,
4.58 REMARK For more information on these topics (a partial
converse of 4.56, the uniqueness of (Г, С^(Г)), etc.)
see Betsch (6) and §7a), in particular 7.5.
4.) 2-PRIMITIVE NEAR-RINGS WITH IDENTITY AND MINIMUM CONDITION
4.59 COROLLARY Let ΝεΤΙ be a 2-primitive near-ring with
DCCL and identity. Then 4.46 is applicable, hence also
2.50 (for ^N), and G has finitely many orbits on Γ,
(2.50(a) and 4.21), which is the same as to say that X
is discrete "on Μβ(Γ) (4.29).
See § 7a) for the information that if a fixed-point-free
automorphism group Η of Γ has finitely many orbits on г then
МС(Г) has the DCCL. See also Kaarl i (2) and Oswald (10).
4.60 THEOREM (Betsch (7)). Let N be 2-primitive on Г with DCC
for the left ideals of N and with an identity. Then N
is equal to one of the following near-rings (notation as
in 4.52):
N + N
N = N.
N„ a ring
о
Maff<r>
Нот0(Г,Г)
dimnr finite
N a non-ring
\ (Г)
мс(г)
G has finitely many
orbits on Г

132
§4 PRIMITIVE NEAR-RINGS
Proof, follows from 4.52 and 4.59. Note that TTr (Γ)
^o
is a near-ring in this case (for it equals N).
4.61 COROLLARY If N has an identity, is 2-primitive on Γ and
if the non-ring NQ has the DCCL and AutN (r) = {id}
о
then either Ν = Μ(Γ) (if N + N ) or otherwise
N = Mo^r^' In both cases· r (and therefore N, toe
is finite. So the DCC implies finiteness!
4.62 REMARK These results illustrate some remarks in the
preface: while the "elements of rinq theory" are rinqs
of Τ i near mappinqs on Γ, those ones for near-rinq theory
are near-rings of arbitrary mappinqs (perhaps with
some restrictions) on Γ.
4.63 THEOREM (Kaarli (4)) If I^S^N and if S/I is 2-primitive
then HN.
Proof. Since I is a 2-primitive left ideal of S, I
iLiS),
holds for some 2-modular left ideal L of S by 4.3
By 3.34, L <HS. Hence I = (L:S)Nn S and
Consequently, I is an ideal of N.
L:S)N 4 N.
See also Kaarli (2) and Ramakotaiah (2). In the latter paper it
is shown that if ΗεΥ). is finite and 2-primitive on Γ, if N is a
non-ring and if
or Ι Ν Η Γ I 2 ; if
Γ | — 1 is a prime then either N = M(r) or N = M (Γ)
Γ is abelian, N = ., ΓφΓ holds in the last case
this result can be deduced from 4.55 and 4.61).

4d 2-primitive near-rings
133
5.) AN APPLICATION TO INTERPOLATION THEORY
4.64 DEFINITION If Γε^ and ΝξΜ(Γ), Ν is said to fulfill
the finite interpolation property if
V SeIN V γχ γ5εΓ, Υ1 + Yj for 1 + j \l ό^.-.,δ,.εΓ
3 ηεΝ V 1e{l....,s): n(Yi) = 6r
There is an obtrusive similarity to the density concepts. In
fact:
4.65 THEOREM Let Ν < Μ(Γ) with N + NQ and NQ not a ring,
Then the following conditions are equivalent:
(a) N is 2-fold transitive on Г*
4 ' о ·
(b) N is 2-primitive on Г with GQ = fid}.
(c) N fulfills the finite interpolation property.
Proof, (a) -> (b): „Г is trivially faithful, 2-fold
transitivity implies 1-fold transitivity and this
in turn that ΝΓ j· {o}. If {ο} + Δ <Ν Γ, take
some δεΔ
Then \j γεΓ 3 noeN0: η0δ = >'· So
Ν δ = Γ and Δ = Г.
о
If G γ = GQ6 but Υ + δ and (say) 5 + o, take
some η„εΝ„ with η„γ = ο Λ η„δ + о. Then
оо о' о '
(ο:γ)« + (ο:δ)Ν , so γφδ in N Γ, hence
0 0 Ο
GQy + GQ& (4.20(c)), a contradiction. Therefore
G0 = {id}.
(b) -> (c): by 4.54 and 4.28(d).
(c) -> (a): trivial.
Cf. Kaiser (1), Lausch (5), Mlitz (12),(13), Pi 1 ζ (25) and
Ramakotaiah (3).

134
§4 PRIMITIVE NEAR-RINGS
4.66 REMARKS
(a) So if a near-ring N of mappings on Γ interpolates at
о and 2 other places then N interpolates already on an
arbitrary (finite) number of places. Compare this with
the corresponding "linear" result in ring theory
((N. Jacobson), Corollary to theorem 1 on p. 32).
This is a "purely interpolation-theoretic" result.
(b) It can be shown that if N fulfills the finite
interpolation property and |r| > 3 then N is a non-
ring.
4.67 COROLLARY Take Г = (IR , + ). Then any one of the following
near-rings and all near-rings containing one of them have
the properties that N is 2-primitive on Г with N 4= Ν ,
G = {id} and NQ not a ring:
N,: = IR [x] , N2: the near-ring of all step functions on IR ,
N3: the subnear-ring of M(IR) generated by the
trigonometric polynomials.
For all of them fulfill the finite interpolation property
which gualifies them for 4.65.
The author hopes that near-rings of interpolating functions
become interesting for approximation theory (because these
functions can be iterated w.r.t. o).
After all that complicated stuff the reader will possibly
agree with the author that the primitive near-rings have
successfully revenged their discriminating name.

135
One fills the trash into some bags
With these one only calculates.
The rubbish which you still can smell
Is often called the "radical".
This beautiful poem dates way back
to 1975. The author is still in hiding.
§5 RADICAL THEORY
This paragraph equals on harvest: the strains of previous
paragraphs are highly rewarded by the fact that many results
of this § 5 are easy consequences of previous ones (cf. e.g.
5.48 or §5 c) ,d)).
A near-ring N might have no faithful N-group of type v. The
next general case is that all N-groups of type ν work together
to get the intersection Π(ο:Γ) to be zero. N is then called
"v-semisimple". Anyhow, this intersection "measures" how far
N is away to be v-semisimple and is called the v-radical
^ (N). It contains all disgusting guys, for factoring out
*\ (N) gives a v-semisimple near-ring N/^ ,,.,.
First we give several characterizations of ^V(N)» using
v-modular left ideals. We get ^0(N)=^1(N)s Jf2(N) immediately.
Between X,(N) and "Ji(N) there is another radical-like
object Χ/ο(Ν)> tne intersection of all 0-modular left ideals
We discuss, when #v is "hereditary"and prove that for
all v, ^v(® N.) = ® ^v(Ni). Also for v + |, 3V(N0)^V(N).
N is v-semisimple iff N is a subdirect product of v-primitive
near-rings. With chain conditions this subdirect product
becomes a finite direct sum and we get (5.31) in special cases
that N is v-semisimple iff N is a finite direct sum of simple
v-primitive near-rinas with DCCL. In 5.32 we get a "Wedderburn-
Artin-like" structure theorem for v-semisimple near-rings.

136
§5 RADICAL THEORY
32(N) contains all nil N-subqroups, "J1/2(N) all nil left
ideals and ^J0(N) all nil ideals. However, in contrast to the
ring case, 22(N) is not necessarily nil if N is finite (cf.5.48).
Finally we consider the nil and the prime radical of a near-
ring.
a) JACOBSON-TYPE RADICALS: COMMON THEORY
1.) DEFINITIONS AND CHARACTERIZATIONS OF THE RADICALS
As usual, let N be a near-ring and νε{0,1,2}. Recall our
convention about the intersection of an empty collection of
subsets on page 1.
5.1 DEFINITION
Ί (Ν): = Π (°:Γ) is called the
«v ΝΓ of type ν
v-radical of N.
5.2 THEOREM ^ (N) =
Proof. 4.3,
П I
I v-pr.id.of N
П (L:N)
L v-mod.
left id.of N
The relations between the radicals are easily described:
5.3 PROPOSITION
(a) %(H) - 3i<N) e 22(N).
(b) If Μεΐ»! then ^(N) = ?2(N).
(c) If N is a ring then 7Q(N) = 71(N) = ^2(N) = ^(N)
(Jacobson-radical of N).
Proof, (a): by 3.7(a).
(b) : by 3.7(c) and 3.19(a).
(c): If ΝΓ is of type \> and N is a ring then one
i r
obvious.
sees as in 4.8 that ,,Γ Is an N-module. The rest is

5a Jacobson-type radicals: common theory
137
If
5.4 THEOREM (Betsch (3)). If ν + 0 then 4 (N) П L.
L v-mod.
left id.in N
Proof. By definition, *1 (Ν) = Π (ο:Γ). But
βν ΝΓ of type ν
,Γ is strongly monogenic, so (ο:Γ) = Π (ο:γ),
ΥεΓ
where each (ο:γ) is = N or a y-modular left ideal
(3.23).
Conversely, let L be a v-modular left ideal of N.
Then 3 ΝΓεΝ<3 3 Υ0εΓ: Γ = NYq λ L = (ο:γ0) (3.23).
Ν/. =Ν Γ (by 3.4(e)) is of type v. Hence the
(o:y)'s are just all v-modular left ideals (or = N)
and the result follows.
This raises the question what happens with the intersection of
all 0-modular (= maximal modular) left ideals of N.
5.5 DEFINITION *11/7(N): = П L ·
αι,ί L 0-mod. left
ideal of N
5.6 REMARK \/zW is often denoted by "D(N)" in the
literature (see e.g. Betsch (3)). Our notation is
motivated by the fact that ^i/?(N) *s in genera 1
only "half of an ideal" (a left, but not necessarily
a two-sided ideal) and by its location:
5.7 PROPOSITION ^(N) ε ^1/2(Ν) Ε ^(Ν).
Proof. ^0(N) = Π (о:Г) П П (ο:γ) «
ΝΓ of type 0 ΝΓ of type 0 γεΓ
^ П П (ο:γ).
ΝΓ of type 0 γεθ^Γ)
These (o:y)'s are (as in 5.4) exactly all 0-modular
left ideals. Hence ?Q(N) - ?1/2(N). g1/2(N) - ^(N)
is a trivial consequence of 3.7(a) and 5.4.

138
§5 RADICAL THEORY
The following result comes from Fain (1).
5.8 PROPOSITION [L ^ Ν Λ ] k£ IN : Lk-^(N) Λ νε {1, 2 }]==> Ls^v (Ν )
Hence ^i(N) and ^2(N) are semiprime ideals.
Proof. If Lk-^(N), but L$Jv(N), then 3 ΜΓεΜ<|: ΝΓ is
of type ν and Lr 4= (°Ь So 3 ΥεΓ: Ly 4= {0}.
Hence ysHg, so yeGj and LY ^ Γ by 3.4(a).
Thus Ly = Г and Lr = Г. Therefore г = Lr =
= L2r = ... = Lkr = (o), a contradiction.
5.9 REMARK If ν = 2 in 5.8, the result remains valid if
L < Γ
5.10 COROLLARY ^Jj(N) contains all nilpotent left ideals and
32(N) contains moreover all nilpotent N-subgroups.
Cf. 5.37 and 5.45 for more results in this connection.
5.11 EXAMPLES The following examples shall show that no two
of X' i?l/2' 7l * <?2 generally coincide, not even for
finite zero-symmetric near-rings. See Betsch (3).
Generalizations can be found in Meldrum (13).
(a) Νχ: = ίίεΜο(24)|f(2) = f(3) = 0). Nj is 1-primitive
on Έ. hence <fi(Nj) = ίθ), but not 2-primi ti ve, hence
ЬУ 4.46(f) ^2(Νχ) = Νχ. So 3j(N) + ^2(N) in
general.
(b) Let N2 := {feMQ(24)|f(2)ε{0,2}}. By 3.8 we know that
N2 is O-primitive, but not 1-primitive on Γ. Since
each map εΝ2 is determined by its effect on 1,2,3,
N2 is the sum of the left ideals L^. = (0:2) η (0:3) ,
L2: = (0:l)n (0:3) and L3: = (0:2)л (0:3). Since
(0:1) η (0:2) η (0:3) = (δ), Ν,
L1+L2+L3.
The map %. ■* Lj with fy(x)
Υ * fv
ί
χ + 1
χ = 1
ι s an
N2-isomorphism. Hence L, =v Щ. Similarly.

5a Jacobson-tyре radicals: common theory
139
= 2 and L? is an Ν,,-qroup °f type 2.
b3 N2 "-4- |l"2
Therefore L2+L3, L^ + L3 and L,+L„ are 0-modular
left ideals. Their intersection Is ^i/?(N2^ = *0^"
Since N contains an identity, ^(N2)= ^o(N?^ (by
5.3(b)). But each N„-group of type 2 is =,, L2 by
3.11(a). Hence ίζ(Μζ) = (0:LZ) = (0:2) + {0} =
Observe that 3j(N2) =^2(N2) = (0:2) is not nilpotent
- in striking contrast to the situation in ring theory!
Compare 5.45!
[с) N3 : = {fEM0(24x22) |f (Δ)£ Δ and (a ,0 ) - (b ,0) ε Δ =*
=*> f(a,0)-f(b,0) ε Δ} with A : = {(0 , 0 ) , ( 2 ,0 )} has id as
identity. All (a,2) with aci, generate the N,-group
Γ := 2.χ22> r has only Ζ.χ{0} and Δ as non-trivial N~-
subgroups, and they are not ideals. Hence ^0(N3) = {(0,0)}
Now Δ and 2.χ{0}/Δ are N,-groups of type 2. The annihi-
lators of (a,0) (ar.2 ), of (2,0) and of (3,0) + A are
maximal modular left ideals of N,, their intersection D
contains 2i/,(N,). But D'- = {0}. In 5.37(b) we will see
1/2'
N.
that this implies DS^./;)i„.
t (0) =>0(N3).
, whence J.
/2'
= D f
5.12 EXAMPLE If N = Nc then ^Q(N) = %/2(N) = ^j(N) =
intersection of all maximal normal subgroups of (N,+) (=Baer-
radical" of (N,+)), while ^2(N) = intersection of all
normal maximal subgroups of (N,+).
(Apply 3.21(c), 3.29, the fact that each 0-modular left
ideal in N = N is 1-modular and 5.2).
2.) RADICALS OF RELATED NEAR-RINGS
To be able to treat ^ , ^, and ^2 jointly (at least for
a while) we introduce the following definition which comes
from universal algebra (see (Hoehnke) and Mlitz (6)).

140
§5 RADICAL THEORY
5.13 DEFINITION A map % which assigns to each near-rinq N an
ideal ?t{N) of N is called a radical (map) if for every
Ν,Ν'εΤ):
(a) *(Ν/Λ(Μ)> = {0}
(b) If heHom(N.N') then h(fc(N)) «= fc(h(N)).
5.14 DEFINITION If fc is some radical map then ΝεΤ? is called
(a) %,-semisimpl e: <=>J2(N) = {0}.
(b) ^-radical : <-> #(N) = N.
If I <1 N and К <ξ N denote {k + I | keK} by K+I/I.
5.15 PROPOSITION If V. is a radical map and N,N' are εΊ7
then
(a) If h: N -» N' and N i s 31-radical then N1 is Ί&-
radical.
(b) If N isft-radical then \/ IaN: N/j is ^-radical .
(c) \/ I«N: ItC M/ x) э^( Μ) +1 /,.
(d) V ΙϋΝ V KEN: ft(N/j) = K/j =» K+I2 4R(N)).
(e) If N is simple then either N is3i-radica1 or3l-semi-
simple.
Proof, (a): by 5.13(b).
(b): by (a).
(c): Consider the canonical epimorphism tr: N + N/I
and apply 5.13(b).
(d): by (c).
(e): this holds because %(N) <3N.
It would have been silly to introduce 5.13 if the *i 's would not
be radicals. In fact, Betsch (3) has shown the following

5a Jacobson-type radicals: common theory
141
5.16 THEOREM For νε{0,1,2} , Ν -> Ί (Ν) is a radical map.
Proof. Clearly Jfv(N) <l N.
Let n+7v(N) be ε^ν(Ν/~ (N)) and let Γ be an N-
group of type v. By 3.14(a), Γ is an N/^ (N,-
group of type ν since *ϊ (Ν) «ξ (ο:Γ). Hence
ηΓ = (n+,/f (N))r = to). Since г was arbitrary,
ηε(1(ο:Δ), where & ranges over all N-groups of
type v. Thus ηε^(Ν) and n+3v(N) = Jv( Ν) ,
so 5.13(a) is shown.
To see 5.13(b), let h be tHon(N.N') and ηε^ν(Ν)·
Im h = :N". Let Γ be an N"-group of type v.
Since N" " N/K . , Γ can be considered as
N-group of type ν (see 3.14(b)). Therefore
пГ = (о). This implies that η(η)Γ = ηΓ = ίο),
Again, Γ is arbitrary, so η(η)ε^ (h(N)).
5.17 REMARK For ^ (Ν) (νε{0,1,2}), 5.15(e) can be improved
if N is simple then either N is Q -radical or v-primitive
(since all (ο:Γ) d N). The near-ring N1 of 5.11(a) is
an example of a simple ^-radical near-ring.
5. 18 THEOREM Let I^N be a direct summand of N. Then
7v(NMs^(I)
holds for all vc{0 ,1/2 ,1 ,2} (let N = NQ for v=1).
Proof. Suppose that N = I+J. Then N/J = I. Take ve{0,1,2}.
Each I-oroup of type ν is an N-group of this type by
3.14 (b). Let Pjv be the class of these N-groups. Now
'N> = - o/?vPev(o:r)N- vCl (°:r)r and
"ν Ni ot type
v^'^rO t
ype ν(ο:Γ)Ι
0\
'N
^v(I)
Finally, if ν = 1/2 then we get with 3.28 and 3.33
*} 1/2(N)n I = (_Π L )rtl =_Л (ΓηΙ)5_Π ΓλΙ =
Le/0(N)
ίε/0(Ν)
Lcy„
1/2
(I).

142
§5 RADICAL THEORY
5.19 EXAMPLES (a) Let N be the near-ring N2 of 5.11(b). Letl:=(0:2)
(=N1 of 3.8). Since I is 1-primitive on Z4, ^(1) ={0}. But
^(N) = (0:2) = I (5.11(b)), so ^(1)^(1)^1. Also let Γ
be finite and Δ a non-trivial subgroup. Then
N: = {f£MQ(r)|f(A)EA} has ^ (^( N ) )e^ ( N ) (Kaarli (9)). See
also Ex. 5.32 in Meldrum (13).
(b) Let the notation and situation be as in 4.50. Since N
is 0-primitive, ^n ( N) = {0}. I contains the nilpotent ideal
{f ε I | Vy ε Γ: ί(γ)ε{0,4}}. By 5.37(d) ,4(1) / {0}, whence
5.20 THEOREM Let Ν. (ΐεΐ) be a family of near-rings, νε{0,1,2}
and let N be the direct sum of the N.'s with N = N if
1.
Then
Proof. Let νε{0,1 ,2} . If π
λ<.©Νι
1 ε Ι
© λ<ν-
1 ε Ι
. :Ν-*Ν· denote the canonical pro-
) gives us for all ι'εΐ the inclusion
S?v(Ni)· Hence ^)£0,( ?ν(Ν) s
τ ε I
jections, 5.13
%-(;v(N))
S .© λ<Ν1>'
1 ε I
Conversely, for each ΙεΧ (Ν) there exist v-modular
left ideals L, of N. which are associated with L via
3.33. For ι'εΐ, let X. denote the set of these left
ideals in N.. Since L contains the direct sum of its
L . ' s , we get
1 V(N) - 0 (L:N) Э f\ (Θί(:0Ν.) =
v L£/ (N) L-εΧ- ίεΐ ιεΐ 1
л
Θ /Ί (1^:1^)2 Θ
ιεΐ L^
ίεΐ L .ε/ (Ν . )
ι ν ν ι '
4:Ni) - .© 7ν(Νι·)-
1 ε Ι
It is not known to the author if 5.20 is also true for direct
products and for v=l/2. Anyhow, one can deduce from 5.20 that
N
N = У" I implies that for all νε{0,1,2} we get 7(N'
α ε A
i:i
(I ). It is easy to see that the corresponding result
does not necessarily hold if the I are "only" left ideals.

5a Jacobsontype radicals: common theory
143
5.21 THEOREM (Kaarli (4)) Let S be an invariant subnear-ring
(see 1.31) of N = NQ. Then
72(S) = ?2(N)n S.
In particular, this holds if S is an ideal in N. Again in
particular, we get for every near-ring N=N :
72( 72(N>> = ?2<N)·
Proof. Let sr be of type 2. Then г ^ S/L, where L is a
2-modular left ideal of S. By 3.34, L «N S . Every
N-subgroup of S/L is, as an S-subgroup, trivial.
Hence „r is of type 2, from which we get ^-o(N) 2
— J 2 ^ N ) n ^ ' T'le converse inclusion also holds, since
every N-group of type 2 is an S-group of type 2.
This makes us curious if similar results hold for other radicals
as well. In this area we get into contact with the general
radical theory of universal algebras (e.g.Mlitz (7)) or even of
categories (see e.g.Holcombe (7)). Let all near-rings until 5.23 be
zero-symmetric. <t as in 5.13 is called a Kurosh-Amitsur radical
if 2Z.( N) = N <=> every non-zero homomorphic image Μ of N has some
ideal I+{0} with %.( I ) = I -
From general radical theory it is known that this holds iff ft(N)=N
and h(N)=M implies &(M)=M, if Ж( Я(Н))= Ц(Н) and if I<N, 3£(I) = I,
and <C (N/1 ) =N/1 implies 7£(N)=N for every nr. N = NQ. "U. is said to
have a hereditary semisimple class if Η Ν, N Ц- semisimple,implies
that I is iiL-semi si mpl e. The main papers in this area wh i ch concern
us here are Betsch-Wiegandt (1), Holcombe (7),(15), Holcombe-
Walker (1), Kaarli (2 ) , (4 ) , (8 ) , (9 ) , Mlitz (7),(11), Wiegandt (1).
5.22 THEOREM Let all near-rings be zero-symmetric and Ж. be as
in 5.13 defi ned on ^7 .
(a) (Kaarli (2)) ^ (νε{0,1,2}) is Kurosh-Amitsur iff it
is "idempotent" in the sense that ^( 7V(N)) = 7V(N)
holds for every Νε f] .
(b) By 5.21, ^2 is Kurosh-Amitsur. Also, ^3 is Kurosh-
Amitsur, where ^{H) is defined to be the intersection
of the annihilators of all N-groups г of type 3 (see the
last lines of p. 80) .

144
§5 RADICAL THEORY
(c) By example N 1) in the "Near-rings of low order"
(Appendix), ^. is not Kurosh-Amitsur; in Kaarli (9) it is
shown that Ί is not Kurosh-Amitsur either.
(d) If Я is Kurosh-Amitsur such that Щ N) =N =v N={0} for all
zero-near-rings but with i£(N)=N for some N = N ф{()} then
Ίλ cannot have a hereditary semisimple class (Betsch-
Wiegandt (1)). But ^ ancl 1-х have hereditary semisimple
classes (Kaarli (4), Hoicombe-Walker (1)).
We conclude our troublesome trip to relatives of N by a
consideration of the behaviour of ^υ(Ν) on the one hand
and *J\>(No^' 'Jv^c' on the other nand· Our first result is
an immediate consequence of 2.18:
5.23 COROLLARY \f νε{0 ,1,2}: ^v(N) = Q v( N) )0 + Qv(N ) ) C .
This is not very much, indeed. It would be fine to be able to
compute ^(N) via ^ν(Ν0) and J„(NC) (5.12!), perhaps
as *L(N> = 3v<No>+3v<Nc>' simi1ar t0 5·20·
This is not the case (see also 9.77):
5.24 EXAMPLE If N = {fεΜ(24)|f(0) = f(2) = f(3)}. One can
show that ^2W = Ν, ^2(Ν0) = NQ (5.11(a)!), but
^2(NC) consists only of the maps which are constant
=0 or =2.
From that one sees that there is no obvious simple connection
between ^V(N),?V(NQ) and ^Nc). But ?V(N0) is always in
5.25 PROPOSITION \/νε{0,1,2}: Ί (ΝΛ)=(*1 (Ν)) ·, in particular,
Proof. It suffices to prove the "in particular", for
^V(NQ) is trivially contained in NQ.
By
3.18(a). %(H0) - П (о:Г)
4 N Г of type υ
П (о:Г) .°3V(H).
ΝΓ of type ν
See 5.67 (t ) for ( 72(N))

5a Jacobson-type radicals: common theory
145
3.) SEMISIMPLICITY
Throughout this number, let ν be ε{0,1,2}, unless otherwise
indicated.
5.26 DEFINITION N is v-semisimple: <=> N is *J -semi simpl e .
5.27 EXAMPLE N is v-primitive -> N is v-semisimp!e.
5.28 COROLLARY
(a) Each direct sum or direct product of v-semisimple
near-rings is v-semisimple.
(b) If V ι ε I: Ν.ε?) then © N. is v-semisimple <=>
ι ε I
<=> \/ i ε I: N, is v-semisimple.
Proof. 5.20.
5.29 THEOREM (Betsch (3)). N is v-semisimple <=>> N is isomorphic
to a subdirect product of v-primitive near-rings.
Proof. Consider the set of v-primitive ideals and apply
5.2, 1.58 and 4.2(c).
5.30 THEOREM Let N have the DCCI (OCCL). Then N is v-semisimple <=■>
<=> N is isomorphic to a subdirect product of finitely
many v-primitive near-rings with DCCI (DCCL).
Proof. —>: The family (I ) . of all v-primitive ideals
J v α'αεΑ r
of N has Π I = {0}. We claim that it suffices
αεΑ
to take finitely many I 's to qet a zero inter-
o.
section. If not, take some Ι (αεΑ). Since
ao
Π I = {0}, there is some α,εΑ: Ι η I =1
αεΑ α l ao al ao
Continuing in this way we get a chain I =>I η I =». ..
3 J э α α α,
οο 1

146
§5 RADICAL THEORY
which does not terminate and we arrive at a
contradiction. Hence N is isomorphic to a subdirect
product of finitely many v-primitive near-rings.
The rest follows by 2.35.
<=: follows from 5.29.
5.31 THEOREM (Betsch (3), Blackett (2)). Let N = NQ and
νε{1,2>. Then N is v-semisimple with DCCL <=> N is a
finite distributive sum of ideals which are v-primitive
simple near-rings with a right identity and DCCL.
Proof. ">: by 5.30, 4.46(e) and (d) and 2.52(b).
<-: By 5.29, N is v-semisimp!e. Using 2.35(b)
repeatedly (by induction) one sees that N has the
DCCL.
5.31 has many interesting corollaries (which mostly stem from
Betsch (3)), which represent the "non-linear" version of the
celebrated Wedderburn-Artiη-structure theorem.
5.32 THEOREM If N = NQ is v-semisimple (veil,2}) with DCCL
then
(a) N has a right identity (so 3.43 is applicable).
(b) N is completely reducible, so all of 2.50 is at hand
(for N).
• · .
(c) N = (L11+...+Llr|i)+(L21+...+LZn2)+...+(Lkl+...+Lknk).
where for 1eil,...,k}, L-j ι > · · · >Ц n b are pairwise
N
.: = £*L. .-isomorphic left ideals of N. and N.-groups
of type v. All L·.; are also simple left ideals of
N and N-groups of type v. Each N-group of type ν is
N-isomorphic to one of them; 2.50 can be applied for
„N.
(d) N has only finitely many classes of non-N-isomorphiс
N-groups of type v.
(e) Every ideal of N is again v-semisimple.

5a Jacobson-type radicals: common theory
147
Proof, (a) and (b) follow directly from 5.31.
(c): by 5.31, 4.46(b), 2.49, 2.48(e), 3.41(b) and
3.11(a).
(d): by (c).
(e) : by 5.31 and 2.55(a).
See also Deskins (2), Kaarli (2),(4) and Chao (1). For the
following result cf. 5.18 and Ex. 6.40 in Meldrum (13).
5.33 THEOREM (Kaarli (4)). Let HcHQ and I й N. Then ^(I)i },(N)л I
and (if N has the DCCN) *} Q (I ) 2 ·£0 ( Ν ) η Ι.
Proof. We only show the statement for ^. if N has the DCCL.
N/m, ,„·, is 1-semi simp 1 e with DCCL. Since
I + il<N>/jl(N) *Ν/^(Η). Ι+7!(Ν)Λ,ι(Ν) is 1-seml-
simple by 5.32(e); the same applies to I/*t /fj)0j
by 2.8. 5.15(c) and 5.16 tell us that
<ϊΐ(Π + ('31(Ν)Λΐ)/^ (Ν)ΛΙ - {0}, whence ^ (I) -
= l(N)n I.
Some decompositions of N induce decompositions of Nr:
5.34 THEOREM (Betsch (3)). If veil,2} and N = NQ has DCCL
and is v-semisimple and if ,,Γ is monogenic then
k#
(a) Nr = Д*Д1 with Δ1· <^ Γ and ^ of type v.
(b) Each (ο:γ) is either = N or a finite intersection
of v-modular left ideals.
(c) If ΝΓ is of type 0 then it is of type v.
s
Proof, (a) By 5.32(c), N = l'l. where each L. <L N
i=l 1 Ί *
is an N-group of type v. Let Γ be = Νγ . Then
Γ - Νγ0 = ( Σ 4>Υ0 = ! (L^o)· E3Ch 4*0 ^ Γ

148
§5 RADICAL THEORY
by 3.4(a).
If LiYo + {0} then Li \ LiYo and L^ is
of type v, hence simple. As in (b) —> (c) of 2.48
one can choose some subset S of {l,...,s}
with Γ = I'(LlY ).
ieS
к
(b) If ηε(ο:γ) (γεΓ) and γ = £ δ^ then
к к
о = ηγ = η( Ι 6^) = Ι ηδ Ί· by 2.30. Since the sum
of the Δ-'s is direct, V ie{l,...,k}: ηδ. = о.
к
Hence ηε Л (о : б .).
i = l 1
If δ1εθ()(Δ1 ) then (о : 6^) = Ν.
If δ^θ^Δ^ then δ1 -Ν Ν/(0.δι), hence (ο:&^)
is v-modular.
(с) Use 5.34(a) and the fact that ,,Γ is simple
(so к = 1).
See Choudhari (1) (no. 3.34) for more characterizations of
*| (N) (especially via quasi-regul ari ty). If ΝεΤΙ л ft, is
2-semisimple with DCCL and if N = ©Ik (Ik finitely many
simple ideals - 5.31) then the center of (N,·) is isomorphic
to the direct product of the centers of Π|<>') (as semigroups).
In this case, if Ik = Μ Q(r) (4.60), the center of (Ik.·)
Go
is a group, isomorphic to the center of G (see Holcombe (2)).

5b Jacobsorvtype radicals: special theory 149
b) JACOBSON-TYPE RADICALS: SPECIAL THEORY
*·) X and ^1/2·
5.35 THEOREM (Ramakotaiah (1)). ^0(N) = (^1/2(N): N)·
Proof. (^1/2(N): N) - ( П L:M) = П (L:M) = ^(N).
Le«C0 Le*0
5.36 THEOREM (Ramakotaiah (1)). *L(N) is the greatest ideal
of N contained in ^i/oC1)·
Proof. Let I be another ideal of N which is in ^i/?^)·
If at is the set of all 0-modular left ideals of N
then V IcX: I*L.
Hence V Lejt: Is(L.:N), so Ι ΐ fl(L:N) = X(N).
Observe that in the finite case this also follows from 3.27(b)
and 3.25.
5.37 THEOREM (Ramakotaiah (l),cf. Kaarli (4), Chao (1)). Let N=NQ.
(a) ^}i/"(N) is the greatest quasiregular left ideal of N.
(b) Ί}1/2(Ν) contains all nil left ideals.
(c) *I0(N) is the greatest quasiregular ideal of N.
(d) 30(N) contains all nil ideals.
Proof. (a)l) We show that ifi/oiN) is quasiregular.
Let ζ be ε^1/2(Ν). If z£Lz (3.35) then Zorn's
lemma (3.22 ! ) guarantees the existence of a
(by z) modular left ideal L, maximal for having
z£L.
If L' is another left ideal containing L then
zeL'; since Μ ηεΝ: n-nzeL', L' = N. Hence L
is a maximal left ideal, so Jf,/2(N)?L and we
arrive at a contradiction.
Therefore Λι/?^) is quasiregular.

150
§5 RADICAL THEORY
2) Now let Q be a quasireqular left ideal. If
31/2(N) = N then clearly 0-Ji/2(N)· So assume
that Cjw?(M) + N. Let L be a modular (by e, say)
maximal 1 eft i deal.
If Q^L then Q + L = N. So 3 ςεΟ. 3 JleL: e = q + Л.
\/ ηεΝ: ne-nq = n(q + i.) -nqeL. Hence
V ηεΝ: n-nq = n-ne+ne-nqεL+L = L. This shows that
L (■*■ N) is modular via q. By 3.38(c), q cannot be
quasiregular, a contradiction. So Q=L.
Hence Q ε П L = *}. .-(Ν).
L O-mod. 0i,i
in N
(b)
(c)
(d)
by (a) and 3.38(b).
by (a) and 5.36.
by (c) and 3.38(b).
Another intersection of big things is contained in ^2(fl):
5.38 THEOREM If N = N , the intersection Μ of all maximal
N-subgroups is quasiregular and contained in ^_(N).
Proof. Take ζεΜ. Assume that z|L . Then there is
some TT <N N containing L which is maximal w.r.t.
not containing z. As in part 1) of the proof of
5.37(a) one shows that Μ is a maximal N-subgroup,
so ζεΤί, a contradiction.
The rest will be obvious from 5.44.
5.39 THEOREM (Betsch (3)). Let ΝεΊ? have the DCCL. Then
s.
(a)T,,2(N) = ^ <==> Ν = I*Li where the Li ' s are modulai
left ideals and N-groups of type 0.
(b) In this case, N contains a right identity and 2.50
is applicable for „Ν.

5b Jacobson-type radicals: special theory
151
Proof, (a) =>: If ^1/2(N) = t0^· the intersection of
the 0-modular left ideals = {0}. As in the proof
of 5.30 it suffices to take finitely many of them,
say Kj,...,K., minimal for having intersection = {0}
k.
Apply 2.50(g) to see that N = £ L., where
i = l
Li ■
У К. ™N N/K are N-groups of type 0.
<=: If N = I'l. as indicated, {Κ,,.,.,Κ.} (as
i = l k
above) are 0-modular left ideals with Π K. = {Ob
i = l λ
Hence ^1/2(N) = {0}.
(b): by 3.27(d) and (a).
5.40 THEOREM (Ramakota i ah (1)). Let Νε7?0 ha/e the DCCN.
Then
(a) ^i/2(N) i s n1Ίpotent.
(b) N is 0-serni s i mpl e <=> N has no non-zero quasi regular
ideal <=> N has no non-zero nil ideal <=> N has no
non-zero nilpotent ideal.
(c) Each prime ideal -f- N is O-primitive.
Proof, (a) and (b) are immediate consequences of 5.37
and 3.40.
(c): Let PON be prime. By 2.104, TT: = N/p has
no non-zero nilpotent ideals and is therefore 0-
semisimple by (b). Since N =)= Ρ, ΤΪ" + i^b s° Я
has DCCN (2.35) and O-primitive ideals 7^ ,. . . ,Trk
k
П
i = l
with П F,- = Ш (5.31). Hence 7]72...?k = {ϋ}, so
(since {ϋ} is prime) some 7. = {7J}. So {ϋ} is
O-primitive and the result follows.
In (4), Kaarli generalized (a) to a wider class of near-rings.

152
§5 RADICAL THEORY
2·> 1ι
5.41 REMARKS If Νε7?ο, ^(N) contains of course all nil,
nilpotent and all quasireqular left ideals of N (5.37),
but not necessarily all nil N-subgroups of N (but ^о(^)
has this property - see 5.45):
In Ν = ίίεΜ (Z4)|f(2) = 0} we have ({0,2}:Z4) <N N
nilpotent, but ^(N) = ί0}· Also, ^j(N) is not nil
in general (see 5.11(b)). On the other hand, *J}i(N)
contains one more item:
5.42 THEOREM N = N , DCCL. Let M(N) denote the intersection
о
of all maximal ideals of N (cf. Mlitz (1), (2)). Then
(a) JjtM) ξ M(M).
(b) If ΝεΊ^ has the DCCN then ^(N) = ^2(N) = M(N).
Proof, (a) Let Ρ '<l N be a 1-primitive ideal. Then N/P
is 1-primitive with DCCL, hence simple by 4.46(e).
Hence Ρ i s maxima 1.
(b): by 4.47(b).
In Kaarli (4) there is an example I AN
^(NlMMO). See also Example N1) in
1ow order).
5.43 PROPOSITION If NeHj has a minimal NQ-subqroup then N is
not 2-radi cal .
Proof. Let Μ <Ν Ν be a minimal NQ-subqroup. Then ^.M
is of type 2. If NM = {0} then Μ = {0} since
Νε??ι , a contradiction.
Hence ?2(N)=N.
Now we look who is contained in <fo(^).
= NQ with fy.l) = {0}, but
the Appendix (nr . 's of

5b Jacobson-type radicals: special theory
153
5.44 THEOREM (Ramakotaiah (1)). If Ν = Ν , $2(N) contains
all quasi regular N-subgroups.
Proof. We proceed similar to 5.37(a). We may assume that
$2(N) + N. Let Q <N N be quasi requl ar. If 0^г(М)
then there is some 2-modular left ideal L with Q^L.
Let e be a right identity modulo L. By 2.15, we see
that L+Q <N N and L«=L+Q. L is a maximal N-sub-
group (3.29), so L+Q = N. Let e = :l+q (HeL, qeO).
If ηεΝ then ne-nq = n(£+q)-nqeL. Hence
V ηεΝ: n-nq = n-ne+ne-nqcL+L = L and L is modular
by q. By 3.38(c), q is not quasiregular and we arrive
at a contradiction.
Cf. Chao (1). From 3.38 we deduce
5.45 COROLLARY If N is zero-symmetric then ^?(N) contains
all nil(potent) N-subgroups.
Recall from 5.11(b) that ^JpW ^s not necessarily nil (not
even for finite zero-symmetric near-rings with identity (not
even under the assumption of distributive generation - see
Laxton (4))). 5.48 will characterize the case that ^p^^ is
nilpotent. See also Beidleman (2).
*J„(N) is the intersection of all 2-primitive left ideals,
while <fl/2^) is </г^) intersected with all 0-, but not
2-modular left ideals. C7?(N)/V, ,,,> swallows up all 0-modular
c <h/2^"J
ones (Laxton (6)):
5.46 PROPOSITION Let N = Nn have the DCCL. Then ^J,(N)/« ,ы,
о и с ii/z^n>
is zero or a finite direct sum of N-groups of type 0.
which are not of type 2.

154
§5 RADICAL THEORY
Proof, ft: = N/N ,.,. is a finite direct sum У'Г,. of
h(H) k i-i '
Τϊ-groups of type 2 by 5.31. N =» У Г,.=». . .«Г. ={0}
i=2 1 K
is a principal sequence. Assume that ^b^ ^s
contained in some 0-, but not 2-modular left ideal L.
of N. Then N/Ai ,N, = L/л .f). =» {0} can be refined
to another principal sequence (2.40) and the first
factor NA, ,MV/ -M N/
%W/u
32(м>
-,, .J/. is simple, so
N-isomorphic to some TJ\ . But Γ. is of type 2 and
N/*l iu\/ is not, so we arrive at a contradiction.
Hence for each 0-, but not 2-modular left ideal L we
get L + <J2(N) = N and consequently
N/L=N L+|f2(N)/L 1N JZ{N]/Ln^(||)i
This shows that Ln^2(N) = :L' ^s a 0-modular left
ideal of ^2(N). In ^2{N)/,1 (N)* a11 theSe
L'/ai /mi's are 0-modular left ideals with "trivial"
d 1 / 2 С N J
(=*31/2(N)) intersection, so by 2.50 *J2(N)/m ^
is a finite direct sum of N-groups of type 0, but
not type 2.
From this we deduce
5.47 THEOREM (Laxton (6)). Let ΝεΊ^ have the DCCN and let L
be a left ideal . Then
Le^2(N) <=> 3 К 4t N: K«=L, К nilpotent and L/K is zero
or a finite direct sum of N-groups of type 0, but not of
type 2.

5b Jacobsorvtype radicals: special theory
155
Proof. =>: Lr>^ 1/2(N) =: К is nilpotent by 5.40(a).
L/K = L/L^1/2(N) ~N L+3l/2(N)/31/2(N) ^
^o *3?(Ν)/<-ι ,μ(· By 5.46 and 2.55, L/K is a finite
* <}l/2w
direct sum of N-qroups of type 0, but not of type 2.
<=: L+J2(H)Ajz(N) ΪΗ U^{H)0 L. By 5.45, K-J2(N)rtL.
Hence L/,
L/.
Ъ<М>Л
L/K is a finite
Q2(N)0 L)/K
direct sum of N-groups of type 0, but not of type 2,
so the same applies to L/*. ,.,ν . by 2.55, and
hence also to L+^2(N )/^ {N) <Ι^ N/^
,(N) '
M/»
is a direct sum of N-qroups of type 2, so
'JZ<N)
L+32(N> = Ъ^ and L-^2(Nb
From this result one can construct a decomposition of ^2^^
into a nilpotent and a "totally nilpotent" part (Laxton (6))
in special cases. For more general cases, see Laxton-Machin (1)
and Scott (1).
Now we characterize the case that ^2^) ^s nilPotent·
5.48 THEOREM (Ramakotaiah (1)). Let Νε7?0 have the DCCN.
Then the following conditions are equivalent:
(a) ^2(N) is nil.
(b) ^2(Ν) is impotent.
(c) ^2^^ ^s quasire9ul ar ·
(d) fc(N) = 1l(N> =3l/2(N> = VN>·
(e) Each N-group of type 0 is of type 2.
(f) V I £ N: I is O-primitive <=> I is 1-primitive <-> I is
2-primi ti ve.
(g) Each 0-modular left ideal of N is 2-modular.
(h)V {0} + I < N: I is prime -> I is 2-primitive.

156
§5 RADICAL THEORY
Proof, (a) <"> (b) <=■> (c) holds by 3.40.
(c) <-> (d): by 5.37(c).
(d) -> (e): If ΝΓ is of type 0 then ,N/ J
is of type 0 by 3.14(a) (since ^(N) = ^0 (N)s (о :Г)).
Since N/j ^ is 2-semi si mple (5.16), „ ;j.r
is of type 2 by 5.34, so ^Г is of type 2 by 3.14(b).
(e) —> (f): is immediate.
(f) -> (d): trivial .
(f) <=·> (g): by 4.3.
(f) <-> (h): by 5.40(c) and 4.34.
Cf. also 5.61(b), Chao (1), Kaarli (4), Scott (8),(12).
Finally, we are going to describe 2-semisimple near-rings with
chain condition more closely.
5.49 THEOREM (Choudhari-Tewari (1), Oswald (3)). Let Νε770
have a right identity. Then the following statements
are equivalent:
(a) N is 2-semisimple with DCCL.
(b) N is a direct sum of finitely many N-simple left ideals.
(c) If ΝΓ is monogenic then Ϊ 4 $Ν Γ] Ε ^ Γ: Γ = Δ + Ε л
Л Δ л Ε = {ο}.
(d) \/ Μ <Ν Ν 3 L ί Ν: Ν « NtL Λ NU = ίΟ}.
(e) Each exact sequence {0} ■+ Mj + M? (Mj.Mp <N N) splits
(definitions as usual).
(f) V Mj.Mg <N N, M^M2 V ΝΓ εΝ«} V ИеНошм(М1,Г)
3 KeHomN(M2,r): Τ\/μ - h.
(g) N satisfies the DCCN and has no non-zero nilpotent
N-subgroup (cf. Blackett (2), where near-rings with
this condition are called "semi simple").

5b Jacobson-type radicals: special theory
157
f. (a) =-> (b): by 5.32.
к
(b) => (a): By 2.50, N has the DCCL. If N = ['ι,,
i = l
Ь N-simple left ideals, then K-: = У L,
2-modular left idea
hence ^2{N) = {0}'
are
2-modular left ideals and (as in 2.50(q)) П K. = {0}
i = l 1
(b) "> (c): If N = У/ί. as above, then by (a)
i = l λ
and 5.34(a) there is a subset S of {l,...,k} with
r= £'LiYo (Y0£9l(r)b a11 LlY + {o} and of
i eS
type 2.
If Δ £Ν Γ, take some maximal element Γ (Zorn!) in
t:= {E Зц Γ|Δ лЕ = {о}}. If Δ+Ё + Г, 3 ieS: L^^
$Δ+Γ. But L^ is of type 2, hence l.у η (Δ+Τ) = {о}
Therefore Δ л(ί.γ +Γ) = {ο} which contradicts the
maximality of Г. Hence Δ+Γ = г and (c) is shown.
(c) =-> (d) is trivial since ,,Ν is monogenic (by e).
(d) "> (e): Let {0} + Mj t M2 be exact. By (d)
3 L <1Л Ν: f(Mj)+L = Ν Λ f(Mj)AL = {0}. Then
the "projection" ρ: N ■+ f(M«) defined by
p(n) = p(f(m, )+{.): = f(m,) is an N-homomorphi sm.
— -1 —
P/u =:p· Then f op: M„ * Mi is an N-homomorphism
with f" opof = idM . Hence {0} ■+ M, -* M~ splits.
(e) => (f): Let M,,M2,r,h be as in the statement,
and let г: Μ, -*■ M2 be the injection map. Then
{0} - Mj -i M2 splits (say by g: M2 ■+ Mj). Then
W: = hog: M2 -*■ Г does the required job.
(f) -> (e): Let {0} - Μχ - M2 be exact. The identity
map idf/M \ can be extended to an N-homomorphism
h: M2 ■+ f(M.). Then clearly f" oh is a splitting
N-homomorphi sm M2 ■+ M,.

158
§5 RADICAL THEORY
(e) -> (g): First we show that N has the DCCN.
Let N = Μ =»Mi^o13· * * ^e a ctia''n °f N-subgroups of N.
\/ ΐεΙΝ : {0} * Μ. 4. M^ (l1 the injection maps)
splits.
Let g. : Μ. , ■+ M. be corresponding splitting N-
homomorphisms. Then h,: = g, is a splitting N-
homomorphism for {0} ■+ M, ■+ N, h~: = g2°9i °ne for
{0} * M« -*■ N, et cetera.
If L.: = Ker h^ then L. d» N and (as easily seen)
N = M.+L· with М-л L· = {o}. Furthermore,
LrL2e··· '
But ,,Ν is completely reducible (2.48(e)), finitely
generable (since eeN), so endowed with the ACCL
(2.50(e)) which causes L,=LpC=... to stop after
finitely many steps. Hence the same applies to
0 1
Now we show that N has no non-zero m'lpotent N-
subgroups. Let Μ be such one. As before,
3 L <lt Ν: Ν = L+M, LrtM = {0}.
Let e be = *·0 + πι0. («· eL, m εΜ). As in 3.43, mQ
turns out to be a right identity for M, hence Μ
cannot be nilpotent and the proof is accomplished.
(g) -> (b): (Blackett (2)). Let MQ be a minimal
N-subgroup. By 3.52, 3 eQ = β*εΝ: NeQ = MQe0 = MQ.
By 1.13, N = Noeo+(0:eQ) = Mo+(o:eQ) with
Mon(0:eo) = {0}.
If Μ = N there is nothing to prove. So let Μ 4= N.
Hence N=(o:eo) + {0}.
Either (o:e ) is a minimal N-subaroup or it contains
(by applying the above considerations to (o:e )
instead of N) another smaller N-subgroup of the
form (o:e )nL where L is some left ideal of N.
The DCCN assures that after finitely many steps
we arrive at a minimal N-group M1 which is the
intersection of (o:e ) with a left ideal of N,
hence a normal subgroup. Take some idempotent eieN

5b Jacobson-type radicals: special theory
159
with Nej = M1e1 = Mj. Hence ( 0 : eQ) = M1 + (o : e1) (as
groups). Repeating this procedure with (o:e^)
(if necessary) yields Ν = Μ +...+Mk where M.
are minimal (hence N-simple) N-subgroups of N.
Now by 5,40 ^i/?(N) = ^°^' so N is the dlrect sum
s
I'l* of left ideals of type 0 (5.39). The M.'s
i = l
of above are N-groups of type 2, hence N-isomorphic
to some L ^'s.
Therefore N is the finite direct sum of left ideals
which are N-groups of type 2.
5.50 REMARKS
(a) As Choudhari-Tewari (1) have shown one can add to
5.49 the following condition if Νείλ :
(i) Each N-subgroup of N is monogenic, projective
(definition again as usual) and generated by
an idempotent.
See there for the proof (c) =■> (i) => (e) . Cf. Chao (1).
(b) If N = NQ is 2-semisimple then the DCCL or DCCN imply
all other chain conditions of ACCL , ACCN, DCCL, DCCN.
This follows from 4.46(d), 5.49(d) and (g). Cf. Oswald
(3) and Scott (1). It is not known to the author if a
2-semisimple near-ring with ACCL or ACCN has all other
chain conditions as well.
(c) As Mason (3) pointed out, there exists no non-trivial
injective N=N -group. See more on that in his papers
(3) and (4), in Prehn (1), Banaschewski-Nelson (1),
Maxson (8) and Oswald (10). In particular, see 9.264.

160
§5 RADICAL THEORY
с) THE NIL RADICAL
5.51 DEFINITION The sum of all nil ideals of N is called the
nil radical of N and denoted by 7}(N) (by *I_i(N) in
Ramakotaiah (1) and Polin (2)). Cf. Gojan (1).
5.52 THEOREM
(a) 7)(N) is the greatest nil ideal of N.
(b)7)(N) is the smallest ideal I of N such that N/I
has no non-zero nil ideals.
Proof, (a): by 2.101(b).
(b): Let π: Ν ■+ N/«,...4 =: TT be the canonical
projection.
If Τ <| ff, Τ nil, look at I:= π_1(Τ) <| N.
If ιεΐ then 3 ke IN : тт(тк) = тт (i)k = TS (zero in IT),
hence ikeKer τ = 7)(N). But Y|(N) is nil, so
-i кг.
3 ЛеIN : (i ) =0 and i is nilpotent. Hence I is
nil, therefore I?7I(N) and we get Τ = {ϋ}.
Now assume that N/I is without non-zero nil ideals.
By 2.103, I+tt(N)/j is nil in N/I, so I+D(N)^I
and we arrive at 7)(N)?I.
5.53 COROLLARIES (Ramakotaiah (3), Meldrum (7))
(a) f) is a radical map (in the sense of 5.13).
(b)7)(N) s7I(N0) «e 30(N0) ΐ ^0(Ν).
(c) N is ΤΊ-semisimple iff N has no non-zero nil ideals.
(d) Each constant near-ring is 7l-semisimple.
Proof, (a): by 2.100 and 5.52(b).
(b): by 2.99, 5.52(a), 5.37(d) and 5.25.
(c): by 5.52(b).
(d): by (b).

5d The prime radical
161
It is clear that for rings fi(N) coincides with the usual nil
radical of rings. Ύ] is subhereditary on direct summands:
5.54 THEOREM (cf. Maxson (1)). If I <l N is a direct summand
then 71(1) s TUN) о I.
The proof follows from 2.12.
5.55 REMARK T)(N) is also identical with the "upper nil radical"
U = s(0) of Van der Walt (1). See this paper for a
characterization of T)(N) via "s-systems". See also
Be idleman (9).
d) THE PRIME RADICAL
5.56 DEFINITION The intersection of all prime ideals of N is
called the prime radical of N and denoted by ·9°(Ν)
(other notations: ^_2(N) , L-r(N), m(0)). Cf. Gojan (1).
Again, this is just the usual prime radical in the case of
ri ngs.
5.57 PROPOSITION f is a radical map.
Proof, (a) ^(N) <) N.
(b) If h:N -* IT and F 3 TT is prime then
P: = h (F) is prime in N by 2.64 and 2.17(a),
, as
showing that η(?»(Ν)) =Τ(Ν).
(с) If F is a prime ideal of IT: = ^tf>in\ tnen
in (b), тт~ (F) =:P is prime in N.
Conversely, if Ρ <l N is prime then π(Ρ) is prime
in тт.
Hence if "x is in each prime ideal of ΤΓ then each
xeh" ({"χ}) is in each prime ideal of N, so xeJ^N)
and χ is zero.
Therefore ^°(N/g»(fn) is zero·

162
§5 RADICAL THEORY
The connection to 2.93 is given by
5.58 REMARK ff(N) =tf»({0}) and this is a semiprime ideal.
5.59 PROPOSITION ff(N) is a nil ideal and contains the sum
of all nilpotent ideals.
Proof: by 2.105.
From this we can locate ^(N):
5.60 COROLLARY f>( N )?ϊ?( N )^Q ( N )^)1/г ( N)*^ ( N )«=JZ ( M)
(and all inclusions can be strict).
The first two inclusions can even be strict in the case of rings.
5.61 THEOREM Let ΝεΊ^ have the DCCN. Then
(a)T(N) = 0Г»(М) =^0(N).
(b)AJ2(N) is nilpotent (cf. 5.48) <=->f°(N) =^(N) =
- ... -12<N>·
Proof, (a) follows from 4.34 and 5.40(c).
(b): "<-" is trivial.
"->": follows from (a) and 5.48(d).
5.62 THEOREM (Maxson (1)). If I 3 N is a direct summand then
$·(!) ? fl»(N) П I.
This result follows from 2.63.
2.69 yields
5.6 3 EXAMPLE Each prime (e.g. each constant) near-ring is S° -
semi simple.
More generally:

5e Concluding remarks
163
5.64 PROPOSITION N is £*-semi simple iff N is isomorphic to a
subdirect product of prime near-rings.
This is a direct consequence of 1.58 and 2.67.
5.65 PROPOSITION Each ψ -semi simple near-ring has no non-zero
nilpotent ideals.
This follows from 2.104 or from 5.59.
See more on that in Scott (1), Holcombe (2) and Ramakotaiah-Rao (5).
e) CONCLUDING REMARKS
5.66 SUMMARY We summarize some properties of our radicals
(we include ^w? although it is not a radical map).
Radical Π *)г ~ ϊΐ ~ 3l/2? *30 ' * ? V
VL(H) quasireqular
K(N)2all quasiregular
N-subgroups
R(M)?all quasiregular
left ideals
щн) nil
R(N)?all nil
N-subgroups
fc(N)?all nil left
i deals
B(N)?all nil ideals
%(N)?all nilpotent
ideals
$(N) semiprime
ft(N) is the greatest
quasiregular left ide<
&(N) is the greatest
quasi reqular i deal
R(N) is the greatest
nil i deal
-
( + )
( + )
( + )
( + )
+
+
+
1 -
_
-
.
( + )
( + )
+
+
+
.
_
( + )
( + )
-
.
( + )
+
+
+
( + )
_
( + )
.
-
.
+
+
+
.
( + )
( + )
.
+
_
+
+
+
β
β
+
(♦)
.
+
.
-
+
+
.
.
_

164
§5 RADICAL THEORY
"+" means "yes"
II _ II
means
'no'
"(+)" stands for "yes, if
Νε7)0)" (otherwise
unknown to the author)
If Ne7)j has a minimal N -subgroup then all radicals are
+ N.
If ΝεΤ>0 has the DCCN then ^2(N) is nil iff all radicals
are equal .
5.67 SOME MORE REMARKS
(a) See Beidleman (1),(3),(8),(11) about the connection
between $2(N) and "(strictly) small" left ideals.
Similar considerations can be found in Riedl (l),Chao (1
and Mlitz (1), (2). They adopt a lattice-theoretic
point of view (the intersection of all maximal ideals
(...) = sum of all "small" ideals (...)). Cf. Oswald (5).
(b) Ramakotaiah (1) showed that each biregular near-ring
(3.49) is 0-semi s imple.
(c) The author suggests not to use the notations 'J.i·'!}.?
for 'Π and Ψ, respectively, because these are not
Jacobson-type radicals.
(d) Ramakotaiah (1),(3) also defines a radical "^.3(M)"
contained in ?°(N), as the intersection of all ideals
I such that N/I has no nilpotent ideals, it follows
from Theorem 8 of Van der Walt (1) that ^_3 =^.2·
See also the paper by Choudhari (1) for other
characterizations of Ρ(Ν) (such as the intersection of
all semiprime ideals (cf. Maxson (1)) and of ti(N).
(e) See Freidman (1), Bhandari-Saxena (2) and Plotkin (2)
for a "Levitzky-type" radical.
(f) Ramakotaiah (4) also defined a radical corresponding
to the Brown-Mc-Coy radical (^-radical) in ring
theory as the intersection <jf(N) of all maximal
modular ideals. See also Choudhari-Tewari (3).

5e Concluding remarks
165
If ζεΝ is called "G-regul ar" if the ideal generated
by {η-ηζ|ηεΝ} equals N then <$(N) turns out to be
the intersection of all ideals I of N, such that N/I
has no G-regular ideals. N/tt/fn nas n0 G-regular
ideals and is a subdirect product of simple near-rings
with a right identity.
(g) Laxton (3) defined one more "radical-1ike" ideal S(N)
of N as the intersection of all "s-primitive ideals".
He showed that ^1/2(Ν) «= S(N) «= ^(N) if Νε7?0
and gives an example of a dg. near-ring with
31/2(N) «= S(M) ·= ^(N). See also Beidleman (7),(8),(9),
Hartney (2),(4) and Meldrum (5),(13).
(h) Another radical was defined by Deskins (1) (see also
Williams (1)). If N = N has the DCCN then semi-
simplicity w.r.t. this radical is equivalent to
2-semisimplicity, and this in turn to semisimplicity
in the sense of Blackett (1), (2) (see 5.49).
(i) Beidleman considered in (2) the "radical subgroup"
R.(N) as the intersection of all maximal N-subgroups
in near-rings ε7)0 ·
By 5.38 we know that in this case RS(N) s ^f.(N).
Beidleman proved e.g. that R (N) =^}2(N) <=> ^ 2 (N)
is quasiregular (in his sense - see 3.37(c)).
Cf. 5.48(d).
(j) The "quasi-radical" Q(N) = f]L, where L ranges over
all maximal left ideals, was also considered (by
various authors). If N has a right identity then
Q(N) = ^}i/2(N) (3.29). This and more radicals can
be found in Choudhari (1).
(k) Gorton (1) called an N-group ^Γ to be of class X
(λ a non-zero cardinal number) if
V Δ?Γ, |Δ|<λ \/feM(r) 3 ηεΝ \/ γεΓ: f(y) = ηγ.
N is called λ-complete if N has a faithful N-group
of class λ. A radical CX(N) is defined as the
intersection of all (ο:Γ), where Г is an N-group of
class λ.

166
§5 RADICAL THEORY
He showed that if N is λ-complete on Γ then N =>„ r,
and that N is 1-complete iff N is faithful (a base
of equality - 1.91). If Ne7)Q then ^2(N) ^ C^N).
Defining C.-modular left ideals as those modular left
ideals L such that N/L is an N-group of class λ
(cf. 3.28) yields a result similar to 5.2.
Also, he gave several examples.
(1) Maxson (1) proved that there is not such a fine
connection between injectivity of N-groups (defined
as usual) and semi simplicity as in the ring-case.
He showed that if each N-group is injective then
22(N) = {0}, but gave an example that the converse
does not hold.
(m) Ferrero developed a radical theory for "p-singular
near-ri ngs" in (18).
(n) A radical (corresponding to *J2) for N-groups was
considered by Beidleman in (1), (3) and (4) and by
Choudhari in (1).
(o) Van der Walt (1) called an ideal I of N a nil radical
if I is nil, but N/I has no nil ideals any more.
He proved that the sum of all nil radicals of N equals
17(N), which is the greatest nil radical of N, while
the intersection of all nil radicals (the smallest
nil radical) coincides with ^(N). Therefore he
called 77 {f>) the upper (lower) nil radical of N.
(p) Mlitz (2), (3) and Polin (2) generalized this radical
theory to what they called "πι-Ω-near-ri ngs".
(q) See also other papers of Mlitz for a radical theory
in universal algebras. However, these radical concepts
turn out to be "too less near-ring-specific".
(r) Another attempt to get a radical theory for zero-
symmetric near-rings was made by Scott in (4).
He used a method similar to that of (Oivinsky) for
rings. As an example he studies the Baer-1ower-radical ,
which turns out to be = V{^) =*).^W f°r near-rings
with DCC on N-subgroups. Cf. Holcombe (3),(8) and Kaarli
(4),(7).

5e Concluding remarks
167
) It is easy to see that if N is a zero-symmetric near-
ring with identity and the DCCN and if e is some idem-
potent in N then \ (N)e = Ne ^^(N) for all
νε{0,1 ,2].
If Ne is a minimal non-ni1 potent N-subgroup of N and
if ^2(N) is n"ilP°tent then ^2^N^e ""s the 9reatest
proper N-subgroup of Ne, so Ne/-, i^\ is an N-group
of type 2 (it is harder to see that all N-groups of
type 2 arise in this way).
See Lausch-Nbbauer (1), where these results are
formulated and proved for dgnr.'s - but they are valid
in the general case.
) Let, for the moment,Μ denote the set of all "strictly
maximal" (cf. 3.29) ideals of n N„ (i.e. those ideals
of
N Nc
"o L
N N which are at the same time maximal N
-subgroups of N ). Routine arnuments nive the followinn
information on ( ^J „ (N ) )
If L e X 2 (N
2s
then
1?^'
L«V Nc
or L rt Ν ε Μ
(for if Msfj Nc contains LnH then L + M <,
vi h e η с е
.+M
N and Μ
Conversely, if Μ ε .Д then II tllt^JII)
Hence 40(Ν)λ N = Π м.
i2 C ΜεΛ
See also 5.12, 5.23, 5.24 and 9.77.
2Kn) η ι if
) In Meldrum (13) it is shown that 31(1)
I is a direct summand of N and that
Kl{ Θ Ν.) = Θ ZR.(N.). In here, 7Я = Ρ or 4 = 7).
ι'εΐ
ιεΐ
In Angerer-Pilz (1) it is shown that there exists a
near-ring N of order 32 with
>0(N) с ?1/2(N)= 2](N)c ?2(N)' and 32 is the smallest
order such that these four radicals are different.
(See also Meldrum (13)). Also, the following results
from Angerer (1) concerning radicals of "small" near-
rings are mentioned:

168
§5 RADICAL THEORY
(α) if (N,+ ) is simple then either ^(N) = i0}s
22(N) = N or J-1/2(N) = {0}, ^(N) = N.
(0) If |N| is the product of two primes or if (N,+)
is cyclic or non-abelian of order 8 then
Ϊ0(Ν) =2,(Ν).
(Ύ) If |N| is the product of three different primes
then ^Q(N) = 3-1/2(N).
(δ) If the normal subgroupsof (N,+ ) form a chain then
either 20(N) β ?1/2(N) с ^ (Ν) = #2 ( N ) = N or
VN) =*1/2(Ν)β *1(N) = >2(N) or
2o(N) = 2h/2(N) = 2MN) * ^2(N) or a11 radicals
coincide.
(w) See the "Near-rings of low order" in the Appendix for
the radicals of near-rings on most groups of order s8.

PART III
SPECIAL CLASSES OF NEAR-RINGS
§6 DISTRIBUTIVELY GENERATED NEAR-RINGS
§7 TRANSFORMATION NEAR-RINGS
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
§9 MORE CLASSES OF NEAR-RINGS
To keep this monograph within a reasonable size
we will only cite, but not give proofs of some
statements which lie a little bit away from the main
stream of discussion (but might be equally important)

170
§6 DISTRIBUTIVELY GENERATED NEAR-RINGS
In this paragraph we discuss these types of near-rings which
are still more "ring-like" than zero-symmetric near-rings.
In fact, every dgnr. is c\· If N is a dgnr. then the ideals
of J are exactly the normal N-subgroups, but this nice
feature does not seem to help a lot. For instance, all near-
ring radicals can still be different (even for finite dgnr.'s).
Abelian dgnr.'s are rings.
We also discuss the open problem of embedding a zero-symmetric
near-ring into a dgnr. .
In the case of near-ring homomorphisms those ones deserve
particular interest which carry the distributive generators
into the ones of the image. These "(N,D)-(N',D')-homoiTiorphisms"
are characterized. Although the dgnr.'s form no variety, it is
possible to speak about "free near-rinqs distributively
generated by a given semigroup". N-groups Г are studied which
have the property that the distributive qenerators of N act
"distributive" (= as endomorphisms) over Г.
Finally we study the structure of dgnr.'s: 2-primitive finite
dg. non-rings with identity are just the Μ (r)'s for a
finite, non-abelian invariantly simple group Γ. In the finite
case, MQ(r) = Ε(Γ) iff Γ is of this kind.

6a Elementary
171
L±
ELEMENTARY
N is distributively generated (dg. , better: distributively
generable) if there is a subsemigroup D of (Nj,·) generating
(M.+).
6.1 NOTATION If D generates N we denote this by (N,D).
6.2 EXAMPLES (see Holcombe (3) for generalizations)
(a) If (Γ,+)ε^, define Ε(Γ) by the set of all finite
sums Σσ-e^, where σ^εί-Ι, + Π and e^eEnd Γ.
Ε(Γ) is a subnear-ring of Μ(Γ), distributively
generated by (End Γ,ο) and called the "endomorphi sm
near-ring on Γ" (see 1.15).
(b) (H. Neumann (1), (2); Frbhlich (1), (2)).
Let (Γ .+) be a reduced free group with generators
{e,,...,e } =: Ε (i.e. each map Ε ■+ Γ can uniquely
be extended to an endomorphism on Γ ; Γ is then
the free group in some variety of groups).
Dei
by
Define for the set End Γ two binary operations Θ,
(Φχ © Ф2)(е1): = Ф1(е1)+ф2(е1.)
(φ, · Ф2)(е·): = ф,(Ф2(е.)) (and extend from Ε
Γη)·
Then (End Γ ,®,·) turns out to be a dgnr., generated
to
Tht
by
0: = {Ф1Л1Ф1Л(ек)
L о i +
}
Remark that + and θ are different if
abeli an , for e.g.
i s not
(ф1+ф2)(е1+е2) = ф1(е1+е2)+ф2(е1+е2) = Ф1(е1)+ф1(е2)+
+ф2(е1)+ф2(е2), while (Ф1®Ф2)(ej+e2) = (Ф1®Ф2)(е1)+
+(ф1®ф2)(е2) = ф1(е1)+ф2(е1)+ф1(е2)+ф2(е2).

172
§6 DISTRIBUTIVELY GENERATED NEAR-RINGS
In Frb'hlich's papers, + is referred to as the
"addition of the first type" and © as the "addi tion
of the second type".
(c) Similar to (a), the near-rings A(r) and Ι(Γ),
defined as the subnear-rings of Μ(Γ) generated by
the automorphisms (inner automorphisms) of (Г,+),
are dgnr.'s.
6.3 REMARKS
(a) Е(Г), А(Г) and Ι(Γ) will be studied in §7c).
(b) The End Γ 's were introduced and studied by H.
Neumann in (1) and (2). Her results on these types
of near-rings include:
End Γ contains no identity, but all φ fixing some
e. and sending the other e.'s into zero are
distributive and can be viewed as "relative units".
There is a 1-1-correspondence ψ between the set J
of all fully invariant subgroups of Γ and the set
In of all ideals of End Γρ by
ψ: *n + Xn
A - {φ | V ιε{1,. . . ,n}: «(e^eA}
All homomorphic images of End Γ are also some
End Γ 's. Each End г is the homomorphic image
m η r η
of End Φ , where Φ is the free group on η
generators.
Similar results hold for the near-rings of the kind
® End Γ , which are also dg. (see 6.9(d)).
ηεΙΝ
(c) See Fitting (1) for the problem, which automorphisms
of a (non-abelian) group have the property that their
sum is an automorphism again. Cf. also Heerema (1)
and Robinson (1) for similar questions.
(d) See Plotkin (2) for the connection between the
representations of a group Г and those of Е(Г).
(e) See all papers of Dasic for generalizations of the
concept of a dgnr.. Cf. also Meldrum (13).

6a Elementary
173
Now we study some elementary properties of dgnr.'s. Note, that
if N is dg. by D then each ηεΝ is a finite (ordered) sum
η = Eoidi with o.j = ±1, d^D.
6.4 PROPOSITION Let N be dg. by D.
(a) V ηεΝ V deD: d(-n) = (-d)n = -(dn).
(b) Νεη,.
(c) V η,η'εΝ V dεD: d(n+n') - dn+dn' Λ (-d)(n+n') =
= (-d)n'+(-d)n = -dn' - dn.
(d) If η = b.d. and n' = Taldi then
The proof is accomplished by easy computations and
therefore omitted.
6.5 PROPOSITION (Seth-Tewari (1), Meldrum (13)). Let N be dg.
by D and Γ an N-group with diy+y1) = dy+dy' for all dεD,
γ,γ'εΟ. If Δ£Γ then the N-ideal Δ generated by Δ is given
by all finite sums of the form Σ (γ · +σ·d . δ-γ·) with γ·εΓ,
σ i ε { 1 ,- 1 }, d.ε D and δ ε Δ.
Proof. The set of all finite sums of the form Eo.d.j. is a
subgroup Δ_ of (Γ, + ). Έ is then just the usual normal
closure of ^ i η (Γ , + ).
To see that Έ <h, Γ, consider η(ό"+γ)-ηγ, decompose
η as η = ^oidi and I as 6" = ΐ {r^*a^a\ &j "Yj ) and
proceed as usual. (The next result shows that it
suffices to show that N&=£.)
See Meldrum (13) that 6.5 is not valid without the d(y + y' ) = dy + dy'-
assumpti on.
Near-rings generated by an inverse semigroup of distributive
elements are treated in Mahmood-Meldrum-0 ' Carol 1 (1) and Meldrum (13).
Examples of d.g. near-rings of low order ca" be found in the
appendi χ.

174
§6 DISTRIBUTIVELY GENERATED NEAR-RINGS
b) SOME AXIOMATICS
6.6 PROPOSITION Let N be dg. by D and Γ be an N-qroup.
(a) If Δ is a normal subgroup of (Γ,+) then
Δ <!Ν Γ <=> Δ <Ν Γ.
(This is one step towards the situation in rinqs,
since the ideals of МГ are just the normal N-sub-
N
groups.
,2
(b) Η is abelian <=> N is distributive. +)
(c) N is abelian <=■> N is a rinq.
(d) If ΗεΎί then N is distributive <=> N 1s abelian <=>
<=> N is a ring.
к
Proof, (a) If η = Ι σ^εΝ, γεΓ and 6εΔ <„ Γ then
к к
η(δ+γ)-ηγ = J σ^^ό+γ)- J σ^γ = a1d1(i+y) +
+...+okdk(6+Y)-okdkY-...-a1dlY.
Since dk(5+Y)-dkY = d^ + d^-d^ = d^Seb and (usinc
6.4) {-dk)(6+Y)-(-dk)Y = (-dk)Y+(-dk)5-(-dk)Y =
= -dkY-dk6 + dkYcA, we see that in any case
CTkdk(6+Y)-okdkYEA.
Proceeding in this way yields Δ <!^ С.
The converse follows from 1.34(b) and 6.4(b).
(b) =>: If N is abelian then for η,η',η"εΝ,
η = Σσ-d. we get n(n'+n") = Σσ^. (n '+n") =
= Σσ . (d .n '+d . η") = Σσ .d.n'+Ea.d^n" = nn'+nn".
<=: Conversely, the proof of 1.107(c) shows that
2
N = Nd implies N to be abelian.
2
(c) =>: If N is abelian, the same applies to N .
So N is distributive and therefore a rinq.
<—: trivial.
+ 2
)"N abelian" stands for "Va.b.c.deN: ab + cd = cd + ab".

6b Some axiomatics
175
<d) follows from (b) and (c).
The next result examines the role of identities in dynr.'s.
6.7 THEOREM (Ugh (1)). Consider the dgnr. (N,D).
(a) If D contains a left (right, two-sided) identity e
then e serves as the same for N.
(b) If N contains exactly one left (or right) identity e
then e is two-sided.
Proof, (a) If e is a left identity of D then
\/ η = Eo-d^eN: en = βΣσ^ = Σβ(σ^) = Ea^ed^) =
= laid- = n, and similar for right identities.
(b) Assume that e is the only left identity of N. Then
V ηεΝ V χεΝ: (ne-n+e)x = nex-nx+ex = nx-nx+x = x,
hence V ηεΝ: ne-n+e = e. Therefore V ηεΝ: ne = η
and e is two-sided.
Suppose now that e is the unigue right identity of N.
Again, take η and χ = Σσ^ arbitrary εΝ.
x(en-n+e) = Eaidi (en-n+e) = Σσ^-e = (Ea-d^e = xe.
As above, en-n+e = e, so e is again two-sided.
6.8 REMARKS
(a) Observe that 6.7(b) holds for general near-rings in
the "left-case". But see Ligh (1) for examples that
in all other cases 6.7 does not hold for general
near-ri ngs .
(b) See Ligh (12) for a proof of "A finite dgnr. N is
commutative <=»> all zero divisors are central". This
does not hold in the infinite case.
(c) Ligh (10) gives characterizations of all dgnr.'s N
with (N,+) = Sn (n > 5) or (N.+) a dihedral
group D2p (pcP\{2}).
(d) If (N,+ ) is nilpotent and N dg. then N/*t2(N) is a ring
(Gringlatz ( 1 ) ).
(e) See Feigelstock (1),(2) for simple dgnr's.

176 §6 DISTRIBUTIVELY GENERATED NEAR-RINGS
c) CONSTRUCTIONS OF DISTRIBUTEVELY GENERATED NEAR-RINGS
Dgnr.'s have no particular stench, so it is not quite easy
to recognize them amonq other nr.'s. The next result miqht
help irt some cases.
6.9 THEOREM
(a) If Μ ^ N and Μ is dg. then N is not dg. in general.
(b) If Μ < N and N is dg. then Μ is not dg. in general.
(c) Every homomorphic image of a dgnr. is dq. .
(d) Every direct sum of dgnr.'s is a dgnr. .
(e) Every direct summand of a dgnr. is itself dg..
Proof, (a) Take some ΝεΆ with N 4" N · Then N is not
dg. by 6.4(b) , but M: = {0} is
(b) See Lyons-Malone (1) for an example of a subnear-
ring of Ε(S^) which is not dg. .
(c) If h: N -·+· N' is an epimorphism and if N is dg.
by D then a routine check shows that h(D) έ N^ and
h(D) generates N1.
(d) Let the near-rings N. (i ε I) be dg. by
Ο,- =: ί d i j | j ε J ^}. Define ^ .: = (. . . ,0 ,di j ,0,. . . )e
ε @ N1 = : Ν, where d^ stands in the i-th
i εΐ
component. Put {Ϊ..|ΐε1 Λ jeJj} =: D.
It is easy to show that D = N..
If η = ( . . . ,n · ,. ..)εΝ, decompose each п.. as
п. = У σ. . d. . , where each j.-eJ,- and the sum
i ^ ij1 U,- ι ι
is a finite one.
Then η = Υ У σ · · d". i (this is anain a finite
lei J, 1Ji Ui
sum). Consequently, D generates (N,+).

6c Constructions
177
(e) follows from (c).
From 6.9(b) we see that the class of all dgnr.'s is no variety.
Hence "distributive generation" cannot be defined by "equations"
(see (Gratzer)). This brings up the question about the smallest
variety which contains all dgnr.'s (the variety "generated" by
the dgnr.'s). We state without proof (it uses a lot of universal
a!gebra)
6.10 THEOREM (Meldrum (1)). The following varieties coincide:
(a) The variety generated by all nr.'s of the type Ι(Γ).
(b) The variety generated by all dgnr.'s.
(с)П0.
So every zero-symmetric near-ring can be embedded into some
"descendant" of a dgnr. (see (Gratzer), pp. 152/153). For
finite nr.'s Νε?λ we get more:
6.11 THEOREM (Malone (6)). Every finite zero-symmetric near-
ring can be embedded into Ι(Γ) = Ε(Γ) = Μ0(Γ), where
Г is a certain finite non-abelian simple group.
Proof. Embed N into some Μ (Γ) (l.B8(c)) with n>3;
embed Γ into some S (e.g. via Cayley's theorem).
Next, embed S into the alternating group A ? in
the following way:
(1) If nt? i s even , let
?: = ( 1 2 η n+1 n+2 ) .
π(1) ττ(2) . . .π(η) n + 1 n + 2
(2) If πεΡη is odd, let
-. _ / 1 2 η n+1 n+2 ),
π(1) π(2)...π(η) n+2 n+1
The map h: IP * A _ is a group monomorphi sm, hence
π -* ff
an embedding map. A _ is a finite simple non-

178
§6 DISTRIBUTIVELY GENERATED NEAR-RINGS
abelian group containing properly a homomorphi с imaae
of Sn. By 1.99, Nc»M0(r)e»M0(An+2). We will see
in 7.46 that M0(An+2) = E(An+2) = I(An+2).
6.12 COROLLARY (Malone (6)). Every finite Νε7)0 can be
embedded into a finite dg. non-ring with identity.
Several questions remain open:
Is every zerosymmetriс near-ring embeddable into some dgnr.?
Is every dgnr. embeddable into some dgnr. with identity ?
Is every E(r) embeddable into some Ι(Γ)?
Is every dgnr. embeddable into some Ε(Γ)?
And so on. Cf. also Heatherly-Malone (2), l .90 and 6 .35(к).
6.13 REMARK In (1) and (3), H. Lausch developed an extension
theory (via homological algebra) for dgnr.'s. For group
dgnr's see Mahmood (1) and Meldrum (4),(13). Categorical
considerations are in Mahmood (1)-(4) and Mahmood-Meldrum (1)·
See also Heatherly (12), John (2).
d) DISTRIBUTIVELY GENERATED NEAR-RINGS WITH FINITENESS
CONDITIONS
6.14 THEOREM (Ligh (3)). Let N + {0} be a dgnr. with DCC on
monogenic N-subgroups. Then
(a) NeTJj <=> N* contains an element which is no divisor
of zero.
(b) V ηεΝ*:(Νε'Μ, and neN is invertible) <=> η is no
zero divisor.
Proof, (a) "=->" is clear. So assume that χ (+ 0) is not
a divisor of zero. Now Nx?Nx э... . Therefore
3keIN: Nxk = Nxk + 1 = ... . This implies that
3eeN: x-xk = e-xk+1. So (x-ex)xk = 0.
Hence x-ex = 0 and we get ex = x.

6d Finiteness conditions
179
Also, (xe-x)x = 0 and thus xe = x.
So Μ meN: (me-m)x = 0 whence me = m.
Now take some arbitrary ηεΝ. Decompose χ as
χ = Σσ-d·. Then x(en-n) = Za-d^(en-n) =
= Eo^d-n-d.n) = 0, implying that en = n.
(b) "->" is clear again.
Let η =)■ 0 be no zero divisor. Then Ucfl, by (a).
As in (a) , 3 keIN : Nnk = Nnk + 1 = ... .
So 3 πιεΝ: nk = lnk = m-nk+1. This implies that
(l-mn)n =0, so 1 = mn. Also, (nm-l)n = 0,
so nm = 1 and η is invertible.
6.15 REMARK (Liqh (13)). If N is a finite simple dg. near-ring
then (N,+) is a perfect group (i.e. N coincides with its
commutator subgroup). See also Feigelstock (2).
There are several connections between chain conditions,
solvability of (N,+) and "weak distributivity" (see
Frb'hlich (1), Oef. 4.3.1). We state without proof the
following collection of results (see also Beidleman (11)).
6.16 THEOREM Let N be a dgnr.
(a) (Frb'hlich (1)). If (N, + ) is solvable then N is
2
weakly distributive. If N = N, the converse also
holds. See also Mason (1).
(b) (Beidleman (4)). If N is finite and if Μ^ε,.^ then
(Γ.+) is solvable iff ^Γ is solvable (i.e. Г has
a normal sequence (2.37) with abelian quotients).
(c) (Beidleman (4)). If (N,+) is solvable and N has
the DCCN then емегу maximal left ideal is modular and
contains the commutator subgroup of (N, + ). 'JJ?^
is nilpotent and N/* ,,,, is a rina. Also, N has a
certain kind of ACC.
(d) (Ligh (3)). If (N,+) is solvable such that not all
elements are divisors of zero. Then the DCCL implies
the ACCL.

180 §6 DISTRIBUTIVELY GENERATED NEAR-RINGS
e) "FREE" DISTRIBUTIVELY GENERATED NEAR-RINGS
Since the dgnr.'s do not form a variety, there is no guarantee
for the existence of "free dgnr.'s". Moreover, this concept
does not seem to be appropriate for this class of near-rings.
We are now going to define a similar concept. First of all we
need a "refined" version of homomorphisms between dgnr.'s.
6.17 DEFINITION Let (N,D), (N'.D') be dgnr.'s. A homomorphism
h: N -*■ N' is called an (Ν,Ρ)-(Ν' ,D' )-homomorphi sm if
h(D) ? D'.
6.18 EXAMPLE Each dgnr.-homomorphi sm N ■* N' is an (N,N.)-
-(N',Ni)-homomorphism.
6.19 PROPOSITION (Frbhlich (2)). Let (N,0), (N'.D1) be dg.
and let h: (N, + ) ·* (N', + ) be a group homomorphism and
a semigroup homomorphism (D,·) ■+ (D1,·)·
Then h is an (N,D)-(N',D')-homomorphism.
Proof. It only remains to show that V η,η'εΝ: h(nn') =
= h(n)h(n').
Let η = Σσ^ύ^, η' = Σσ ^ d t. Then, using 6.4(d),
h(nn') = h(Jaf (Jajd^j)) = ^ (Jaj h (d i d^-)) =
= ^i(^jh(di)h(dj)) = (Jaih(d1 )) - (Σ<» j h (dj )) =
= h(n)h(n').
We are now going to define something like a "free near-ring
dg. by a given semigroup (D,·)".
We use a slight modification of a method due to Frb'hlich (4)
and Meldrum (2). Cf. also Zeamer (1).
6.20 DEFINITION Let (D,·) be a semigroup and V a variety
of groups. Denote by (Fr, v, + ) the free group in V on
(the set) D.

6e "Free dgnr.'s"
181
FD у consists of all finite sums Ζσ^ά-, where equality is
determined by V. If e.g. V = Ц, then "equality" is "formal
equality".
Defining (Εσ^ ) ■ (Eojdj ) : = Jo 1 (Jaj d i d j ) yields
6.21 THEOREM Let ^ be a variety of groups.
(a) · is wel1-defi ned.
(b) (,rr)v»+»*) =: F is a nr·» d9· ЬУ D· whose additive
group belongs to V .
(c) For every dgnr. (N'.D1) with (N',+) ε У every semigroup
homomorphi sm D -+■ D' can uniquely be extended to a
(F,D)-(N ', D ')-homomorphism.
(d) Every dgnr. (N,D) with (N,+) ε V is a (F,D)-(N,D)-
homomorphic image of (F,D).
Proof, (a): holds by the definition of equality via laws
in V.
(b): By a routine but somewhat nasty calculation one
sees that (^n«>+·') 1S a near-ring. By construction,
F is free over D iη V, so {F , + )cV and D generates
(F.+).
(c): By definition, every map f:D -♦ D' can uniquely
be extended to a homomorphism h:(F,+) ■+ (N,+).
If f is moreover a semigroup homomorphism, h is an
(F.D)-(N',D')-homomorphism by 6.19.
Considering the diagram (ι is the inclusion map)
and remembering group theory (or
making a routine diagram argument)
gives the information that h is a
group-epimorphism. Now h/D = idD>
whence h is a (F,D)-(N,D)-epi-
morphism by 6.19,
See also John (1), Mahmood (l)-(4), Meldrum (13) and
Rhabari (1),(2).
Representations of groups via free dgnr.'s are studied in
Meldrum (4) and (13).

182
§6 DISTRIBUTIVELY GENERATED NEAR-RINGS
f) D-GROUPS AND ( N , D) -GROUPS
Like nr. homomorphisms of dgnr.'s, the concept of N-qroups can
be "refined" for a dgnr. (N,D): we want the elements of 0 to
"distribute over Γ" (this appeared already in 6.5).
6.22 DEFINITION Let (N,D) be a dgnr. . ΝΓε^ is called
an (N,D)-group if V γ^,γ2εΓ V d£D: d(y1+Y2) = dYj+dY2.
6.23 DEFINITION Let (D,·) be a semigroup and (Γ.+) a group.
Γ is called a D-group if a multiplication ·: Οχ Γ ► Γ
(d,Y) -* dy
is defined with V γ^,γ2εΓ \/deD: d(Y^ + Y2) = dY1 + dY2>
6.24 REMARK So if (N,D) is dg. then ΝΓ is an (N.D)-group
iff Γ is a D-group (w.r.t. the restricted multiplication
of ΝΓ).
Now let (D,·) be a semigroup and FD w be the "free nr. dq.
by D in V" as in 6.21, where V is some variety iη Ц.
6.25 THEOREM Every D-group TcV is an (FQ ^,D)-group.
Proof. If Ea^eFg ^ and γεΓ« define (la^d-)y. -
= Σσ,- (d.Y).
Again this is well defined and checking the (Ρηφ'^)"
group axioms creates no problem.
Again, let 11 be a variety of groups and (N,D) a dgnr.
We consider the class -N Q. y§ of all Γεί? which are
(N,D)-groups.
Let Ω be the family of operations
(+,0,-)u (ωη)
η'ηεΝ
of type
(2,0,I)w(l)neN .
Let^Cbe the class of (universal) algebras of this type.

6f D-groups and (N,D)-groups
183
Let СЛ be the variety determined by all laws which define, for
Tcj{, (Γ, + ,0,-) to be a group zV and by all laws
(ωη+η,(x). ωη(χ)+ωη,(χ)) (η,η'εΝ)
(ωη(ωη,(χ)) , ωηη,(χ)) (η,η'εΝ)
(Ud(xi+xj), ω(1(χ.)+ω£|(χί)) (dcD).
Then clearly
6.26 THEOREM <J = (H . „(J; so the latter class is a variety.
From universal algebra we now get
6.27 COROLLARIES There exist all free (N,D)-groups; they are
unique up to (N ,D)-isomorphisms; each (N.D)-group is
the (N,D)-(N ,D)-homomorphiс image of a free (N.D)-qroup.
6.28 REMARKS
(a) Meldrum (2) used these "free nr.'s dg. by D in V "
and the free (N,D)-groups in a suitable non-abelian
variety V to show that not every dgnr. (N,D) has a
faithful (N.D)-group (not even in the finite case).
Therefore not every dgnr. (even not every finite dgnr.)
can be embedded into some Е(Г) in such a way that
all dcD remain distributive on Г (= become
endomorphisms on Г).
Observe that we know from 6.11 that every finite dgnr.
can be embedded into some Е(Г), if one does not insist
that all deD remain distributive.
Meldrum also constructed in (2) "nearest" dgnr.'s
(N.D), (J4. D) with faithful (N ,D) - ({Η,ΰ)- )groups
such that (N,D) is a (N,D)-(N,D)-homomorphiс image
of (JT.D) and (^,D) is a (N,D)- (H ,D)-homomorphic
image of (N,D).
Moreover he considered the "Dorroh-type" adjunction
of an identity 1 to a dnnr. (N,D) (one has to
adjoin 1 to D) (cf. (Kertesz), Th. 3.13).
See also Meldrum (7), (10)-(13).

184
§6 DISTRIBUTIVELY GENERATED NEAR-RINGS
(b) For more information on (N.D)-groups see Frohlich
(2), (4). In (4), Frohlich described free sums and
products, orthogonal sums, free bases and projectivity
in the case of (N ,D)-groups. It turns out that the
situation is similar to the ring (-module) case.
(c) Frohlich also studied categories of N- and (N,D)-
groups in (5) and developed a "non-abelian homological
algebra" via these groups in (6) - (8).
g) STRUCTURE THEORY
We start with a result on generators in N and N/I.
6.29 THEOREM ((GaschUtz), Lausch (4)). Let Ν (ΝΓ) be fg.,
N a dgnr, and Ι (Δ) be a finite ideal. Moreover, let
N/I (Γ/Δ) be the N-subgroup generated by {ϊί, ,. . . ,?k>.
Then V ie{l,...,k} Зе^е?·: {ej ek} generates
the N-subgroup Ν (Γ).
Proof. As in (Gasch'jtz) (where it is proved for groups;
this proof carries over to groups with operators -
see Lausch (3)).
This result can be used to prove 6.31:
6.30 DEFINITION If ΝεΜ. then I(N): = {ηεΝ|η is invertible
+)
in (N,·)} denotes the "group kernel" of (N,·).
6.31 THEOREM (Lausch (3), Lausch-Nobauer (1), Scott (1)).
Let NeWj be a finite dgnr. and let h:N ■+ Л be a nr.
homomorphism. Then
h(I(N)) = I(h(N)).
+ )
The elements of I(N) are also called the "units" of N.

6g Structure theory
185
Proof, (a) If iel(N), then 3 jeN: ij = j i = 1. Hence
h(i)h(j) - h(j)h(i) = h(l). so h(i)eI(h(N)).
(b) Conversely, if ΤεΙ(η(Ν)) then {T} generates
the N-subgroup h(N): take h(n) = [aih(di)eh(N)
and 7 = ^h(d^) with J-T = h(l). Then
(^Ja^h(d.)h(d^))T = h(n).J-T = h(n).
So by 6.29 there is some i ε I with h(i) = Τ and
such that the N-subgroup generated by i eguals N. So
there is some jeN with ji = 1. Hence by 1.113,
i is invertible.
6.32 REMARK See Lausch (4) for some more general versions of
6.31.
Next, we visit primitive dgnr.'s. with OCCN and get
6.33 THEOREM (La/ton (2)). Let ΝεΜ(Γ) be a finite dg. non-
ring with a left identity. Then the following conditions
are equivalent:
(a) N is 1-primitive on Γ.
(b) N is 2-primitive on Γ.
(c ) N is s imple.
(d) Ν = Μ (Γ) and moreover Γ is a finite, non-abelian,
invariantly simple group.
Proof, (a) <-> (b) <-> (c) follows from 4.47(a).
(b) => (d): Assume that N is 2-primitive on Γ. By
4.6(b), N has an identity. By 4.60, N = Мго(Г). If
Γ is abelian, N is abelian by 1.49, so by 6.6(c) N
is a ring. Hence Γ is non-abelian. Since N is finite,
the same applies to Г. Г is monogenic, so
3 γοεΓ: Ν/(ο:γ ) "Μ Γ Ьу 3-4<е)· So every deNd
is an endomorphism of Г. Since ,,Γ is N-simple, it
cannot contain a non-trivial subgroup invariant under
all deN^, whence Γ shows up to be invariantly simple.

186
§6 DISTRI8UTIVELY GENERATED NEAR-RINGS
Now Aut„(r) is finite and fixed-point-free, so it
consists either of {id} alone or contains a fixed-
point-free automorphism of prime order. The paper
(Thompson) tells us that (r,+) is nilpotent. But
Γ is invariantly simple and therefore abelian, a
contradiction. So AutN(r) = {id} and Μ (Γ) = Μ (Γ).
(ι
(d) =-> (b): If Ν = Μ (Γ) then N is 2-primitive on
Γ by 4.52(b).
This theorem has some interesting conclusions (see 7.46).
We now collect some results concerning radicals of related d.g.
near-rings. Proofs and more on this can be found in Kaarli (3)
and (4).
6.34 THEOREM (Kaarli (4)). Let N be a dgnr, I3N and Η <N N.
(a) If qel is quasiregular in I then q is quasiregular in N.
(b) 2υ(Μ) 2}ν(Ν) η Μ for v = 0 and υ = \.
(c) If 11 й Ip s ... й Ik 3 N and ISI, then
6.35 REMARKS
(a) Surprisingly (or unfortunately), 6.6(a) does not force
the various radicals of a dgnr. to coincide (not even
for finite dgnr.'s). See several papers of Laxton and
Beidleman. Also, ^2(Ν) is not necessary nil in this
case. See also Scott (11).
(b) For dgnr.'s ΝεΤ?, , Beidleman (8) defined "strictly
primi ti ve" ideals as 2-primitive maximal ideals.
The intersection of these ones contains 7?(rj) anc*
equals ^2^') ΐη the case of DCCN (this follows from
4.47(b)).
(c) Laxton (3) contains an example of a finite dgnr. U with
the property that Jf,/2(N) is no ideal, while N has
nilpotent left ideals (but of course no nilpotent idea!;.

6g Structure theory
187
(d) Deskins (2) contains more information on the eNe's,
where eeN is some idempotent.
(e) In (4), Tharmaratnam calles a dgnr. N a division dgnr.
if EnCL(N,+ ) = AutN(N ,+) о {5}. A finite dgnr. without
non-trivial left ideals is a division near-ring, for
instance. For a finite division dgnr. N which is not a
ring there is some finite, non-abelian simple N-group Г
with N=M (Г). This establishes a 1-1-connection between
isomorphism classes of finite division dg. near-rings.
(f) See Tharmaratnam (1 ) , (2 ) , ( 3 ) and (4) for "topological
dgnr.'s": a topological nr. N (def. as usual - see
Beidleman-Cox (1)) is called a topological dgnr. i f
N. generates N topo1ogiс a 11 у .
If the topological nr. N is a dgnr. then N is a
topological dgnr., but the converse does not hold in general.
Tharmaratnam also described topological (N,D)-groups
and the structure of topological dgnr.'s, especially
that of a 2-primitive complete topological dgnr..
(g) See Laxton (4) and Laxton-Machin (1) for the behaviour
of prime ideals in dgnr.'s.
(h) Plotkin (1),(2) transferes the concept of a dgnr. to
uni versal algebra.
(i) See also §7 с ).
(j) N is called a generalized dgnr. (gdg.nr.) if N.oN
generates (N,+). Dgnr.'s, constant nr.'s and many
polynomial nr.'s are of this type. The variety generated
by all gdg.nr.'s is Ύί . Every finite nr. can be embedded
in a finite gdg.nr.. For finite gdg.nr.'s N with identity
and N not a ring, "1-primitive", "2-primitive", "simple"
and "N = M(o)(r^" witl1 r finite, invariantly simple,
non-abelian are equivalent. See Pilz-So (3).
(k) By 5.19(a) and 6.34, not every Ν ε% can be embedded in
a dgnr. as an ideal.

188
§7 TRANSFORMATION NEAR-RINGS
This chapter contains results on near-rinqs of group mappings
(the "elements of near-ring-theory" of 4.62) and of near-
rings which are related to these (§7 d)). We will mainly be
concerned with the ideal structure of these classes of near-
ri ngs .
We start with Μ°(Γ): = MH ί6}(Π = (ΜΗ(Γ))ο, where Η is some
fixed-point-free group of automorphisms of the (additively
written) group Γ. Mu(r) is shown to fulfill all conditions
of 2.50 iff Η has finitely many orbits on Γ. In this case,
Ми(П is simple and the finite topology on Μ°(Γ) is discrete.
We also answer the question, under which conditions Μμ (Γ,)
1
and M„ (Γ~) are isomorphic, using semi-linear transformations
as in ring theory. The automorphism group of the M°(r)-qroup
Γ is just Η itself.
Turning to Μ (Γ) in b) we show that for Μ (Γ) the followina
are equivalent: all conditions of 2.50, ACCL, DCCL, finite
generation and finiteness of Γ. All minimal left ideals of
MQ(r) are shown to be the (ο:Γ\{γ)) for γεΓ*. The maximal
ones are all (ο:γ) (γεΓ*) and some others, which are less
easy to characterize. Concerning ideals we show that Μ0(Γ) and
(if |r| «f 2) М(Г) are simple near-rings. There are no subnear-
rings strictly between Μ (r) and Μ(Γ).
In c) we study mainly Е(Г). Е(Г) is 2-primitive on Г iff Г
is invariantly simple. In this case, Ε(Γ) = Μ (Γ). Similar
results are obtained for A(r) and Ι(Γ). Ε(Γ) has all
conditions of 2.50 if Γ is the direct sum of finitely many
minimal fully invariant subgroups. Ε(Γ) is simple iff Γ is
invariantly simple. Aut I(r) - r.
Finally we study near-rings of polynomials R[x] or Γ[χ] over
a commutative ring R with unity or a group Γ and their associated
near-rings P(R), Ρ(Γ) of polynomial functions. We show that

7a MS (Γ)
189
P(R) ■= M(R) iff R is a finite field. If F is a field, F [x]
is simple iff F is infinite. If F is finite, but + Zz, F [xj
contains exactly one maximal ideal: (peF[x]|p induces the zero
function on F}. If F = z2» there are exactly 2 maximal ideals.
If F is finite but char F + 2» each ideal of F [x] is a
ring-ideal of F[x] and hence quite well-known, provided
that char F + 2.
P(R) is simple iff R is a field 4" гг and г Μ is simP1e
iff |Г|>1. Г[х] = М(Г) holds iff Г = Z2 or Г is a finite,
non-abelian simple group. We continue and close with nr.'s
of polynomials and polynomial functions on Ω-groups.
a) Mg(T)
Now we are going to decompose Μ,,(Γ), where Η is some fixed-
point-free automorphism group of (Γ.+).
In contrast to 3.43, we first decompose the identity and then
get a decomposition of Μ„(Γ). Before doing so, we have to
fix some notation. A categorical approach is in Holcombe (4).
7.1 NOTATION Throughout this section 7a) let Г be a non-zero
group and Η a fixed-point-free automorphism group of Γ.
Let Γ = {о} у U B · be a partition of Γ into a disjoint
ιεΐ λ
union of orbits of Γ under H.
Denote (for i ε I) by e^ the uniquely determined map
(4.28(a)) of ΜΗα{ό}(Γ) with
ίγ for γεΒ.
1
о for γ^Β1
(so e. is like the identity in B. and о elsewhere).
Moreover, we abbreviate sometimes M„ ,.,(Γ) by Μ„(Γ)
or simply by M.

190
§7 TRANSFORMATION NEAR-RINGS
7.2 THEOREM (Betsch (7)). As promised, let Μ°(Γ) =: Μ.
(a) {e . | i ε I > is a set of orthogonal idempotents.
(b) All Me, = Π (o:B,) =: L, are left ideals and M-
j+1 J
groups of type 2 which are M-isomorphic to г and fulfill
γεΒ1
y*b1
LiY -
f Γ if '"1
< ; e. is a naht identity
I. to) if -1" 1
for L,
<«>«*„ J L,
(d) If L: = J L, = l'l, then L = M°(Γ) iff Η has
ιεΐ Ί ίεΐ Ί и
finitely many orbits on Г; in this case, 1 = У е,.
lei
(e) Every non-zero invariant subnear-rinq S of Μ contains L.
Proof, (a) is established by an easy computation.
holds because of
(b) Me.γ =
f Γ if γεΒ.
Ι ίο) if γψΒ.
4.28(a), whence Me. ί Ο (ο:Β·).
j'+i
Conversely if me Π (o:B.) tnen m = me.eMe.,
j+i J
so Me, = П (о:ВЛ. By 1.43(a), L.· <L M.
1 j+i J 1 *
The map f : Me, ■+ Г is an M-epimorphism for
me, ■+ me-γ
γεΒΓ
Ker f = Me^iory). But (ο:γ) = (o.'B^, so
Ker f = Π (o:B4) = {0}, and f is unmasked to
γ jel 3 γ
be an M-isomorphism from L- to Γ. Since Μ„(Γ)
is 2-primitive on Γ by 4.52(b), the same applies
to Ц \ Γ.
(c) is settled by the M-i somorphism f: Ми(Г) ■+ π Ц
sending m into (...,me· ,. . .)
iel

7a Μβ(Γ)
191
(d) IT L - = ® L- holds iff I is finite. Now apply
iel Ί ιεΐ
2.30.
(e) If iel and S as described, (ο:1^) = (ο:Γ) by
1.45(b). Suppose that Ц-лБ = {0}. Then SL^L^S =
= {0}, so S?(o:L·) = (о:Г) = {0}, a contradiction.
So Цп S + {0}, whence Ц-л S = Li by (b), so all
L^S and therefore L«eS.
Ramakotaiah (7) showed that the L·'s are exactly all minimal
left ideals of Μ^(Γ). He also characterized in this paper
all maximal left ideals (also in terms of the finite topoloay
in 4.26).
The following result generalizes Theorem 5.7 of Betsch (7)
(notation as above). Cf. Ramakotaiah (3).
7.3 COROLLARY The following statements are equivalent:
(a) M = L.
(b) MM fulfills all conditions of 2.50.
(c) Η has finitely many orbits on Γ.
Proof: apply 7.2.
7.4 COROLLARY If Μ fulfills the conditions of 7.3 then Μ has
no non-trivial two-sided invariant subnear-rings. In
particular, Μ is simple.
This follows from 7.2(e) (simplicity can also be derived from 4.46).
More on simplicity of мЯ(Г) can be found in Meldrum (12).
7.5 COROLLARY (Betsch (7)). Let the non-ring ΝεΤ) fttlj be
2-primitive on Γ. Then the followina conditions are
equi valent:
(a) Ji is "finitely completely reducible" (all conditions
of 2.50 are valid) .
(b) N = М2(Г) and G has finitely many orbits on Г.
(c) N — M~(r) and the finite topolooy on M~(r) is discrete.

192
§7 TRANSFORMATION NEAR-RINGS
Proof. 2.50, 4.60, 7.3 and л.29.
There is an intimate connection between the lattices
all Η-invariant subgroups of Г and У(М) = {S<M|SM?S>
"right-invariant subnear-rings"of M).
We mention without proof:
7.6 THEOREM (Laxton (2), Betsch (7), §8). If Η has s£IN
on Γ* then the map f: $Η(Π - $(M) is a lattice
Δ -»■ (Δ:Γ)
phism with f'1: jf(M) ■+%{?) given by S - sr.
Moreover, 1^Н(Г)| = |tf(M)| < 2s.
Holcombe (6) suggested the following
7.7 DEFINITION Every choice B: = {b.jicl} of representatives
b i ε Β η- is called a Η-base. dim„(r): = |B| is called the
Η-dimension of r.
This comes from the easy-to-prove (cf. 4.28)
7.8 PROPOSITION (Holcombe (5)).
(a) \/ γεΓ* 3 ιεΐ 3 heH: γ = h^).
(b) Each map В + Г can be uniquely extended to a map
ε ΜΗ(Γ).
Holcombe formulated 7.8(b) more generally: "Every map В ■+ Г',
where Г' is another group on which Η operates (Γ1 is an
"H-group") can be extended to a unique map Γ ■+ Γ' which
commutes with H".
So Γ is in a kind "free" on B.
Η operates on Μυ(Π in a natural way. From 7.8(b) we get
7.9 THEOREM (Holcombe (5)). If dimH(r) = sclfi then
dimH(MH(r)) = (s+l)s-l.
%(П of
(the

7a Μβ(Γ)
193
7.10 REMARK In (7), pp. 92-97, Betsch studied the distributive
elements 0: = (Μ°(Γ)), of М°(Г) and "monomial matrices"
over D (i.e. matrices over D which contain in each column
at most one non-zero entry - cf. also Frbhlich (3)). D is
shown to be embeddable into the semigroup (End (M,+)j°)
if Η has finitely many orbits on Γ. (D,·) is anti-
isomorphic to the monoid of all feEnd(r) which commute
with all heH. Cf. Deskins (2).
Now we consider the following problem: when are M„ (Γ.) and
M^ (Γ2) isomorphic ?
For rinqs, this problem is solved in the followinq way (see
(Jacobson) p. 45 and p. 79): If h: HomD (Vj ,νχ) + HomD (V2,V2)
is an homomorphism (vi>v? vector spaces over the division rinq
rinqs Dj.Dp. respectively) then h is an isomorphism iff there
is some 1-1-semi-1inear transformation t: V, * V2 such that
V φεΗοπίρ (V1,V1) : η(φ) = ίφί"1.
We follow in some way Jacobson's discussion and start with
7.11 THEOREM (Holcombe (4), Ramakotaiah (6)). If H^Hg are
(as usual) fixed-point-free qroups of automorphisms on
Γ then M° (Γ) = M° (Γ) <-> Hl = H2.
Proof. We only have to show "->".
Suppose that H,=H2, and take some h2eHAH2. Take
γεΓ* and consider the orbits
, Β , containinq γ
with respect to H,,H2. Then Β, = Η,γ
and
B2 = Η2γ.
Clearly η2(γ)εΒ2,
3 h^HjHHg:
'l(Y) =
but
h
n?(Y)iBi (since otherwise
2(Y).
so
since Η,
1 - Up Э I II U С lip
fixed-point-free, a contradiction).
4.28(a) guarantees the existence of some m l£M° (Г)
with m^(y) = h2(y) and V δφΒ1: π^(δ) = о. Hence
m1(h2(y)) = о.
But о = m^h^r)) = b2(ml{y)) = η2(η2(γ)) since
ι s
mierVr> = мн2<г).
a contradiction,
Thus
Hence Η
h2(Y)
1 = H2·
o, whence γ = о,

194
§7 TRANSFORMATION NEAR-RINGS
7.12 REMARK Observe that Мн (Г) = Мн (Г) <-> M° (Γ) = M° (Γ).
7.13 COROLLARY (Ramakotaian (5)). Aut
МЙ(П
(Г) = Η.
Proof. If Η': = AutM ,гч(Г) then H^H'. H* is by 4.52
shown to be fixed-point-free with Ми(П = М„,(Г).
So Η = Η' by 7.11.
Next we consider M„ (Γ,) and M^ (Γ2), where H^ are fixed-
point-free on r. (i = 1,2).
7.14 DEFINITION ScHom(Гj,Г2) is called a semi-linear
homomorphi sm if 3 s: H,>-** H~ V Yi£ri V η,εΗ,:
S(h1(Y1)) = s^HS^)).
If S 4= ό, s i s uni quely
determined and called the
isomorphism associated with
S^. We will also speak about
the semi-linear monomorphism
(S,s).
7.15 THEOREM (Ramakotaiah (5), for the finite-dimensional (7.7)
case also Betsch (7) and Holcombe (5)). A near-rinq
homomorphi sm f: H° (Γ,) ■+ Μ? (Γ„) is an isomorphism <=>
<—> there is some semi-linear isomorphism S:T, ■+ Г~ with
V meMS (Г. ): f (m) = SomoS"1.
Hl 1
Proof. We abbreviate M° (r^) by M,- (i ε{ 1,2}), and
keep this notation for i.
<—: If (S,s) is a semi linear isomorphism Tj * Γ2
then the map f: M,
is an isomorphism:
SmS
To see this we first show that f maps M, into M2
Clearly f(m)eM0(r2). So take γ2εΓ2 and h2eH2>

7а М8(Г)
195
3 Y^Tj 3 h^Hj: S^) = γ2 л sfhj) = h2. Also,
h^om = mohj. So Sfh^mfy^)) = s (hj) (S (m(Yl))) =
= hz(S(m(Yl))) = hgiSomoS^CStYj))) = h2 (SomoS-1 (Y2 ))
On the other hand,we can compute S(h, (rn(y. )) ) in a
second way:
S(h1(m(y1))) = SfmfhjiYj))) = (SomoS-1)(S(hj(γχ))) =
= (SomoS^Hs^HSiYj))) = (SomoS_1)(h2(Y2)).
This holds for each γ2εΓ2, so we get
h?o(SomoS ) = (SomoS- )°h2; hence SomoS" εΜ~.
It is easy to show that f is an isomorphism.
—>: Assume now that f: M, ■+ M2 is an isomorphism.
(a) M, can be considered to be 2-primitive on Γ2,
does the required job.
(mltY2) - f(mj)Y2
since Mjxr2 -*■"*■ r2
(b) We show that there is an isomorphism S: Г, ■+ Г2
with \j т^М^: f(mj) = Som^oS" .
By 7.2(b), M, contains a minimal left ideal. By (a),
M, is 2-primitive on Г^ and on Γρ. 4.56(a)
assures that Γ. and Γ2 are M«-isomorphic (by S,
say). So by (a) \j Υ1εΓ1 \/ m^Mj: Sfm^Y^) =
= mi ts ίΎχ)) = f (mi)(s(Yi)) · whence Sorrij = ffm^oS
or f(m^) = Som.oS" .
(c) Now we claim that \/ h.eH,: Soh.oS" εΗ2>
Clearly Soh^s'^Endf Γ2).
h, commutes with all m^cM,. If f(m^)=m2 we get from
(b) and 7.13 m^iSoh-oS"1) = f(m. )°(Soh.°S~1) =
1 1 1 - 1
= Som.oS oSoh.c=S ' = Som.joh.joS ' = Son^m^S
= (Soh^S-1 )°m2. Hence S°h 1 °S" 1 e AutM (Г ) = Η2·
(d) Next we observe that s: Hj ·* Η
ι s an
h1 + Soh^s
-1
isomorphism, a fact which can be seen by the usual
procedures.

196
§7 TRANSFORMATION NEAR-RINGS
(e) Finally, we have to check the semi-1inearity
condition 7.14 for (S,s): take some ΥιεΓι and
ιεΗι = АиЧ'РЧ
some ίι,εΗ, = Aut„ (Γ) . Then S(h,(Y,)) =
■1.
Ί
= (Sohl0S χ)(5(Ύι)) = sthjJtStYj))
The proof is now complete.
From that we can deduce interesting results about the
automorphism of near-rings of the type Мм(Г):
7.16 COROLLARY (Ramakotaiah (5)). If Μ = Μ°(Γ) and feEnd(M)
then feAut Μ <=> there exists a semi-linear automorphism
S on Γ with f(m) = SomoS for all meH.
This follows from 7.15 by specializing M^ = M2
M.
7.17 THEOREM (Ramakotaiah (5)). Let G be the qroup of semi·
linear automorphisms on Γ and G': = Aut Мц(Г).
Then G/r u = G*.
li ft Π
Proof. Define a: G -* G' as follows: if SeG, there is
some feAut Μ°(Γ) with \/ πιεΜ°(Γ): f(m) = SomoS"1
(by 7.16). Observe that this f is unique. Put
a(S): = f.
First we prove that α is a homomorphism. To do this,
take S.TeG. a(ST) =: g, a(S) =: fj, a(T) =: f2.
Then for all πιεΜ°(Γ) g(m) = (ST)m(ST)"1 =
= STmT'V1 = Sf2(m)S"1 = f1(f2(m)).
Hence g = fjfo» implying that a(ST) = a(S)a(T).
Now a is an epimorphism: if fefi' then there is
some SeG with \/ теМ?.(Г): SomoS" = f(m). Thus
a(S) = f.
Finally we compute Ker a.
If SeKer a then \/ теМ°(Г)
-1
So SeAut
м°(г)
= id(m) = SmS
(Г) = Η (7.13). Hence ScGa H.
Conversely, each element of GAH is in Ker a.
This shows that G/G „ = G'.

7Ь M(DandM0(D
197
Clay (14) determined the group U of units of М^(Г): U is
isomorphic to the wreath product of G with the symmetric group
on the index set I of the orbits of Г under G. He also pointed
out the intimate connection between U and the general linear
groups in linear algebra. He also defined a "determinant
function" on U. See also p. 376.
In (3), Ramakotaiah showed that м[}(Г) is a ring iff Г is a 1-
dimensional vector space over the skew-field H.
Мм(Г) for H=End(r) is studied in §9 h). Don't forget to read this
chaoter as wel1!
b) М(Г) AND М0(Г)
There are a lot of things which we can get by specializing
Η = {id} in the previous section. By 1.13 it is justified
to consider primarily Μ (Γ). We start by considering left
ideals in Μ (Γ). Cf. Holcombe (4).
7.18 COROLLARY (Heatherly (1), (4), cf. also Frbhlich (3),
Ramakotaiah (7)).
{6 if γ = 6
о if γ 4= δ
then {е^беГ*} is a set of orthogonal idempotents.
(b) All М0(Г)е6 =: L& = (о:Г\Ш are left ideals and
M0(r)-groups of type 2 (hence minimal Μ (r)-subgroups)
generated by e- and Μ (Γ)-isomorphiс to Г.
<·> v> - ,:r.i«·
(d) If L: = I' L6 then L = Μ (Γ) iff Γ is finite.
(e) Every non-zero Invariant subnear-ring S of Μ_(Γ)
contains L.
Proof: 7.2.

198
§7 TRANSFORMATION NEAR-RINGS
Hence M0(r) is a 2-primitive nr. on Γ with Identity and a
minimal left ideal (see §4d3)). 7.3 and 7.18 give
7.19 COROLLARY (Heatherly (3), M. Johnson (6)). The following
are equivalent:
<·) »o'r> ■ ,р. ·L·
(b) M0(r) has DCCL.
(c) Μ (Γ) has ACCL.
(d) Μ (Γ) is completely reducible into finitely many
mi nimal 1 eft i deals.
(e) Μ (Γ) has only f.g. left ideals.
(f) Μ (Γ) is finite.
(g) Γ is finite.
Clearly (by 7.4) Μ0(Γ) is simple in this case. However, we
will extend this result to the arbitrary case (7.30).
But first we examine the left ideals more closely.
7.20 THEOREM (Heatherly (3)) Let L be a left ideal of MQ(r),
(a) tf γεΓ: Ly = {o} or Ly = Γ.
(b) If Δ: = {δεΓ|L5 = Г} + 0 then
Σ L,
όεΔ
Proof, (a) If Ly 4= ίο} then 3 JUL: ly 4= o.
tf meM (Γ): mjleL, whence {mty\ meMQ( Γ)}
fortiori Ly = Γ.
But
Γ and a
then
(b) It suffices to show that if L6 + {o}
LrSL. This trivially holds for |Γ|<2 since then
]Μο(Γ)|<2. So assume that |r|>3.
Suppose that fc*EL, (δεΔ). Denote £Л<5) =: θ.
Choose some SleL with 1(δ) = θ = Л.(6). This is
possible by (a).
Take m.neM (Γ) with m(y)
( θ γ = θ
Ιο γ + θ

7b Μ(Γ) andM0(D
199
(so m = eQ of 7.18(a)), and n(6) = o, but for
γ + δ η(γ) Ι {θ,θ- ϋ,(γ)}.
Then I: = m(η + ί.) -mneL and for γ ψ б we qet
ϊ(γ) = η(η(γ)+ί.(γ))-πι(η(γ)) = 0-0 = о = *6(γ),
while 1(6) = m(n(5) + i.(6) )-m(n(6) ) = m(e)-m(o) =
= θ = ls(&).
So Л, = Τεί and we are through.
7.21 COROLLARIES (Blackett (1), Heatherly (3), M. Johnson (1),(3)).
(a) The L ,'s (δεΓ*) are exactly all minimal left ideals
of М0(Г).
(b) Every left ideal of Μ (Γ) which is contained in
L = У Lx is isomorphic to a direct sum of suitable
6εΓ* δ
L6*S·
(c) L cannot be a non-trivial direct summand of M0(r).
Proof, (a) is a consequence of 7.20.
(b) follows from (a) and 2.55.
(c) If L' 4t М0(Г) is such that L+L' = MQ(r) and
L' + {0} then (by 7.20) 3 δεΓ*: L6 = LnL1, a con-
tradi cti on.
Since we have been very successful in determining all minimal
left ideals of Μ (Γ), w<
readily get some of them:
left ideals of Μ (Γ), we turn to maximal left ideals. We
7.22 EXAMPLE (Heatherly (1), (3)). For every γεΓ*. (ο:γ)
is a maximal Μ (r)-subgroup (hence also a maximal left
ideal) of MQ(r).
Proof. By 3.4(e) and 7.2 we get N/ .
SN Γ £Ν LY:
where Ν = Μ (Γ). Now apply 3.4(h),

200
§7 TRANSFORMATION NEAR-RINGS
But woe:
7.23 PROPOSITION (Heatherly (3)). If Γ is infinite then L
(as in 7.18(d)) is contained in a maximal left ideal of
Μο(Γ), but not in any (ο:γ) (γεΓ*).
Proof. The first statement is settled by 1.53(a) since
Μ0(Γ)εΊΊ1 and by 7.18(d). Now if Ls(o:y) then
L -(ο:γ) and e (γ) = ο, whence γ = ο.
So there are other maximal left ideals beside the (o:y)'s,
which implies more trouble for us. But, fortunately, M. Johnson
has solved this problem. See also Ramakotaiah (7) and (8).
7.24 NOTATION For meM0(r) call ίγεΓ|πι(γ) = о} =: Zm (the
"zero set of m").
We state without proof
7.25 PROPOSITION (M. Johnson (3), (6)). Let L be a left ideal
in MQ(r). Then
(a) leL <=> (oiZ^L.
(b) \/ tj.lgcL 3 *eL: 1% л 1% = 1%.
7.26 DOTATION Let £(Г) be for the moment = % =
= U йг Мо(Г)| \/JUL: 1% is infinite}.
Now we are in a position to characterize the maximal left
ideals of MQ(r).
7.27 THEOREM (M. Johnson (6)). Let L be a left ideal of MQ(r).
L is maximal <-> ( ^ΥεΓ*: L = (ο:γ)) ν (L is maximal in
Proof. =>: Suppose that \/ γεΓ: L 4= (°:γ). Assume moreover
that 3 £εί: ζ? is finite. For γεΓ consider ey of
7.18(a). Since L is no (ο:γ), all e cL by 7.20(b).

7Ь М(Г) and М„(Г)
201
Assume that |Z,| = nelN. We claim that 3 kcL: Zk
= {0} and prove this by induction on n.
(a) This is trivial for η = 1.
(b) Suppose that n>l and the statement holds for
n-1.
so
Now £ + e ει and Z0i
γ Я.+е
lZ*+e I - "-1·
Υ
■ ζΛ{γ}'
L = Μ0(Γ) by 7.25(a) since (o:Zk) = (0:0)
So
= Μ (r)=L. This is a contradiction. Hence ίε£(Γ)
and since L is a maximal left ideal, L is maximal
in £(Γ).
<—: In view of 7.22 it suffices to consider maximal
elements of£. Let К 3 М0(П properly contain L.
Then 3 keK: IZ. | = ηεΙΝ . Again we use induction to
show the existence of some k^eK with Zk = {0},
η = 1 is trivial again.
If n>l, take some ΥεΖ... γ + о. Then e ει
= (ο:Γ\{γ}) by 7.18(b). L + (o : Л (γ) )e£, so
ί+(ο:Γ\(γ}) = L since L is maximal inJ&. Hence
e eL and so k+e εΚ with
"k + e
= n-1 as before.
Hence
K = MQ(r) and L is shown to be a maximal left
ideal of MQ(r).
Since ^(Г) is empty if г is finite we get fror
7.28 COROLLARY Let Г be finite. Then
(a) The minimal left ideals of Μ0(Γ) are exactly the
Ly's. (γεΓ*).
(b) The maximal left ideals of Μ_(Γ) are exactly the
(o:Y)'s (γεΓ*).
(c) The left ideals of Μ (r) are exactly the (o:u)'s
(ΔΕΓ).
In Scott (14) it is shown that 7.28(c) also holds for f.g. left
ideals in any Μ (Γ). These left ideals are even generated by a
single element.

202
§7 TRANSFORMATION NEAR-RINGS
For in this case [ L, = Ι (ο:Γ\{δ}) = (ο: f] (Γ\ {δ}) = (ο:Γ\Δ).
δεΔ δεΔ δεΔ
We can generalize 7.28(c) easily to the infinite case:
7.29 THEOREM (M. Johnson (3)). Every L «^ Μ (Γ) is the sum of
annihilator left ideals.
This is a consequence of 7.25, for L = J (o:Z.).
Now we extend our result that Mg(r) is simple if Γ is finite
(7.19).
7.30 THEOREM (Berman-Si1verman (2), Meldrum (13)).
Μ (Γ) is simple for every group Γ.
Proof. We may assume that Γ is infinite. Take I <!M (Γ),
I + {0}. If πιεΜ0(Γ), call | {m(y) | γεΓ} | =: rk(m)
the rank of m.
(a) First we remark that for each γεΓ* there is a
maximal set Α?Γ with respect to the property that
А л (A+y) = if.
This follows by Zorn's lemma.
(b) The set A of (a) satisfies |A| = |r|. To see
this, consider Au(A+y).
If Ay(A+y) = Г, there is nothing to prove.
If δεΓ is not in Αυ(Α+γ), consider A': = Au(5}.
Since A is maximal, Α' η (Α'+γ) + 0· But
А'л(А'+г) = (A u{6}) η ((Α+γ)υ {δ+γ}) = ΑΛ{δ+γ}
by the algebra of sets and the assumptions.
So 3 αεΑ: α = δ+γ, whence δεΑ-γ.
Thus Γ = Α α(Α+γ) у (Α-γ). Since |A| = |Α+γ| =
= |Α — γ| and the fact that Γ is infinite we get
IH = |A|.
(c) Now we show that I contains some i with
rk(i) = |r|. Take γεΓ* and A as in (a) and (b).

7b Μ(Γ) andM0(D
203
Since I f {0}, there is some γ,εΓ with γ := .Ηγ,) f 0.
Define g,f,hcM0(r) by g(6): = |vi JjJ «+J
f(6)
/{ fo
1 о fo
г 6εΛ
r δ±Α
ind h(«): = |γ f,
lo f
or 64=0
for 6=o
Then i: = fo(id+h)-foidel , since h = j°gel
V αεΑ\{'ο}: i (α) = f (α + h (α) )-f (α) = -α.
Hence rk(i) > |A\{o}| = |A| = |r|, thus proving
that rk(i) = |Г|.
(d) Now we prove the theorem.
Let m be arbitrary εΜ0(Γ). Since rk(m) < |r| = rk(i)
(i as in (c)l there is some infective map
f: т(Г) ■+ ι'(Γ). For each γεΓ choose one γ'εΓ
with ι(γ') = f(m(y)). Denote this correspondence
Υ ■* Υ1 by g. We may assume that f(o) = о and
i(o) = o.
Define the map r as follows:
'm(y) if 3 γεΓ: 6 = f(m(y)) 4s о
о otherwise
■"(■5):
{:
Then for all δ
we get:
( r° i °g) (<5) =
= r(i(g(6))) =
= r(i(6')) =
= r(f(m(6))) =
= m(6)
Hence r°i °g = m
and mc I.
ИГ!
This shows that I = Μ (Γ} and the proof is finished.

204
§7 TRANSFORMATION NEAR-RINGS
Of course one is led to the question whether Μ(Γ) is simple
or not. We try the simplest example:
7.31 EXAMPLE M(Z2) has one non-trivial ideal, namely МС(Г)
But this is the only case that М(Г) is not simple:
7.32 THEOREM (Berman-Si1verman (2)).
There are no proper subnear-rings between Μ (Γ) ancf М(Г).
Proof. Let N be a subnear-rinq of М(Г) with N=M (r).
Let η be εΝ\Μ (Γ). If γ is arbitrary εΓ and if
m denotes the map which is constant = γ then
d m0£M0(r) witn m0(n(°)) = Y· But m0n =: η'εΝ,
and m = -η'+η'εΝ (where n' = n'+n! as in 1.13),
γ ο v oc '
showing that N э МС(Г). Hence N = Μ (r).
7.33 THEOREM (Berman-Si1verman (2), Meldrum (13)).
М(Г) is simple iff |r| + 2.
Proof. Let I be an ideal of М(Г). Then IQ: = Inl (r)
(2.18) is an ideal in Μ (Γ). Hence IQ: = {0} or
I0 = M0(r)· If !o = {0} then I-Mc(r)· Examining
1.27 β) shows that I = {0} if |r| + 2.
If lo = fVr) then l = Μο<Γ) or l = Μ(Γ) by
7.32. Since М0(Г)МС(Г)ЕМС(Г), М0(Г)#М(Г).
It remains I = М(Г).
Cf. Adler (1) and Blackett (1). Other substructures have also
be«n considered. We mention without proof:
7.34 THEOREM (Berman-Si1verman (2)). The only two-sided invariant
subnear-rings of М(Г) are Μ (Γ) = IfeM(r)|rk(f) = 1},
(ίεΜ(Γ) |rk(f)<K0> and ίίεΜ(Γ ) | rk(f )<&} with #0<Ν<|Γ|.
The ones of Μ (Γ) are just the intersections of the. ones
above with Μ (Γ).

7b М(Г)апс1М0(Г)
205
7.35 REMARK See Heatherly (3) for a discussion of the right,
ideals of Μ (Γ).
7.36 COROLLARY By 7.6, the lattices of all subgroups of a finite
group Γ and of all right invariant subn<
are isomorphic (cf. also Heatherly (4)),
group Γ and of all right invariant subnear-rings of Μ (Γ)
7.37 COROLLARY AutM /ΤΜ(Γ) = {id}.
Proof. 7.13.
The next corollary gives one half of 1.99 once again in a
different version.
7.38 COROLLARY MQ(Γ) and Μ (Δ) are isomorphic (say by f)
iff there is some isomorphism S: Γ + Δ with
V т£Мо(Г): f (m) = SomoS-1.
Proof. 7.15.
We now turn to the automorphisms of Μ (Γ).
7.39 THEOREM (Ramakotaiah (5)). A homomorphism f : Μ (Γ) - Μ (Γ)
is an automorphism <=> there is some automorphism S of
(Γ, + ) with V πιεΜ0(Γ): f(m)= SomoS"1.
Moreover, Aut (Γ.+) - Aut (Μ (Γ),+,·)·
Proof. 7.38 or 7.16 (for Η = {id}). See also Nbbauer (12).
7.40 REMARKS
(a) See Clark (1) for examples of Μ(Γ) with Г = 1 ,
Г " V Г " V ■
(b) See Blackett (4), (5) and (6) for examples of simple
subnear-rinqs of M(IR) and M(C) and for "dense"
subnear-rings of M(IR) .
(c) Beidleman (11) proved for a finite group Г that Г is
nilpotent (solvable, supersolvable) iff the same
applies to (M (r),+).

206
§7 TRANSFORMATION NEAR-RINGS
(d) Malone-McQuarrie (1) showed that if Γ is torsion
without elements of order 2 then (Μ (Γ),+) is
(uniquely) halvable.
(e) M. Johnson (3) characterized the cases when left ideals
and normal Μ (r)-subgroups coincide in Μ (г). ^е
know already that this happens e.g. if Г is a finite,
non-abelian invariantly simple group (6.33 and 6.6(a)).
M. Johnson showed that if Г is finite then left ideals
and normal Μ (r)-subgroups coincide in MQ(r) iff
Γ is not abeli an.
For infinite groups Γ this happens iff
3 γεΓ: [Γ: Γ,(γ)] = |г|, where Γ,(γ) is the centralizer
of γ.
Or, equivalently: this coinciding happens for some
group Γ iff every normal Μ (r)-subgroup of Μ (Γ)
is the sum of annihilator left ideals of Μ0(Γ)
(7.29).
(f) Μ (Γ), Γ finite, has an involution iff Г is neither an
elementary abelian 2-oroup nor isomorphic to TL^ (Scott (13'
c) E(r), А(Г) AND I(D
In this section we will examine similar items for the dgnr.'s
Е(Г), А(Г) and Ι(Γ), generated by all endomorphisms
(automorphisms, inner automorphisms, respectively) of Γ, as we did
for Μ(Γ) and M0(r) (one- and two-sided ideals, simplicity,
radicals, automorphisms, etc.). As in b) we will get the best
results for the case that Г is finite.
A lot of further information on these near-rings can be found in
Meldrum (13), ch. 10 and 11. See Holcombe (3) for a categorical
approach.
First we define our objects in question.
7.41 DEFINITION Let Г be a group. Similar to Е(Г) we define
МП (ЦП) as the near-rings additively generated by
all automorphisms (inner automorphisms) of Г.

7с Е(Г), А(Г) and I(Γ)
207
7.42 REMARKS
(a) Let End(r), Aut(r) and Inn(r) denote the
semigroups of all endomorphisms (automorphisms, inner
automorphisms) of Γ.
(b) Ε(Γ), Α(Γ) and Ι(Γ) are dgnr.'s.
(c) Similar to 6.4(d), Α(Γ) (Ι(Γ)) consists of all
finite sums Ισ|<αι, with ak = +1 and all a.eAut(r)
к
(аке1пп(Г)).
(d) I(r) - А(Г) s Ε(Γ) Ε Μ0(Γ).
6.6(c) gi ves us
7.43 PROPOSITION If Г is abelian then Е(Г), А(Г) and Ι(Γ)
are rings.
Hence we will be interested in non-abelian groups Γ.
7.44 PROPOSITION The Ε(Γ)- (Α(Γ)-, Ι(Γ)-) subgroups are
exactly the fully invariant (characteristic, normal)
subgroups of Γ.
Proof. 6.24.
This impli es
7.45 COROLLARY
(a) Е(Г) is 2-primitive on Γ <=> Γ is invariantly simple.
(b) Α(Γ) is 2-primitive on Γ <=> Γ is characteristically
simple.
(c) Ι(Γ) is 2-primitive on Γ <*=> Γ is simple.
Now we characterize in which cases Ε(Γ), Α(Γ) and Ι(Γ)
coincide with M0(r) and get as a generalization of results
of Johnson (1), Frbhlich (3) and Maxson (14)

208
§7 TRANSFORMATION NEAR-RINGS
7.46 THEOREM Let Γ be a non-abelian, finite group.
(a) Ε(Γ) = Μ (Γ) <-> Γ is invariantly simple.
(b) A(F) = Μ (Γ) <-> Γ is characteristically simple.
(c) Ι(Γ) = Μ (Γ) <=> Γ is simple.
This result follows from 7.45 and 6.33. Observe that if Γ
is invariantly simple then Ε(Γ) is a non-ring: Ε(Γ) is
2-primitive on г and can be no ring by 4.8. The same applies
to A(r) and Ι(Γ).
The last theorem can be expressed as a "purely group-theoretic"
result:
7.47 THEOREM Let Γ be a finite non-abelian group. Then each
map Γ * Γ which fixes о can be expressed as a finite
sum Eo^.d. with d^Endfr) (Aut(r), Inn(r)) <=> г is
invariantly simple (characteristically simple, simple,
respecti vely).
Ramakotaiah (9) contains information about the infinite case.
7.48 COROLLARY Let Г be as 7.47.
(a) If Г is characteristically simple then А(г) = Е(Г) =
= м0(г).
(b) If Г is simple then Ι(Γ) = Α(Γ) = Ε(Γ) = MQ(r).
Thus Ε(Γ), Α(Γ) and Ι(Γ) can be viewed to be characterized
by 7b) if Γ is as in the cases of 7.46.
The next step is to consider the case that Γ is a non-abelian
group which is the direct sum of fully invariant (characteristic,
normal) subgroups. Without taking the hands out of the trouser
pockets we deduce from 7.44 the following result:
7.49 COROLLARY Let Γ be the direct sum of minimal fully
invariant (characteristic, normal) subgroups. Then Γ is
a completely reducible (2.48) Ε(Γ)- (Α(Γ)-, Ι(Γ)-) group.

7с Е(Г), A(Dand Ι(Γ)
209
Now we will try to decompose Е(Г) (...) itself. We start
with Е(Г) and fix our notation. Our discussion follows partly
and generalizes Johnson (1).
7.50 NOTATION Let Г be the direct sum of fully invariant
subgroups Ф, (i ε I) (don't forget 7.44!). Then each γεΓ
can uniquely be decomposed as γ = [ ψ. with φ^εΦ-
i εΐ
(note that this sum is actually finite).
If πιεΕ(Γ) and i ε I then
m<1>:r
Finally, let Ε{ι)(Γ): = (π/ 1 > |mEE(Γ)}.
γ = £ φ, - т(ф )
J-εΙ J '
7.51 PROPOSITION Let Г be as in 7.50 and let m,n be εΕ(Γ)
and i ε I.
(a) πιεΕ(ΐ)(Γ) <=■> m = m(l).
(b) nr 1 ' = т°тг. , where π. is the projection Γ ■+ Φ^ .
(c) (m+n){i) = m{i)+n{i) and (mn){i) = mn(i).
(d) Ε(1)(Γ) - E^.) by m
<1> - т<1>/л - т/л
Proof, (a): If тгЕ^^Г) then 3 η^^εΕ^^(Γ): m = n{l).
But then for all γ = Ι φ^εΓ we obtain
ίεΐ J
J'Uy) = m(^(E4j) = т(ф.) = n^^.) = η^^Σφ^,) =
= n^'fy), whence m^1^ = ip ' ' = m.
The converse is trivial,
(b) - (d) follow easily.
Now we improve Proposition 5.7 of Johnson (Γ
7.52 PROPOSITION \j i e I : Ε^(Γ) «3Ε(Γ).
Proof, (a) If m^1 'εΕ^1 '(Γ) , consider πιεΕ(Γ) and
decompose m as m = I^kek with all e.pEnd Γ. All
e^^End(r) by 7.51(b). Usina 7.51(c) we obtain

210 §7 TRANSFORMATION NEAR-RINGS
гЛ1) = Σσ^1), whence πι^^εΕ(Γ).
This shows that E{l)(Γ) ? Ε(Γ) .
(b) Applying 7.51(c) again shows that Ε^^(Γ) < Ε(Γ).
г
(c) Now let n = Ι σ-ί.εΕ(Γ) (all f.eEnd Γ) and
j=l J J J
take some m^1^εΕ^1^(Γ).
Then n+m'^-n = a ^f j + . . . +apf г + пЛл '-a f r~ . . . -Ojf 1.
But if γ = Ι φ· = φ.+ψ where ψ = У фк then
i ε I k-fi
(cjrfr+mO) -σΓίρ)(γ) = σΓίΓ(Φι.+ψ)+Γη{ι)(φ1+ψ)-σΓίΓ(Φ1+ψ)
= σ/Γ(Φ1)+π.(1)(Φ1)-σΓ.ίΓ(41) = (orfr+n(i)-arfr){i)(y),
since ί_(Ψ) ε У Ф. commutes with ш^^фЛсФ,.
k*i
This shows that of+m{l*-σ f„εΕ^Ί*(Γ).
Г Г Г Г '
Proceeding in this way yields η+ηκ 1^-ηεΕ'^ '(Γ) and
E^'fT) is normal and by (b) and 6.6(a) we know that
Ε<^(Γ) *t Ε(Γ).
(d) Finally consider again η = Уо,-^ and
j л J
m^ = lakek^ ' · Then by 6-4(d) ·
m^'n = la ί ycr.eV^f.. Now if γ = Φ^+ψ as before,
к K j J K J
ek1)fj(v) = e^ffji^J + fji,») = е^>(^(Ф|)) -
= (eJ^fjJ^^Y). whence ejj^fj ε Е^)(г) and hence
So Е^'(Г) has no chance any more to escape from
bei ng an i deal .
The following result is implicit in Johnson (1).

7с Е(Г), А(Г) and I(Γ)
211
7.53 THEOREM Let Γ be finite, non-abelian and the direct sum of
finitely many minimal fully invariant subqroups Φ. (i ε I).
Then, as near-ri ngs ,
Ε(Γ) = ΓΕ(1)(Γ) i ® E($i).
ιεΐ ϊεΐ
Proof. If теЕ(Г), m = I m^', for 2.30 implies that
i ε I
V γ = Σ^εΓ: m(y) = т(Гф1) = Ет{ф1 ) = Ет^^ф^ =
= Em1(γ).
If m^1 'εΕ^^(Γ) η ( У E(j^(r)) then for each φ1εΦ1
m' (φ<) = о; consequently nr ' = ό and the sum
is direct. The rest follows from 2.30, 2.28, 7.51(d),
7.43 and 7.46(a).
Using 4.46 and 7.53 one gets
7.54 COROLLARY If Γ is as in 7.53, Ε(Γ) is finitely completely
reducible as a near-ring as well as an E(r)-group (2.50).
Ε(Γ) is simple iff |I| = 1, i.e. iff Γ is invariantly
simple.
So the structure of (left) ideals of Ε(Γ) seems to be clear
for a group Γ as in 7.53 (see 2.50 and 2.55).
7.55 COROLLARY Let the finite non-abelian qroup Γ be the direct
sum of minimal fully invariant subgroups Φ,,...,Φ^. Let
Φ. «j be abelian and **+ι»···»*k not· ^nen
л t k
E(r) - § End($.) <S § Μ (Φ.)
1-1 1 j=t+l ° J
(where Εηΰ(Φ·) denotes the endomorphism ring on Φ-).
Proof. 7.53, 7.43 and 7.46(a) (observe that the minimal
fully invariant subgroups Φ· are invariantly simple)

212
§7 TRANSFORMATION NEAR-RINGS
It is harder to get similar results for A(r) and Ι(Γ),
since aeAut(r) does not imply or ЧЕ^(Г) to be an
automorphism in general. At least we get the following
7.56 THEOREM Let Г be the direct sum of non-abelian minimal
fully invariant (minimal characteristic) subqroups
$. (jeJ). Then Ε (Γ ) <=> Π Ε ( $ . ) = Π Μίφ.) and
J jeJ J jeJ ° J
А(г) =* Π Α(φ,) = Π Μ UH), respectively.
jeJ J jeJ ° J
Proof. An embeddinq map can be h: Ε(Γ) ■+ Π Ε(Φ·)
" jeJ J
m ► (...,m/ ,...)
j
and likewise for the other two cases.
Observe that the restriction of an endomorphism is
an endomorphism and that of an automorphism.
For Ε(Θ Γ■) see Fong (2) and Fong-Meldrum (1).
ιεΐ η
7.57 COROLLARY If Γ is as in 7.53 then
Э2(Е(П) = ^(Е(Г)) = ... = ^(Ε(Γ)) = {0}.
So the situation is quite clear if Γ is the direct sum of
finitely many minimal fully invariant subgroups.
If this is not the case, life is much more complicated (e.g.
if Γ is the dihedral group with 8 elements).
We mention only two results without proofs (but with hints)
in the finite case:

7с Е(Г), A(Dand Ι (Γ)
213
7.58 THEOREM (Johnson (1), (2)). Let Γ be a non-abelian finite
group. Then
(a) Е(Г) is simple <-=> Γ is invariantly simple.
(we know <=; conversely, if Δ is a proper fully
invariant subgroup of Γ then (ο:Δ) is a proper
ideal of Ε(Γ)).
(b) 3Ζ(Ε(Γ)) = ^(Ε(Γ)) = ... »?(Ε(Γ)) (but not
necessarily = {0}).
C^2(EСг)) turns out to be nilpotent. Now apply 5.61(b)).
For more on that see Lyons (5),(6), Lyons-Mel drum (1), Meldrum
(6),(7),(13).
We continue with a result due to Frohlich (3).
7.59 THEOREM (Ramakotaiah (5)). Let Γ be a finite simple non-
abelian group. Then Aut(I(r)) a Aut(r) (as groups) and
each hcAut(I(r)) is of the form m -»· ama" , where aeAut(r).
Proof. 7.39 and 7.46(c).
If we are going to leave finite groups г it becomes pretty dark.
In (3), Frohlich mentioned that one can see (by comparing the
cardinal numbers) that if г is an infinite simple group then
Ε(Γ) < Μ (Γ). Introducing a suitable topology in Μ (Γ) gives the
result that Μ (Γ) is the completion of E(r). More can be found
in Ramakotaiah (9).
We conclude this section with some remarks.
7.60 REMARKS
(a) An excellent survey on special E(r)'s can be found in
Meldrum (13).
(b) (Malone-McQuarrie (1)). Let Г be halvable. If for all
m,ncE(r) (А(Г), I(r)) U^-J-J = 0 then Е(Г)
(Α(Γ) , Ι(Γ)) is a ring.

214
§7 TRANSFORMATION NEAR-RINGS
(c) (Haxson (14)). If Е(Г) is a ring and Г is fg. then
Г is abelian iff сг/г\Г is monogenic. (This can
actually happen; so Е(Г) can be a ring although Г
is not abelian! - Compare 7.46). If Г is a finite
p-group and Е(Г) is a semisimple ring then Г is
abelian.
See also Lyons (5), Malone (9 ) , (10 ) ,(12) , McQuarrie (5).
(d) (Chandy (2)). Call a group Г an L-group iff Ι(Γ) is
a ring. Cf. Scott (9).
Then Г is an L-group <=> all conjugated elements
commute <~> the centralizer of each γεΓ is a normal
subgroup. If Γ is an L-group and nilpotent of class
<3 then (Ι(Γ),+) is nilpotent of class 1.
(e) (Chandy (3)). I(r) is a commutative ring iff г is
ni1 potent of class 2.
(f) (Beidleman (11)). If Г is finite and solvable (nil-
potent) then the same applies to (Е(Г),+).
(g) (Fain (1), Lyons (3), (4), Lyons-Malone (2), Johnson (1)).
If D is the dihedral group with 2n elements (n odd)
then E(0 ) = A(D ) = I(D ). See a more detailed
description of these dgnr.'s in Mel drum (13). Lyons-Malone (2)
contains also a description of all dgnr.'s definable on
D . See Meldrum (13) for informations on Ε(QJ , where DM
is the infinite dihedral group. Ε(Γ) for free r's are
studied in Zeamer (3),(4),(5) and Lockhart (4).
(h) See Lyons (4), for a proof of the fact that if Δ is a
fully invariant abelian direct summand then Ε(Δ)
embeds in Ε(Γ) as a direct summand.
(i) Ι(Γ) has the DCCL iff Γ is finite (Scott (10)).
Ε({γ}) is studied in Scott (12).
(j) A density result for E(r) is in Kaarli (5).
(k) Call ecEnd(T) normal if \/ aelnn(r): ea = ae.
Let Ν(Γ) be the nr. generated (additively) by all
normal endomorphisms of Γ. By a routine check we get

7d Polynomial near-rings
215
1
the following result (Heerenia (1), Plotkin (3), 6.1.3.2)
Ν ( Γ ) is a r i η g .
(1) (Β.Η. Neumann (2)). Let a be a fixed-point-free
automorphism of Γ of order 3 and Aa the subnear-rinn
of Α(Γ) generated by a. Then A is a rinq, too.
(m) See Scott (1) for much information about the invertible
elements in certain subnear-rings of E(r). He also
proved that if I(r) has the DCCL then Γ is finite!
(n) (Scott (8)). Let Γ be finite, αεΑιΐί(Γ), α fixed-
point-free and N the nr. generated by a. Then (N,+)
is nilpotent iff (Γ,+) is nilpotent.
(o) See also 7 d) 6) and 6.35(d) .
d) POLYNOMIAL NEAR-RINGS
POLYNOMIALS AND POLYNOMIAL FUNCTIONS
O.K., you are right: polynomial near-rinns on some alnebra A
are no transformation near-rinos on A in neneral. But they are
so close to this subject and necessary to treat near-rinns of
polynomial functions that we include them into §7.
Polynomials can be defined over any algebraic structure A and
any set X (with ЛлХ = 0) of " i ndetermi nates" in any variety
^containing A. Denote the set of these polynomials by A(X,1T).
Roughly spoken, A(X,V) is the "set of all words in AuX
where equality is defined in accordance with the laws defininq
V". In fact, A( Χ ,φ") is defined as a "suitable" factor algebra
of the word algebra over AuX inU, hence becoming also an
algebra ofV, the "polynomial algebra". Another possible
characterization of A(X,V) is the following: A(X,T>) is the
free union of A and the free alnebra F over X inV.
See Lausch-Nbbauer (1) (whole book, particularly pp. 12/13) for
a detailed exposition.

216
§7 TRANSFORMATION NEAR-RINGS
In some instances for these polynomials (= equivalence classes
of words over AuX) there exist "normal forms" (see Lausch-
Nbbauer (1), pp. 22 ff.)> e.g. for the case of commutative
rings with identity and for groups.
We will restrict our considerations to the case of a single
"variable" X = {x} and to two varieties: the one of all
groups (Й) and the variety%of all commutative rings with
identity 1.
Then we have the following polynomial algebras (in normal forms)
(see Lausch-Nbbauer (1), 1.8.11 and 1.9.11):
7.61 NOTATION If RcH and Γ = (Γ, + )ε« then
η "
(a) R[x]: = { I aVlnelfT , ai£R, a_ + 0} и {0}.
i=0 υ ι μ
(b) r[x]: = {γ0 + η1χ+γ1 + η2χ+.. .+Yr_1 + nrx+Yr|r£INo , γ^Γ,
η.εΖ* and \/ tc{ 1,. . . , r-1}: ytf o),
Our interest in polynomials stems from the easy-to-establish
7.62 PROPOSITION Under addition "+" and substitution "°" of
polynomials (definition as usual, cf. Lausch-Nbbauer (1),
p. 77) (R[x]. + »°) and (r[x], + ,o) are near-rinns.
We will continue to denote these near-rinns by R[x] and
r[x]. Throughout this section R will mean an element of ft.
7.63 DEFINITION
(a) If peR[x] then p: R -* R is called
r ■+ por = : p( r)
the polynomial function induced by p.
P(R) : = {"p | peR[x] } .
(b) Similarly we define 'p for ρεΓ[χ] and Р(Г).
Trivial, but good to note is

7d Polynomial near-rings
217
7.64 PROPOSITION If Γ is abelian then every polynomial function
Γ>εΡ(Γ) is of the form ~p: Γ ■+ Γ with γΛεΓ and
ηεΙΝ .
Υ0 + ηγ
^οεΓ
7.65 PROPOSITION The correspondence h: ρ ■+ ~p is a near-ring
homomorphism. Hence P(R) and P(r) are subnear-rings
of M(R) (М(Г), respectively).
Proof: straightforward.
7.66 REMARKS
(a) P(R) and P(r) are called the near-rinos of polynomial
functions on R (Г, respectively).
(b) Throughout this section, h will have the meaning of
7.65. h is not necessarily an isomorphism R [x] -♦ P(R):
Take R =: 2 and q: = χ · (x-1 )·...· (x-(p-1)).
Then ρ =f 0, but h(p) = ~p = ό (zero function on TL ).
Anyhow, Ker h ά R[χ] will play a decisive r51e.
Similar considerations apply to groups (if e.g.
Γ = 23, q = 3x behaves as above).
Hence one cannot identify polynomials and polynomial
functions (as one is used to do over IR ) in general.
But:
7.67 REMARK If (R,+ ) is torsion-free ( (Aczel)) or if R is an infinite
field (use the theory of linear equations and the Van-der-
Monde-determi nant) then h is an isomorphism R [x] ->· P(R1
Later on, we shall also consider Ν£x] (Ν a near-rinq),
V [x] (V a vector space) and so on to get more near-rinqs. But
now we remain at R [x] and Γ [χ], since most phenomena can
already be seen there and discuss R[x] first since R [x]
seems to be more familiar than Γ [χ].

218
§7 TRANSFORMATION NEAR-RINGS
2·) RCx]
We start with some elementary, yet fundamental properties. Let,
asoften, [p] be the degree of ρ ( [θ] : = 0).
Observe that R[x] is an abelian near-rinq with identity x.
η ·
If we write ρ = Τ a,x , we allow a = 0 if ρ = 0 (7.61(a))
i=0 Ί π
to avoid separating cases. See also Mc Quarrie (1).
7.6B PROPOSITION (Lausch-Nbbauer (1), p. 134).
(a) \/ p,qcR[x]: [poq] < [p] · [qj (with equality iff R is
an integral domain).
(b) If peR[x] is invertible (w.r.t. o) then [p] = 1.
If R is a field, the converse also holds.
(c) If R is an integral domain then each pcR[x] with
[p] > 0 is right cancellable.
η . m .
Proof, (a) If ρ = I a.x , q = £ b-xJ then
1=0 j=0 J
η ·
poq = £ a-q = a b χ + summands of lower deqree.
(b) follows from (a).
(c) If fop = qop then (f-q)op = 0. By (a),
[f-q] = 0, so f-q is a constant с Hence f = q+c
and so qop = fop = qop+c, whence с = 0.
7.69 DEFINITION Call pcR[x] indecomposable if ρ = fog
implies [f] = [p] or [g] = [p] .
7.70 PROPOSITION (Lausch-Nbbauer (1)). Let R be an integral
domai η and peR [x].
(a) [ρ]εΡ =-> ρ is indecomposable.
(b) If [p] > 1 then there exist indecomposable polynomials
p1,...,PscR[x] with ρ = pjop2o. . .ops.

7d Polynomial near-rings
219
Procf. (a) is trivial.
(b) follows by induction on [ρ] , using 7.6B(a).
7.71 DEFINITION
(a) Let ρ be called normed if ap = 1.
(b) ρ is called linear if [p] = 1.
Now we can sharpen 7.70(b) and state without proof:
7.72 THEOREM (Lausch-Nbbauer (1)). If R is a field of
characteristic 0 and if peR[x] has [p] > 1 then
(a) p = ί,ορ.ο.,.ορ with a linear, s ε IN and ρ.,..., ρ $
indecomposable and normed.
(b) If ρ = moq1o...oq is another such "prime decomposition"
of ρ then m = l, s = t and the degrees of the P^'s
and q*'s are the same (up to order).
J
(c) There exist only finitely many decompositions of ρ of
the form (a).
This uniqueness can even be strengthened by the deep theorem
2.46 of Lausch-Nbbauer (1).
7.7 3 REMARK See Graves-Mai one (2) for another result of "prime
factor decompositions" in "Euclidean near-domains"
(see 9.60 and 9.67(c)).
3·) fffr],
7.74 DEFINITION Call RzW, polynomially complete (Nbbauer (6))
оr 1-polynomially complete (Lausch-Nbbauer (1))
if P(R) = M(R), i.e. if each map R ■+ R is a polynomial
function.

220
§7 TRANSFORMATION NEAR-RINGS
Of course, 2 , φ, 1R and I are not polynomially complete. On
the other hand, the fact that 22 is polynomially complete is
of high value in mathematical logic.
Perhaps we now feel which R might be polynomially complete and
prove the following interesting result, which will also prove
useful in the sequel .
7.75 THEOREM (Nobauer (6), Lausch-Nobauer (1)).
R is polynomially complete <-> R is finite and simple
(hence a finite field).
Proof. ->: If R is infinite, |P(R)| < |R fx] | = |R| <
< |R|IRI = |M(R)|, so R cannot be polynomially
complete.
Let I be a non-trivial ideal of R. Take some
nonzero i ε I. If 'peP(R) then "p (i )-'p(0 )εΙ , since
I <) R.
Let feM(R) fulfill f(i)-f(0)$I. Then f$P(R)
and again R is not polynomially complete.
<—: This follows from Lagrange's interpolation
theorem (or from the fact that P(R) is 2-primitive
on R).
4.) IDEAL THEORY IN R[x]
Of course, R[x] can also be considered as a ring (R[x] , + ,·),
where "·" means ordinary multiplication.
There is some connection between the ring- and the near-ring
ideals of R [x] :
7.76 PROPOSITION (So (1)). If p,q,rER[x] then r divides (w.r.t.".")
p°(q+r)-poq. In particular, r|p°r for each peR [x].
η
Proof. Let ρ = I a^x1 . Then
i=° n ι n i
po(q+r)-poq = Υ a-(q+r) - У a.q is a multiple

7d Polynomial near-rings
221
7.77 COROLLARY (Lausch-Mobauer (1)). Every ideal of (R[x].+,·)
is a left ideal of (R[x],+ ,o).
See 7.94 for a (partial) converse.
Note that if we write R[x] we always mean the near-ri na-
version (R[x],+,o). It's also high time to fix one more
notation.
7.78 NOTATION
(a) (R[x])0 = (I a^^a^R Λ ηεΙΝ} = ίρ|ρ(0) = 0} =
= : R0[x].
О
(b) (R[x])c = { 1 a.x'la £R} = {ρ | [p] = 0} =: R [x] .
c i=Q Ί ° c
(Of course, Rc[x] can be identified with R, and
we will do so in the sequel.)
(c) The elements of R [x] (Rc[x]) ai"e called the
zero-symmetric (constant) polynomials.
7.79 PROPOSITION (Clay-Doi (2), So (1)). Let L $ R [x] . Then
(a) RQ[x]oL s L.
(b) \/ reR \/ let: ΓίεΙ.
(c) If xeL then RQ [x] = L.
Proof, (a) follows from 1.34(a).
(b) гЛ = (rx)oJUL by (a).
(c) also follows from (a).
7.80 REMARK Observe that R is an R[x] - and an P(R)-group
in the natural way (pr: = ~p(r) if peR[x] and reR).
We will milk this observation in the next number.
Now we take a glance to two interesting ideals of R [x] :

222
§7 TRANSFORMATION NEAR-RINGS
7.81 EXAMPLES (Nobauer (1), Lausch-Nobauer (1)). Let J «3 R.
Then
(a) (J): = J[x] 4 R[x].
(b) <J>: = (J:R)R^X] <1 R[x].
These examples of ideals in (R[x], + ,o) show up to be also
ideals in (R[x]»+.·)· This leads to the followinn
7.82 DEFINITION I «3 R[x) is called a full ideal (Nobauer) or
T-O-i deal (Menqer) or composition ideal (Adler) if also
I <l (R[x] ,+ ,·)·
7.83 REMARK Thus full ideals are exactly the "ideals" of the
composition ring (TO-Algebra) (see 1.117) (R[x] ,+, · ,o ).
In fact, these full ideals of R[x] are much better explored
than the "ordinary" ideals of R[x] . Not every ideal of R [x]
is a full ideal, but this holds in special, important cases
(see 7.93).
If e.g. R happens to be a field then all full ideals are
principal (a well-known result of ring theory) and so quite
easy to overlook. It is not known under which conditions each
ideal of R [x] is "principal" (= generated by one single
polynomial), or at least f.g. .
The ideals of 7.81 show up to be quite important ones:
7.84 THEOREM (Nobauer (1), Lausch-Nobauer (1), Hule (1)).
Let I be a full ideal of RΓχ]. Then there exists
exactly one ideal J of R with (J)sls<j> (namely J = 1лК).
Proof. Let J: = I n R. Then clearly (J) = j[x]?i.
Also, if i ε I and rcR then T(r) = iorelrtR = J,
whence I^<J>.
If J' 3 R also fulfills (J')«=I = <J'> then
(J')=<J>, whence J's<J> and so J'^J. Similarly,
J^J' and hence J = J'.

7d Polynomial near-rings
223
7.85 DEFINITION Let I,J be as in 7.84. Then J is called the
"enclosing ideal" of I (J is not ideal in R[x] in
general! ).
For (much) more information concerning these enclosing ideals
see the comprehensive book Lausch-N'dbauer (1). Cf. also
Mlitz (1).
We will get more powerful results when P. is assumed to be
a field:
5-) F[x]
Throughout this number, let F denote a commutative field.
7.86 PROPOSITION (Clay-Doi (2), Straus (1)). Let L <L, F[x]
with L л F + {0} (cf. 7.78(b)).
(a) F - L.
(b) If [F| > 2 then L = F [xj .
Proof, (a) Let I + 0 be cLr\F. Then take some feF.
By 7.79(b) , f · fc"1·* = feL.
,-1
By (a), feL
(b) If char F + 2, take f:
2 2 2
and also f eL. Hence χ o(x+f)-x oXeL, so
2
x+f eL, whence χει. Use (a) and apply 7.79(c) to
get L = F[x].
If char F = 2, and lF| > 2, then in particular
3 3 2
char F + 3. Hence χ ο(χ+1)-χ 0x = χ +χ+1εί, so
χ +χεί.
-1 3
Take some feF\{0,l) and denote f ·χ by p.
2 2
Then po(x+f)-pox = χ +fx+f eL.
о
Since χ +χεί and F"=L, (f-l)xeL, so by 7.79(b)
and the fact that F is a field we get xeL and again
F[x] = L.

224
§7 TRANSFORMATION NEAR-RINGS
7.87 REMARK Brenner (1) has shown that 7.86(b) does not hold
for F = Z2. See 7.98(b) .
7.88 PROPOSITION (CI ay-Doi (2)).
(a) (F,+) is an F[x]-group and P(F) -group of type 2.
(b) If h (7.65) is an isomorphism (cf. 7.66(b)) then
F[x] is 2-primitive on (F, + ) = F.
(c) P(F) is always 2-primitive on F.
Proof. If f.f'cF, f + 0, then 3 pQeFo [χ]: pQ(f) = f,
namely ρ = f'f χ . The rest is equally obvious.
If the reader is still interested, he is cordially invited to
a nearly complete trip to the ideals of F[x] and P(F).
First we settle the question, for which F F [x] happens to
be simple.
7.89 THEOREM (Straus (1)). Let F be infinite. Then F[x] is
simple.
Proof. If I 3 F[x], I + CO}, take some ίεΐ, i + 0.
By 7.67, Τ + 6, so 3 feF: T(f) = iof + 0. Hence
iofelrtF and 7.86(b) implies I = F[x] .
So the infinite case is settled and we turn to finite fields.
To do so, we first determine all full ideals (7.90) of F [xl
and then we will see (7.93) that, if char F 4= 2, all ideals
of F[x] are full ideals.
7.90 THEOREM (Menger (2), Milgram (1), Lausch-Nobauer (1),
Straus (1)).
Let F be a finite field and I an ideal of (F [xl ι + »·) ·
Since (F[x],+,·) is a PID, I is some principal ideal (p)
of this ring (F[x],+,·). Then the followinq conditions
are equivalent:
(a) I = (p) is a ful1 ideal.
(b) V scF[x]: p/pos.

7d Polynomial near-rings
225
(c) There exist kcIN , η,,.,.,η.εΙΝ with l<n,<n~<...<nk
and m, ,. . . ,m. ε IN wi th
n. n. „
1 m. к ш,
ρ = l.c.m. {(xq -x) ,...,(xq -x) } where |F| = q
nnl m. nn2 m, nnk m.
(then I = ((xq -x) l)n ((xq -x) ')л ... л((хч -χ) κ)).
Proof. (Straus).
(a) =-> (b ) is trivial.
(b) => (c): Let С be the alqebraic closure of the
field F. Let с be in С with p'(c) = 0. Let m be the
multiplicity of the root с Then с is a root of
multiplicity >m of each pos (seFTxJ). Hence
ρ has a zero of multiplicity >m at each element
of F(c) (field extension of F by adjunction of c).
Applying the theory of finite fields we see that ρ
is divisible by (xq -x)m, where η : = [F(c):F] :
V deF(c): (x-d)m/p, hence
(x" -x)'
Π
deF(c)
(x-d)m divides p.
But Π (x-d)"
deF(c)
Starting with a root с with maximal С^(с):^]
arrive successively at (c).
we
:c)
5y 7.77, it suffices to show that (p) is
a right ideal. Let s e R [x] . For m ,η ε IN , (xq -x)mos
induces the zero function. But xq-x is the lowest-
degree zero function in P(F), whence (x -x)|(xq -x)m,
If we do this for η
result.
1<iSk) we get our
7.91 REMARK The representation in 7.90(c) is moreover unique:
see Lausch-Nobauer (1), Ch. Ill, 7.21.
7.92 DEFINITION A polynomial ρ as in 7.90 is called saturated
(Milgram (1)).

226
§7 TRANSFORMATION NEAR-RINGS
7.93 THEOREM (Straus (1)). Let F be a finite field. Then:
Every ideal of F [x] is a full ideal <·=> char F =j> 2.
Proof. —>: Assume that char F = 2. We show the existence
of an ideal I of F [x] which is not a full ideal:
η
Let
and F[x\] : = { I a2i
,2i
i=0
ηε IN , a~ . cF}
Consider I: = (xq + x)2 ■ F Γχ2]+( xq + x)4 · F [x] .
I £ (F[x] ,+ ,· ) since (xq + x)2cl, but χ · (xq + x) 2<j:I.
Let ρ := (xq+x)2=x2q+xZeF[x2]. We show that
I 5 (F[x] ,+ so):
(a) Clearly, (I,+) is a normal subgroup of (F[x],+).
(b) In order to show that I ^^[х] it suffices to
consider u:=x °(r+pt+p s)-x °r with r,s ε F[x] and
t ε F[χ ]. If η is even , u
I. If η is odd then u
' r+pt+p s ) -r11
n.n 2n η
ρ t +p s
r""1 (pt+p2s ) + v
with ve(p2). So u = p(trn 1 ) + (p2rn " 1s + ν ) ε Ι, since
n-
l s even.
(c) In proving I 3 F [x] we have to show that for all
9 9
t ε F [x ] and all s ,r ε F[x] we get w:= (pt+ps)oreI.
Now w = (p°r)(t°r)+(ρ °r)(s°r). Since char F = 2,
t»rt F[x ] and poreF[x ]. Now p|p°r by the same
? 7
argument as in the proof (c) => (a) of 7.90 and ρ |ρ °r
as well. Hence (ρ »r ) (s°r ) ε ρ F [χ] , and
9 9 о
(p»r)(tor) ε pF[x ]F[x J 9 pF[x J. This proves we I.
г г 1
<-»: Let I^F[x]. Then for all iEI and al
ι ρ 2 2 2
we net i-p = 4(χ'°(ι + ρ)-χ °p- (x o( i +0) -x o0)) ε I
7.94 REMARK The proof of "7.93 <-" also shows:
If F is finite with char F + z tnen tne 1eft ideals Of
(Р[х],+ .°) are exactly the ideals of (F[x], + ,·)
(cf. 7.77).

7d Polynomial near-rings
227
7.95 REMARK Straus (1) also showed that if F is finite with
char F = 2 but |F| > 2 and if I <1 F [x] then
F[x ]·I h I and I contains an ideal J of (F[x],+,·) which
9
is generated by {i = i·i|i ε I} ; J contains all i. · i2
(i1,i2el).
7.96 COROLLARY If F is finite then F[x] does not fulfill
the DCC (because of (x)=>(x2)= ... not even the DCCL).
For more information see So (1).
7.97 COROLLARY If F is a finite field of characteristic + 2
then F[x] fulfills the ACC, but not the DCC on ideals.
Hence F [x] cannot be completely reducible (2.50).
So it remains to consider finite fields F with char F = 2
and there in particular F = ZL. As usual in algebra,
characteristic 2 causes a lot of trouble. The ideal structure
of Ζ2[χ] is much more complicated than that of F[x] in 7.89.
or 7.90/7.93.
We get satisfactory results concerning the ideals of F [x]
with char F = 2 only in the case of maximal ideals:
7.98 THEOREM (Clay-Doi (2), Brenner (1)).
(a) If F is infinite, {0} is the only maximal ideal of
F[x].
(b) If F is finite, but £ #2 then Ker h = ipeF[x]|"p = 6}
is the unique maximal ideal of F Γχ] .
(c) 2»[χ] has exactly two maximal ideals:
V: = {peF£x]|r> is constant} and
3
T: = the (near-ring) ideal generated by χ .

228
§7 TRANSFORMATION NEAR-RINGS
Proof. Consider h of 7.65 and the diaqram (observe 1.30).
r -ι (a) is settled by 7.89.
So we may assume that F is finite.
By 7.75, P(F) = M(F). So if
|F| + 2, P(F) is simple and
Ker h is maximal. If |F| = 2
then ΜC(F) is a maximal ideal
in M(F) = P(F) by 7.31. Hence
its pre-image under h (= V) is
maximal in F[x] . The facts that Τ is another maximal
ideal in ZL [x] and the uniqueness statements in (b)
and (c) involve some technical reasoning and we overgo
the proofs. That of (b) is in Clay-Doi (2), while
that of (c) can be found in Brenner (1).
7.99 COROLLARY (Nobauer (6), Hule (1), Clay-Doi (2), Lausch-
Nbbauer (1)).
F [x] is simple <=> F is infinite.
Theorem 7.98 has some applications. 4e mention
7.100 COROLLARY (Clay-Doi (2)). Z[x] contains maximal ideals,
Proof. If ρεΡ , (pZ)[x] <1 Z[x] (7.81(a)), and
2[x]/. ,. r ι = ^D W > which contains at least
one maximal ideal by 7.98. An application of 1.30
gives the statement.
All maximal ideals or all full ideals of TL [x] are not known
(Clay-Doi (2), Lausch-Nbbauer (1), p. 131). One also might
raise the question, which P(R) happen to be simple.
We are happy to have a full answer:
7.101 THEOREM (Nobauer (6)).
P(R) is simple <—> R is a commutative field with |R|>2.

7d Polynomial near-rings 229
Proof. =>>: (a) Let I be a non-trivial ideal of R. Let
h be again as in 7.65. By 7.81(a), I [x] <l R [x] ;
consequently h (I [x] ) = P(I) <IP(R).
Considering the constant polynomial functions yields
P(I) + {s} and P(I) + P(R), a contradiction.
Since P(R) is assumed to be simple, we arrive at
a nonsense.
(b) P(Z2) = M(22) is not simple by 7.31.
<«: Let R be a field. If R is finite + Z2 then
by 7.75 P(R) = M(R) is simple (7.33). If R is
infinite then by 7.67 P(R) = R[x] which is simple
by 7.99.
We get only partial results on the radicals of F [x] :
7.102 REMARKS
(a) Let F be infinite. Then ^2(F[x]) = ... = ?{F[x]) =
= {0}. This holds by 7.88(b). Cf. 7.123 (a).
(b) For any integral domain R, 11(R[x]) = ?*(R[x]) = (0),
for by 7.68(a) R [x] has no nilpotent elements =f 0·
(c) Clay-Doi (2), Mlitz (1) and (3) determined radicals of
some F[x] 's, which do not always coincide with our
<j2(N)»· * · «PCO · But one can get immediate results
on polynomial near-rings over finite fields F:
^2(F[x] ) £ Ker h (by 7.88 (a)) and if char F f 2
then jr1/2F(tx^) = '·· =η(ρ[χ]) = {0} (this follows
from 7.94). See also 7.123.
(d) The situation in the general case does not seem to be
clear. Anyhow, we observe two strange phenomena:
(1) The near-ring-radicals might differ substantially
from the ring-radicals of R [x] (the latter ones
are always = {0} if R is a field).
(2) The smaller F, the more complicated is the
structure of F[x] .
(e) See So (1) for a detailed study of the ideal structure
of R[x] and P(R).

230
§7 TRANSFORMATION NEAR-RINGS
[f) Every finite field can be obtained by forming the
ring (Z [x],+,.) for some prime p, choosing an
irreducible polynomial ρ and forming 2 [x]/(p). One
might wonder which near-fields can be obtained as
(Z [x] ,+ ,»)/I for some maximal ideal I. The answer is
very surprising: none at all ! In (2), Y.S. So has
shown: If R is a commutative ring with identity and
if F := (R [x"] ,+ ,°)/ I is a near-field then F is a
field and moreover a factor field of R ! In this paper
it is also shown that every near-field contained in
R[x] is a field. Also, every planar near-ring (see
8.85) contained in some R[x] is a field.
6.) Γ[χ] AND рГг]
7.103 PROPOSITION (Lausch-Nbbauer (1)). Every normal suboroup D
of r[x] is a left ideal of Γ [χ] .
Proof. Similar to 7.77: if deD <l (r[x], + ) and ρ,ςεΓ[χ]
then with ρ = γ£) + η1χ+γ.+n2x+ . . . +γr_i + nrx + Yr we net
po(d + q)-poq = Yo + n1(d + q)+Y1+. . .+Yr_ j + n,. ( d + q )+γρ-
-Yr-nrq--..-Y0eD.
The definition 7.74 of polynomial completeness is carried over
to r[x] in the obvious way. Similar to 7.75 we net
7.104 THEOREM (Lausch-Nobauer (1)). The polynomially complete
groups are exactly TL and all finite, non-abelian
simple groups.
Proof, (a) As in 7.75, a polynomially complete group Γ
is shown to be finite and simple.
If Γ is abelian and |ri>3 then Γ is some Ζ
(ρεΡ , p>3). Take feM(£ ) with f(0) = 0,
f(l) = 1 and f(2) = 0.

7d Polynomial near-rings
231
If q"eP[r], q~ has the form q": γ ■* Y0+nY (7.64).
From q(0) = f(0) and q"(l) = f(l) we net yQ = 0
and η = 1. But then q~(2) = 2 J- f(2), whence
f =j= q", and Γ is not polynomially conplete.
(b) Conversely, £2 1s easily shown to be polynomially
complete, while for finite non-abelian simple oroups
each ρ,: γ ■* δ + γ-δ (δεΓ) is in Ρ [г] . Consequently
Ι(Γ) ? Ρ [г]. But Ι(Γ) = Μ(Γ) for these ciroups
(7.46).
7.105 REMARK This result is transferred to Ω-groups by Lausch (2).
See part 7.) of this paragraph.
The following result is easy to prove.
7.106 PROPOSITION rQ [x] := (ГЫ )Q = ίΡ = Ύ0 + η1x+...+npx +
r
+ γΓ| Ι γ, = 0} = { Ι (γ.ίχ-γ,)|γ ε Γ} and Ρ (г) = Ι(Γ).
l'=0 111 О
Again, г can be considered as an Γ [χ] - and an P[r]-group
(cf. 7.80). But in contrast to 7.88 we get
7.107 REMARK Γ is as Γ[χ]- and P[r]-group not of type 2 in
general. (Consider an abelian, but not simple oroup and
you have a counterexample.)
But the theory of enclosing ideals still works:
7.108 EXAMPLES Let Δ <| Γε^ . Then
(a) (Δ) (= the ideal of Γ [χ] generated by Δ) <Ι Γ [χ] .
(b) <Δ>: = (Δ:Γ) <1 Γ [χ] .
See Lausch-Ndbauer (1) or Hule (1) for a description of (Δ).
Completely similar to 7.84 we get

232 §7 TRANSFORMATION NEAR-RINGS
7.109 THEOREM (Lausch-Nbbauer (1)). For each I <| г [х] there
is exactly one Δ <1 Γ with (Δ)ϊΙ?<Δ>.
7.110 DEFINITION Again, Δ is called the enclosing ideal of I.
Concerning simplicity we get surprisingly
7.111 THEOREM (Hule (1)). Γ [χ] is never simple (unless
|Γ| = 1).
Proof. Let Γ have at least 2 elements and take some
nonzero γεΓ. Suppose that Γ [/] is simple.
Consider the near-ring epimorphism
g: r[x] »- 22
γ0+ηιΧ+...+nrx+yr - (n1+...+nr)-l
Clearly g \ 0. Hence Ker g = {5} and g is an
isomorphism. Consequently |Γ[χ] | = |Z„| = 2 which
is quite hard to fulfill since г [х] is infinite.
7.112 THEOREM (Mel drum-Pi 1 ζ-So (1)). The only idempotents in
r[x] are χ and the constant polynomials γ in Γ[x] .
For the proof see the paper mentioned in 7.112.
7. 1 13 COROLLARIES Let Γ be a group.
(a) r[x] has exactly |r|+1 idempotents.
(b) If e is idempotent in Г [x] then e=0 or e=x.
(c) If {0}fNir[x] and N has an identity e then e = x.
L : = { Σ γ·+ζ·χ| γ. ε Γ, ζ· εΖ, Σζ· ερΖ } for some
ρ ι
7. 1 14 THEOREM All strictly maximal left ideals of r[x] are:
(a
prime p,
(b) L.:= (Α:γ), where A is a maximal normal subgroup of
Γ not containing the derived group [r,rjand γ ε Γ\Α
or γ = о.
(c) L„:= (B:o), where В is a maximal normal subgroup of
Г containing [г,г]

7d Polynomial near-rings
233
(d) L(<t> ,p): = { Σ γ^. + ζ^ еГ[х] j φ( Σ γ^ ) = Σ ζ. (mod ρ)} fo
prime ρ and φ ε Нот (г, TL ).
г а
The proof can be found in Meldrum-Pilz-So (2). The intersection
of all these four collections of strictly maximal left ideals
gives the ^„-radical of Г [x]. For a group G, let B(G) be the
intersection of all maximal normal subgroups of G ( β (G) is known
as the Baer-radical of G). Then it is shown in the same paper:
7.115 THEOREM If β(Γ) 2 [Г,г] then ^„(r[x]) = (3(Г):Г)
Note that this applies to all solvable groups, for instance,
7. 1 16 COROLLAR
Υ Ύ( Ζ [χ]) = (01 for
each ν.
Concerning ν = -~ we get (again in the same paper'
7.117 THEOREM Ί]/2 (Γ[χ]) = β(Γ [χ] ,+).
7.) POLYNOMIALS OVER Ω-GROUPS
We now see, how the generalizations mentioned at the end of
p. 217 can come true. We start by fixing a variety 1? of Ω-
groups and, according to the lines on p. 216, we restrict our
considerations to the case X = {x}. This is the reason for writing
7.118 NOTATION A^[x] : = A({x},^·) is the polynomial algebra
over A in V. P(A) is the set of all polynomial functions
from A into A.
Even if ^is a "well-behaved" variety, there might not exist
"normal forms" for theelements in A [x] . For instance, if 22 i s
the variety of all rings, we get for R e 1?:
R*[x]
2 2
Q. _1ЛТ, р.л, 2^,,, .p, k^3x +r5x +...,. ^. ,_v
and the situation is even much worse if^is the variety of

234
§7 TRANSFORMATION NEAR-RINGS
near-rings, for example. On the other hand, for the variety
^of abelian groups and Α ε jf we get
A^ [x] = { a + zx| a ε Α, ζε Ж).
It should be noted, however, that A[x]does not depend on the
variety 1? A is taken from. For instance, for the (abelian)
group Ϊ we get in the variety<4 of groups.
whi 1 e
but
Z*[x
Z-*[x]
{zo+z1x+z2+z3X1
{ζο+ζ1χΙΖο'ζ1 ε Z)
, + z x+z ,
η η+1
2 }
Ζ [χ] = {χ
We state without the the simple proof
ζ +ζ.χ|ζ„,ζ. ε Ζ} holds in both cases
οίο ι
7.119 PROPOSITION If V is a variety of \i- groups and А с V
«V
then A [x] and P(A) are near-rings w.r.t. + and r.
The correspondence p^p is a near-ring epimorphism from
A^[x] onto P(A) .
7y
As before, we denote the zero-symmetric parts of A [x] and
P(A) by Ao^[x]and P0(M, respectivly. If A,B n^and h is an
epimorphism form A to В then h extends uniquely to a (near-
ring) epimorphism h* : A [x] -+ Β [χ] with h*/„ = h and
h(x) = x. If h is an ismorphism, h* is an ismorphism, too
(see (Lausch-Nobauer (1), p.15). There exist, however,
near-ring homomorphisms h between polynomial near-rings with
h(x) \ x:
7.120 EXAMPLE (So (1)). Let N = Z~ [x] (in the variety of
commutative rings with identity). Then
h 1 U0 + a1 x+ . . . +anx
p(aQ+a1x+..-+anx
and
h2(ao + a1x + . . -+anx ) = p(a^
.a ) x+ pa
η 'о
define two different near-ring homomorphisms from N to
H with h.(x) = hJx) = px and h. , = h,
In
2/,
"2p
■2p

7d Polynomial near-rings
235
7.121 DEFINITION Let A be any Ω-group.
f εΜ(Α) is called com patiЫ e if for each ideal I of A
we get a. ξ a„ (mod I)^f(a-) Ef(a») (mod I). C(A) is the
set of all compatible functions from A to A.
Hence each ideal of A is an ideal of the C(A)-group (A,+).
In particular f(i) e I for each ideal I of A, each i ε I and
each f e С (A).
7.122 PROPOSITION For each Ω-group A, C(A) is a near-ring with
P(A) < С(A) i Μ(A).
Proof. C(A) < M(A) is easily seen. id. and the constant
functions are compatible, so is each "ρεΡ(Α), since
{idA)uMc(A) generates P(A).
Near-ring theory not only receives contributions from universal
algebra (cf. e.g. 1.60), but also pays something back. We give
such an application and then numerous applications of this
application.
If γ is an element of some group г then the normal subgroup
Γ generated by γ „ consists of all finite sums of elements
о о
of the form γ±γ -γ with -, *. Γ, i.e, Γ
{f(Y0)!f r. Ι(Γ)} =
= {ρ(γ)|ρ ε Ρ (Γ)}. This motivates the following result which
describes generated ideals in o-gr0ups completely.
7.123 THEOREM Let A be an Q-group and a ε A. Then the ideal
(a) generated by a is given by
(a)
If Bs A then the ideal (B) generated by В is given by
(B) = ε (b).
Up(a) [ ρ ε Ρ0(Α)}
beB
Proof. It suffices to show the part for (a
Since (a ) < A , (a )
A,hence (a)з A,
Let N:= PQ(A;
whence { p(a ) | ρ ε Ρ ( A)} = Nac(a).
Conversely, we will show that this Na is an ideal
of A containing a, from which we get (a ) £ N a .

236
§7 TRANSFORMATION NEAR-RINGS
) Because of id, ε Ν, a = i d ( a ) ε Ν a .
i) Clearly, Na is a subgroup of (A , + ) ·
ii) Vb ε A Vp(a ) ε Na :b+p(a)-b = ( b_+p-t>) ( a ) , where
b_ is the constant polynomial function with value b
Now b+p-b_eP(A) and (b_+p-b)(0) = b+p(0)-b = b-b = 0,
so b+p-b eN and b+p(a)-beNa. Hence Na is normal
in (A,+).
(iv) Let ω be an n-ary operation on A and
b15...,beA, p1(a),...,p(a)cNa. Consider
b:= ш(Ь1+р1(a!
,b +p (a'
' η ' η ч
(b.,...,b.
Let q be the polynomial
ω (^ + p^
>kn + pn'" ω ^b1 '
»br
Then q(a
and q(0)=0, therefore b = q(a) ε Na,
One possible application (cf. 2.52 (b) 1 ) is given in
7.124 THEOREM Let the Ω-group A be a subdirect product of
simple Ω-groups A,(i ε I) and suppose A has the DCCI.
Then A is a finite direct sum of the simple Ω-groups A.
(j с Jsl).
Proof. Since A is a subdirect product of the A.'s there
there is a (by the DCCI finite) family of ideals K.
η J
(say with j e {1 ,. . . ,n} ) and f] K.= {0}. Suppose that
j = 1
J
η is minimal with this property. Since A/K - A·,
j η
each K. is a maximal ideal. We show that A = ©Α.,
J
j = 1
It suffices to show that each (0,...,0,a.,0,...,0)cA
whenever a . ε A .. If a ε f] A. but a f A. then a j= 0
J J teJ 1
and (a) = {p(a) Ι ρ ε Ρ (Α)} by 7.123. Hence there is
some "ρ.εΡ (A.) with a,-= p.(a) еИ.сй.
Again by 7.123, a. = p.(a) ε /| Α.. Hence all t-th
J J t+j l
components (with tj=j) of a are zero, whence
(0,. . . ,0,a. ,0,... ,0) = a e A.

7d Polynomial near-rings
237
7.125 THEOREM Let V be a variety of Ω-groups, Ae^, Μ a maximal
ideal of A and a e A\M. Then
(M:a) = {ρ ε Α [χ] | ρ (a) = p»aeM} is a strictly maximal
(see 3.92) left ideal of AV[x] .
Proof. Let N:= A [x]. A is an N-group, so (M:a)i{ N by
1.42. Since a φ Μ, χ (£( Μ: a ) , whence (Μ: a ) j= N.
Now suppose that U -^ Ν is strictly between
о
(Μ : a ) and N. Take u ε U \ (Μ : a ) . Then u ° a 4 M · Now
u-u»a e (M:a)cU, whence u°aeU. Hence Μ <= U η A.
The ideal (11лА) generated by ΙΙλΑ equals A. Since
U<M N, AiSU by 7.119. Hence x-ae(M:a)eU, and
N° v
aeU implies χ г U. Since A^ {x} generates Α [χ] ,
ν
U = Α [χ] , a contradiction.
This enables us to compute an upper bound for the (near-ring)
radical s of АУЫ and P(A).
7.126 THEOREM For ν ε{0 ,1 /2 ,1 ,2 } we get 3V(Α [χ])с ША):A)
where Я.(А) is the intersection of all maximals ideals
of A (the "radical of A").
Proof. l,(AV[x1)6l(AVfx])g /Ί Λ (M:a) =
Μ max. aeA
= ( Γ\ Μ:Α) = ША):А).
Μ max.
7.127 EXAMPLES Let V be the variety of commutative rings with
identity.
(a) Let ReU be semisimple with (R,+) torsionfree.
Then 2v(R^[x] )s(0:R) = {0}, hence ^(R^x]) = {0}.
In particular, we reproved 7.102 (a).
γ
(b) If R is a finite field of order >2 than V2(R[x]) =
Vr
= (0:R).
:c) By 7.98 (с), /)2тигМ)c(o-.22:
;d) See also 7.115 - 7.117.
In Pilz-So (1) it is shown that if R is a field with char R J 2
then ^i/oC [x] ) = {ОЬ In this paper, the following result is
also proved.

238 §7 TRANSFORMATION NEAR-RINGS
7.128 THEOREM Let R be a ring with identity. Then Ρ (R) is
a ring iff R is a Boolean ring. In this case we get for
all ν ε {0,1/2,1 ,2}:
2v(P(R)) = ^(P0(R)) + ^(R) , where "I is the Jacobson radical
of ring theory and $(R) is the intersection of all
maximal submodules of the P_(R) - module R.
The statement concerning ^ (P(R)) will follow from 9.77. We
remain at P(R) and cite a result of (Keller-Olson).
_ „6(10
7.129 |P(Z
!P(Z . J| for к > 2 and | P(Z ]
pk~] p
Μ (Ζ ) | = pp. In there, e(k) is the smallest t ε IN with
ν Ρ
pklt!.
Ί
From P(ZJ = © P(Z . ) for η = p. ...p„ we get
n i=1 Ki ' r
Pi
|P(2 )| by repeated application of 7.129.
There are numerous near-rings between P(A) and M(A).
7.130 DEFINITION Let A be an Ω-group and η e N.
(a) LnP(A):= {f e M( A)| \/ Τ s A , | Τ | < η , \/ ρ ε P( A) : f fj = p/T)
(b) LP(A):= fl LnP(A). The elements in LP(A) are called
nelN n
local polynomial functions.
Hence local polynomial functions can be interpolated by
polynomial functions, and we are back to the topics treated
on pages 133/134 and 219/220. First we state
7.131 PROPOSITION Let A be an Ω-group. Then LP(A) and each
LnP(A) are near-rings with P(A)i LP(A]
L3P(A)i L2P(A) = C(A)S L^U;
LnP(A)<
m(a:
Proof. It is easy to see that LP(A), L P(A) and C(A)
are subnear-rings of M(A) with P(A)cLP(A)cL P(A]
sL (A) = M(A). Let η > 2 and a = b in A.

7d Polynomial near-rings
239
Then there is some ρ εΡ(Α) with f(a) = p(a) and
f(a) = p(b). Hence f(a) ;f(b) by 7.122 and we have
shown that L P(A)cC(A). It remains to show that
L2P(A)2C(A). Let f eC(A) and suppose that a.beA.
Let χ ξ у iff x-yr(b-a). Since a = b we have
f(a) Ef(b), whence f(b)-f(a)e (b-a). By 7.123 there
is some ρ eP (A) with "p(b-a) = f(b)-f(a). Now
q: = p'tx^ahf (a) εΡ(Α) fulfills q(a) = p(0)+f(a) =
= f(a) and ^(b) = "p(b-a) + f(a) = f(b). Hence
fe L2P(A).
Looking back to 7.75 we (re-) define:
7.132 DEFINITION A is called
(a) polynomial 1 у complete if P(A) = M(A).
(b) affine complete if P(A) = C(A).
(c) locally polynomial1 у complete if LP(A) = M(A).
(d) locally affine complete if LP(A) = C(A).
Obviously, (locally) polynomially complete algebras must be
simple. We remark that our definitions differ slightly from the
ones in Lausch-Nobauer (1), since we are only concerned with
polynomial functions in one variable.
In (11), Nobauer characterized compatible functions on the
rings Ζ and Ϊ . We mention without proof.
3 η r
7.133 THEOREM C(Z) = {f:Z+Z|f(x) = Σ с,A(i)(x^1)}, where
i=0 1
с i ε Ζ, A(i) = 1.с.m . of 1 , 2 ,. . . , i.
7.134 EXAMPLE (Nobauer (11)). f:Z^Z, χ -+ ?(x4+x2) is compatible,
but not a polynomial function.
7.135 EXAMPLES
(a) A commutative ring R with identity is polynomial1 у
complete iff R is a finite field (7.75).

240
§7 TRANSFORMATION NEAR-RINGS
(b) By Lagrange's theorem, R is locally polynomially
complete.
(c) The rings Z„, Ζ, , Zft are polynomially complete,
affine complete, not affine complete, respectively
(see Pi 1 ζ - So (1)).
(d) By 7.104, Z„ and finite simple non-abelian groups
are polynomially complete.
(e) From 4.66 (a) we know that LgP(A) = M(A) implies
LP(A) = M(A); hence A is locally polynomially complete
in this case.
(f) A near-ring N is polynomially complete iff N is finite
and simple and if N has either non-abelian addition
or abelian addition with a multiplication depending
on both arguments (Istinger - Kaiser (1)).
We shall improve these results considerably. For that, we
define a concept due to S.D. Scott, which is related to
7.121 (see 7. 140).
7.136 DEFINITION Let N be a near-ring and Γ an N-group
Γ is called compati bl e if for all γεΓ and ηεΝ there
is some m ε Ν with η(γ + 5 ) -η γ = mo for all 6 ε Γ.
N is called compat ible if N has a faithful compatible
N-group Γ (we express this by saying that Μ is compat i ble
on Γ).
This condition means that N admits all horizontal and "many1
vertical translations (see Pilz (6)):
Γ
η(γ)! \

7d Polynomial near-rings
241
with n(5) = η(γ+δ) and m(6) = η(δ)-η(γ) = η(γ+6)-η(γ).
(This picture shows a nr. N > M(r) which is compatible on Г.)
7.137 EXAMPLES M(r), MQ(Г),MC(Г ) , Mcont(r) (for a topological
group Г), Md.ff(P) (see 1.4 (a)), P(R) and Ma(r)
(9.69; Г abelian) are compatible.
7.138 PROPOSITION An N-group Г is compatible iff it is
compatible as an N-group.
Proof. If Г is a compatible N-group then for all η ε Ν
and γ ε Γ there is some m ε Ν with η(γ+ό)-ηγ = mo for
all όεΓ. Decomposing m into m=m +m according to
1.13 gives m 6+m 6 = m 6+m о on the riqht side.
3 о с о с я
Choosing ό=ο yields о =m o+m о. Hence η(γ+ό)-ηγ = m 6
for all 6 ε Γ and Γ is a compatible N-group. The
converse is even easier and omitted.
7.139 COROLLARY If N is compatible on Γ then NQ is compatible
on Γ.
7.140 PROPOSITION Every near-ring N between Ρ (A) and C(A)
(in particular, each member of the chain in 7.131) is
compatible on (A,+).
Proof. N acts on A in the obvious and faithful way. Let
f ε Ν and a ε A. Then q:=f°(a+id)-f°a, where a^ is the
function which is constantly =a , is in N and
g(b) =f(a+b)-f(a) for all b ε A.
Without proof we mention a result on compatible N-groups.
7.141 THEOREM (Lyons-Scott (1)) Let Ν εUQ л Щ be compatible on
ΝΓ. If N has the DCCL then Г is "nilpotent by finite",
i.e. г has a nilpotent normal subgroup Δ such that Γ/Δ
is finite.

242
§7 TRANSFORMATION NEAR-RINGS
Primitive compatible near-rings are studied in Scott (17), where
it is shown that if the near-ring Νεΐ is 2-primitive and
compatible on ΝΓ with ACCL then either N=M (Γ) is finite or N is
sparse in a certain topology (arising from "zero sets"
N
Γ)
S.D. Scott mentions in private conmunications that in this
second case (if Γ is infinite) Γ either has prime exponent or
else Γ is divisible (cf. 9.190(c)).
We will return to compatible N-groups in §9 g).
7.142 THEOREM Let N be as in 7.140 and A (non-zero) simple
Ω-group. Then N is primitive on (A,+).
Proof. Let В be a non-zero subgroup of (A,+) such that
η ° b ε Β for all ncN and b ε Β. Let ω be an n-ary
о J
•bne
operation in A and a. a ε A ,b.
Then a. = p-(b) and b:· = q.(b) for some b ε Β and
p.,q. ε Ρ (A.) by 7.123. Let ρ be the zero-symmetric
polynomial function ζ-»-(ω( p.+ q,.
- ω(ρ1,...ρ
- ω (a 1
whence В
[ ζ ). Then ω(a. + b *
'Pn+V
"an + bn
, a ) = p(b)cB. Hence В is an ideal of A,
A.
A powerful result of S.D. Scott enables us to draw important
conclusions. We will mention this result in 9.170 (g).
7.143 THEOREM Let A be a simple Ω-group. Then P(A) is either
dense in Μ(A) (then A is locally polynomially complete)
or P(A) is dense in Μ ff(A) (in which case A is a vector
a τ τ
space over Horn ,,,(Α,Α) and PQ(A) is a ring.
г о
Proof. By 7.142 and 7.139, PQ(A) is primitive and
compatible on(A,+). By Theorem 9.170, P0(A) must
either be a primitive ring or dense in Μ ( A). Now
4.52 gives the result. In order to extend this result
to some non-simple Ω-groups we need more information.
7.144 PROPOSITION Let A be a subdirect product of Ω-groups
Ai (i ε I
f, cC(Ai
Then for every f cC(A)
with f ( . . . ,ai ,. . . ) = ( .
ι .,...) ε A. If f ε Ρ(Α]
all (... ,ai ,.
i ε I. (For the proof see e.g. Pilz (25)).
there are unique
:.. ..f^a.),...) for
then fi ε Ρ(Α. ) for all

7d Polynomial near-rings
243
7.145 THEOREM Let A be as in 7.144.
If J. denote the annihilator
of A in P(A) then Ρ(A)/J i = P(A.) is isomorphic to a sub-
direct product of the P(A.)'s.
Proof. We assign to every fεΡ(Α) the uniquely determined
in 7.144. This gives a homomorphism φ
fiEp(Al
from P(A) into P(A.) with kernel кегф
{p'p. =o
zero map}
Ji
is an endomorphism:
if ρ- ε Ρ(A.) then p- is a word p.
1 (к) 1
id. ) with a ■ ε A. We replace a:
k)
oH a.
by some
U..,a:
Then ρ:
к)
εΑ and id. by id.,
ч i
iila:
,a:
,id.
PA and
Now the homomorphism theorem does the rest of the
job, together with the remark that Π J- ={0} .
i ε I 1
This shows that each P(A.) is a homomorphic image of P(A) if
A is a subdirect product of the A■'s.
The next result follows from 7.143 and 7.145.
ι ε
7.146 COROLLARY Let A be a subdirect product of the simple
Then P(A) is semisimple and each
ι (if Ρ (A. ) is not a ring) or
is a ring).
Ω-groups A.
Ρ(A.) is dense in M(A^
in Maff(A.) (if P0(Ai)
Now 7.124 gives us
7.147 THEOREM Let A be a subdirect product of simple Ω-groups
Then P(A) is the
Ai (id) such that Ρ (A) has the DCCL.
direct sum of finitely many of the P(A.)'s.
either equal to M(A i
(with dim A. finite)
Each P(A.
ι s
,with A. finite
or to Μ ~ Л А .
атт ι
7. 148 COROLLARY Let A be as in 7.147 such that none of the
P_(A . )'s are rings (c.
and 2-semisimple.
7. 128!
Then P(A) is finite
Finally, we close with some remarkable embedding theorems. The
proofs can be found in Meldrum - Pilz - So (1).

244
§7 TRANSFORMATION NEAR-RINGS
7. 149 THEOREM
[a) For every near-ring N there is a variety^of Ω-groups
and some к zU with N<*A [x].
[b) There exist d.g. n.r's which cannot be embedded in
some Γ [χ] (Γ in the variety ^"of groups).
]c) Every finite near-ring can be embedded in some P(r)
for a finite, simple non-abelian group Г.
[d) For every group Г there is some group Δ with Γ [χ] <5>
Ρ(Δ) (Δ can be chosen as (r[x],+ ))·
[ see 9.71) can
(e) Not every abstract affine near-ring
be embedded in some A [x], where Λ is the variety of
abelian groups. But every near-ring N is abstract
affine iff N is isomorphic to some A [x], where Λ
is a variety of (ring-) modules.
(f) With a similar idea as for (a) (namely by adding
unary operations), one can find for each compatible
nr. N some Ω-group A with Μ = P(A). Hence, by 1.86
and 7.137, every nr. can be embedded in a compatible
one, even in a P(A)-type one (S.D. Scott, private
со mmunication).
For many purposes it would be very valuable to have a better
knowledge of the ideal of all polynomials which induce the
zero function. This is just the kernel of the (near-ri nci )-epi -
morphism which assigns to each polynomial its polynomial function
As we have seen in this chapter, this kernel decides if one can
identify polynomials and polynomial functions. It also has
several connections with the radicals of polynomial near-rings.
In Meldrum-Pilz (1) these questions are further investigated,
but they are far from being solved.
8) CONCLUDING REMARKS
We close this section with some remarks concerning questions
related to polynomial and polynomial-1ike near-rings.

7d Polynomial near-rings
245
7.150 REMARKS
(a) Nb'bauer (6) remarked that for R,Sc#, R [x] = S [x]
implies that R = s (this follows from 7.119).
He also remarked that each subnear-ring of M(F)
(F a field) which contains all constant functions
is automatically simple.
P(R) is directly decomposable iff this applies to R.
(b) If С is a composition ring and D is a map С -* С then
D is called a deri vati on (Mencier (3), Muller (1),
Lausch-Nbbauer (1), Nb'bauer (9)) if for all a.bcC:
(1) D(a+b) = D(a)+D(b) ("sum rule")
(2) D(a-b) = D(a)-b+a-D(b) ("product rule")
(3) D(aob) = (D(a)ob)-D(b) ("chain rule")
Clearly the zero endomorphism on С is a (trivial)
derivation. RΓχ] has also a non-trivial derivation,
namely the usual one: D: ρ -* ρ'. All on R [x]
arc; given by D : ρ -+ r-p', where rcR is idempotent
(Lausch-Nbbauer (1)).
Nb'bauer (6) showed that the composition rinq M(R)
has no derivations except the trivial one. If R is
a finite field, the same applies to P(R) (by 7.75).
If R is an infinite integral domain then Muller (1)
showed e.g. that if (R,+) is torsion-free, the sum-
and the chain rule imply the product rule. Muller
studied also "derivations" in near-rings as well as
"intervations" (see (8)). Cf. also Seppala (1).
(c) Invertible elements (w.r.t. o) are studied in Lausch-
Nbbauer (1) and Suvak ((1), (2)).
Those pcR[x] such that "p is bijective (= invertible)
are called permutation polynomials, were considered
by many authors and are presented extensively in
Lausch-Nbbauer (1).
(d) Clearly R [x] and P(R) are in general non-commutative
near-rings. Those polynomials which commute with a
certain family of others were studied e.g. by
Kautschitsch (1) and Lausch-Nbbauer (1).
Call С ^ F[x] (F a field) a P-chaJn ("permutable

246
§7 TRANSFORMATION NEAR-RINGS
chain" ) if \/ с ε С : \ с | > 0 , \/ к ε И j с ε С: [с] = к
2 3 }.
and <^л°^2 = с2°с1 ^ог а11 с1,с2еС.
Examples: (1) The P-chain of powers ίχ,χ',χ0,
(2) The P-chain of Cebyshev polynomials
} (where t is defined
over F = Ц
and then transferred to F[x] for an
arbitrary field F;
via cos ηφ = t ocos
t, = χ
l + 2x'
.3
t, = -3x+4x~
? 4
t4 = l-8x +8x
Also , t„ot = t m.)
η m nm '
If I is a linear polynomial and С is a P-chain then
С^: = iloCot' | с ε С} is a P-chain, too, called a
conj ugate P-chain.
One can see (the proofs are not too easy - see Lausch-
Nbbauer (1), p. 156 - 159):
(a) If С is a P-chain then С contains to each kcIN
exactly one с with [с] = к.
(β) All P-chains over a field F are conjuoates of
either the P-chain of powers or of the P-chain
of Cebyshev polynomials.
(e) Lausch-Nobauer (1), ch. 5, contains more information
on Γ[χ} and Р(г) . For example, the classes
Ε. (Γ) of all k-place functions oenerated by all
"k-place endomorphisms on Г" are considered ("k-
dimensional composition groups").
These are more examples of dgnr.'s.and results
similar to our 6.33 and 7.46 are obtained.
(f) Heatherly (7) considered FQ[x] (F a field). This
is a near-ring with identity, but without divisors
of zero. F[x] is also not regular §9 f)'
The

7d Polynomial near-rings
247
ideals I. : = { У a,x j ηε IN , a.cF} form a strictly
K i=k Ί ° Λ
descending chain. So F [x] does not fulfill the
DCC on ideals (cf. 7.97).
(g) Nbbauer (6) also considers the near-rings R(x) and
"R(x) of all "rational polynomials" and "rational
polynomial functions". Again R~(x) = M(R) iff R is
a finite field (cf. 7.75). £(x) is directly
decomposable iff R is it (cf. Remark (a)).
(h) The near-rinqs R0[[x]] °f aH formal power series
over Rcfl were considered by Frbhlich (9), Cartan (1),
Kautschitsch (1)- (8) and others.
Frohlich (9) studied Μ:=(Κ0[[><ι Хп-^П' defined
in this set a composition "o" by (f°g)·: =
= f1-(g1,...,gn) (where f^ denotes the i-th
component of fcM.
If f ξ g: <=> all f- and g^ have the same
degree, then one can cefine in Μ/ξ an addition
"+" in that way that (Μ/ξ,+,ο) is a near-ring
of number-theoretic relevance.
Cartan's result was already mentioned in 1.12.
Graves-Mai one (3) looked at the subnear-rino
N = { Σ a2p + 1x2n + 1|a eIR } of lR[[x]].
n>0
N satisfies the right Ore condition (1.64) and is
integra1.
(i) Heller (1) defined generalized polynomials ρ in a
composition ring R by the property that for all f eR
there are η ε IN and constant с ,. . . ,c £ Я with pof =
= с +c,f+...+c fn. There exist composition rings in which
о 1 η 3
every element is a generalized polynomial, but not a
polynomial.
Anyhow, this section seems to be a wide field for -further
research.

248
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
This chapter brings up two important classes of near-rirms.
start with perhaps the most important class, the near-field
A thorough treatment would require nearly a whole book. But
there are several excellent presentations of parts of this
theory (e.g. Karzel (1), Kerby (7) and Wahlinn (6)) so that
we dare to give the theory partly without proofs. First we
characterize those nr.'s which happen to be nf.'s. After
showing that the additive aroup of a nf. is abelian we oive
a super-sonic trip through the relations between near-field
and geometry (incidence groups, coordinatisation of planes,
pianar near-fields).
In b) we deal with planar near-rings. Their structure is
explored (8.90, 8.96), "blocks" aN+b (a =f 0) are defined
and it is shown that a planar finite near-rinn tooether wit
its blocks forms a tactical configuration (N,B). The case
when (N,B) is a balanced incomplete block desion is
characterized in 8.118 and several consequences are deduced
The author thanks Dr. G. Betsch for leaving him unpublished
lecture notes concerning this paragraph.

8a Near-fields
249
a) NEAR-FIELDS
1.) CONDITIONS TO BE A NEAR-FIELD
We start with (cf. 1.15)
8.1 PROPOSITION If N is a nf. then either N - Mc(Z2) or N
is zero-symmetric.
P_ro_o_f .If n c ε N c ' nc^R' then 1 = ncnc = n с' whence
ΙεΝ .
с
So V ηεΝ*: η = In = 1, hence Ν = {0,1}. The
rest is obvi ous.
8.2 CONVENTION In all of our subsequent discussion of near-
fields we will exclude this silly nf. Mc(22) of order 2.
(cf. Malone (2)).
Evidently, every near-field is simple.
We now characterize those near-rinos which are near-fields:
8.3 THEOREM (Li oh (2), Maxson (1), Beidleman (1), Fain (1)).
Equivalent are for Νε7)0:
(a) N is a near-field.
(b) Nd + {0} and \/ ηεΝ»: Νη = N.
(c) N has a left identity and ,.N is N-simple.
(d) N has a left identity and N is 2-primitive on „N.
(e) N has a left identity and N is 1-primitive on »,N.
Proof, (a) =-> (b) is clear.
(b) =-> (a): V a,bEN* 3 a',b'eN*: b'b = а Л а'а = b'.
Thus a'(ab) = (a'a)b = b'b = a + 0, so ab + 0 and
Nisi ntegral.

250
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
Take some άεΝ*. 3 ecN: ed = d.
So (de-d)d = ded-dd = 0. From above, we net de = d
Now let η be εΝ*.
Then d(en-n) = den-dn = 0, whence en = n.
Finally, \/ ηεΝ* 3 η'εΝ*: η'η = e.
This shows,that (N*,·) is a group and (N, + ,·) is
a near-field.
(a) =■> (c) <-»> (d) <™> (e) are obvious (observe 4.6)
j.
(b)
8.4 REMARK (Li qh (2)). Of course, e.o. (c) in 8.3 can be
replaced by (c)': "Nd 4= {0}, V ηεΝ» 3 η'εΝ»: η'η + 0
and NN is N-simple." (For (c)' => (b) =--> (c) => (d) ~>
-> (c)'!)
Without proof we mention the followinn results of Ligh (2) and
(1):
8.5 THEOREM Let N + i0} be a dgnr..
N is a skew-field <=> \j neN* 33 η'εΝ: nn'n = η <=>
<=> V ηεΝ*: Νη = N.
8.6 COROLLARY A finite integral dgnr. is a commutative field.
A dgnr. Ν + ί°> with left identity is a field iff it
i s N-simple.
8.7 THEOREM ΝεΤ) nTIj is a nf. <-> every ηεΝ, η + 1, is qr.
(in Beidlenan's sense - see 3.37 c)).
8.8 REMARK See Andre (3) for a development of a theory of
"linear algebra over near-fields" and "near-vector-spaces"
(cf. also Beidleman (1)). See also Grb'ger (1), Pellegrini
(1) and Rado (1) as well as 7.102 (f). A very good survey
on the applications of near-fields is Karzel-Kist (1).

8a Near-fields 251
2.) THE APDITIVE GROUP OF A NEAR-FIELD
Let the character!stic char N of a near
as usual - (Wahling (6) defines char N: =
gives the same {see 8.23)).
Then one sees as for fields:
8.9 PROPOSITION Let N be a nf. and o(l) be the order of 1
in (N ,+ ) . Then
(a) If o(l) is finite then char N = o(l).
(b) If o(l) is infinite then char N = 0.
(c) char N is either 0 or a prime.
For the following result, cf. and apply 1,5.
JLi°JlL9L0ALTJL°Ji (Karzel (1), Maxson (J). Ligh-Neal (1)).
Let N be a nf. . Then
(a) V ηεΝ: (η2 = 1 <=> пс{1,-1}).
(b) V η,η'εΝ: η(-η' ) = (-n)n' = -nn'.
Proof, (a): "<=" is clear; so let η =1, but η J= 1.
If char N = 2 then (observe that (N,+) is abelian
in this case) (n+l)n = η +n = (1+n)·1j now n+1 Φ Ο,
whence η = 1, a contradiction.
Now let char N be 4= 2, and 1 + 1 =: 2 (+ 0).
Then 2(-l) = (1 + 1)(-1) = -1-1 = -(1 + 1) = ("1)2.
So (-2)"1 = (2-(-l))"1= (-l)"1^"1 = -2"1.
Observe that (2"1(-1 ) + l)(-2) = 1-2 = -1, whence
(2-1(-1)+1) = 2"1. Let m: = 2_1(n-l)+l.
Then m-n = 2"l■(n-1)·n+n = 2_1(n2-n)+n = 2"1(l-n)+n-
-1+1 = 2_1(-l)(n-l)+(n-l)+l = (2_1(-l)+l)(n-l)+l =
= (2_1)(n-l)+l = m = m-1.
nJ-1 gives m = 0, so η-1 = 2·(-1) = -1-1, whence
η = -1.
field N be defined
char Nj - but this

252 §8 NEAR-FIELDS AND PLANAR NEAR-RINGS
(b) It suffices to assume η =j» 0. But then
(n'^-ljn)2 = 1, so by (a) we get n_1(-1 )ηε{ 1,-1} .
η" (-l)n = 1 implies -n = n, so char N = 2.
But then the result is trivial.
If n_1(-l)n = -1, n(-l) = (-l)n, so
\/ η'εΝ: n(-n') = n(-l)n' = (-l)nn' = -nn'.
The following famous result was first shown for finite nf.'s
by Zassenhaus in 1936, for infinite nf.'s by B.H. Neumann in
1940. There exist several essentially different proofs. The
following (due to Karzel) seems to be the most simple one.
8.11 THEOREM (Dickson (1), Zassenhaus (1), B.H. Neumann (1),
Li gh (6), (13), Ligh-McQuarrie-Slotterbeck (1), Karzel (1),
Zemmer (2)).
The additive group of a nf. is abelian and
characteristically s imple.
Proof. By 8.10(b) V ηεΝ: η(-1) = -n. Hence by 1.109(a),
(N, +) is abelian.
Consider for ηεΝ* the automorphism α : N -* N .
χ -* xn
If Μ + {0} is a characteristic subgroup of (N,+)
take mcM*. Let n' be arbitrary in fl*. Then
α , (m) = η'εΜ*, whence Μ = N.
m~V
Recall that from 1.88(f) and (a) one sees that every nf. is
isomorphic to a nf. of bijective mappings (plus the zero map)
on an abelian group Γ.
Standard group theory gives us the structure of the additive
groups of nf.'s:
8.12 COROLLARY (Ligh-McQuarrie-Slotterbeck (1), Heatherly (11)).
Let N be a nf.
(a) If char N = 0 then (N,+) is torsion-free divisible,
so the direct sum of copies of (φ,+).
(b) If char Ν = ρ then (N,+) is elementary abelian, so
the direct sum of copies of (2 ,+).

8a Near-fields
253
The orders of finite nf.'s are the same as those of finite
fields (cf. also number 4.)), for 8.12(b) implies
8.13 COROLLARY If N is a finite nf. then 3 ρεΡ 3 kcIN : |N|=pk
8.14 COROLLARY (Heatherly (2), Liqh (13). Let N be a finite nr.
with Nd + {0} and (N,+) simple.
2
Then either N = {0} or N is a commutative field.
Proof. Let some n'n" be 4 0. Take dcN5.
(0:n") <a (N, + ) implies (0:n") = {0). Hence
dn" + 0 and n'4D-· --· (nEN|dn = 0} <| (N,+), so
D = (0). Consequently, \/ ncN*: (0:n) = (0). By
8.4, N is a nf., hence abelian.
But then (0) + (Nd, + ) 3 (N, + ), so Nrf = N and
N is a finite field.
8.1.5 REMARK See Wahling (6), p. 49, for a characterization
(due to P. M. Cohn) of those groups which can be the
multiplicative group of a near-field. See also Linh (19).
3.) THE CENTER AND THE KERNEL OF A NEAR-FIELD
8.16 NOTATION Let C(N): = {ηεΝ| \/η'εΝ: ηη' * n'n] be the
center of (N,·) and call Nd the "kernel of N"
(Karzel et al.).
8.17 THEOREM (Karzel (1)). The subnear-rinn I of the nf. N
generated by C(N) consists of all sums of elements
of C(N) and is an intenral domain. The subnear-field
of N generated by C(N) is the field of quotients of I.
Proof: straightforward calculations using 8.10 and 8.11.
8.18 COROLLARY Every nf. N contains a commutative subfield F.

254
8 NEAR-FIELDS AND PLANAR NEAR-RINGS
There is a (possibly) different subfield in N:
8.19 THEOREM (Zemmer (1)). If N is a nf. then Nrf is a subfield
of N.
8.20 REMARK If Μ is a subfield of a nf. N then N can be
considered as a vector space over M. It's dimension will
be denoted by dim.H.
8.21 REMARK Clearly C(N) 5 Nrf. More exactly (but without
proof) there is the followinn relation between center
and kernel of a near-field:
8.22 THEOREM (Andre (2), WShlino (2)). Let N be a nf. which
is no proper skew-field. Then
C(N) = П n_1(Nd)n.
ηεΝ*
Moreover, C(N) = Nd iff V ηεΝ*: n-1Ndn = Nrf.
See more on that e.g. in Wahlinn (6).
We close this number with the following
8.23 REMARK If N is a nf. then char N = char Nd (by 8.9(a)).
4.) DICKSON NEAR-FIELDS
Dickson obtained the first proper near-fields in 1905 by
"distortinq" the multiplication in a finite field.
We axiomatize this procedure, trackinn the presentation of
Wa'hling (6). Proofs (or references where to find them) can be
found there.
For this number, confer also the chapter on Dickson near-rinas
in §9d). Unless otherwise indicated, N will always denote a nf.

8a Near-fields
255
8.2 4 DEFINITION A map φ: Ν* -> Aut (N,+ ,·) is called a
η -* φ
couplinn map if \/ ρι,ηεΝ: *п°Фт = Φ, (m\.n
8.25 EXAMPLE φ: η -+ id», is a couplinn map on FJ.
8.26 NOTATION If φ is a coupling map on N then
фт(п)-т if m + 0
0 if m = 0
no . m:
■{
8.27 PROPOSITION If 6 is a couplinn map on N then (N,+,o )
is again a near-field.
(The "coupling property" in 8.24 is just the restatement
cf the associativity of о ).
Φ
8.28 DEFINITION (Ν, + ,<> ) is then called the φ-derivation of
Φ 4
( N , + , ·) and denoted as N .
{φ ΙηεΝ*} is called the Pi ckson-nroup of φ.
N is said to be a Dickson near-field if N is the φ-derivation
of some field F: F* = N.
To the author's knowledoe, all known near-fields (up to 7
examples - see below) are Dickson near-fields.
We give an example of a class of finite and infinite Dickson
near-fields which are not fields:
8.29 EXAMPLE (Zemmer (1)). Let F be a commutative field and
F(x) the field of rational functions (7.113(h)).
•P(x)\. . Pix + rf]-M
φ: F(x) - Aut F(x). given by Фд^^): = ^f
TOO
is a coupling map on F(x) and (F(x),+,o ) is a Dickson
near-field.
For "most" finite fields we net important couplinn maps. But
first we need the followinn

256
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
8.30 DEFINITION (ς,η)εΙΝ
if
is called a pair of Dickson numbers
(a) q is some power ρ of a prime p.
(b) Each prime divisor of η divides q-1.
(c) If q ξ 3 (mod n) then 4 does not divide n.
8.31 THEOREM Let (q,n) be a pair of Dickson numbers.
Let F be the (Galois-)field GF(qn) = GF(p*n) with qn
elements, (F*,·) is cyclic. Let g be a oenerator and
\ Let
let Η be the subgroup of (F*,·) generated by
α be the (Frobenius-) automorphism f
of (F,+,·)
q2-l
qM
• ,Hg
q-·
}
Then F*/H can be represented as {Hg, Hoq"'
■k-l
= ak ε Aut (F,+,·)· If * : F* - F*/H
is the canonical epimorphism then ψ: = λ π is a coupling
map on F.
K^1
Let X(Hgq_1 ):
N:
{,η
F9 = (GFip"·"),*^.) is a nf. with C(N) = Nd =
= {xcF| VfcF*: фг(х) = x) = RF(q).
The number of non-isomorphiс Dickson near-fields derived
in this way (by different choices of g) is —|—*■ , where
Φ is the Euler-function and i is the order of ρ (mod n).
Their multiplicative groups are isomorphic. For more
information see LUneburo (1).
8.32 THEOREM By taking all pairs of Dickson numbers, all finite
Dickson near-fields arise in the way described in 8.31.
This makes the question, which (finite) near-fields are Dickson
near-fields, even more interesting. Of course, it mioht be hard
to visualize Dickson near-fields with naked eyes. So we use an
instrument (see e.g. Wahling (6)):
8.33 THEOREM ("Zassenhaus-criterion"): A finite nf. N is a
Dickson nf. iff G: = (N*,·) is metacyclic (i.e. [G,G]
and G/o r-\ are cyclic).
LG>GJ -1 q
In this case, G has two generators a,b with b ab = a^.
where q = IN
di

8a Near-fields
257
It was Zassenhaus, too, who determined all finite nf.'s:
8.34 THEOREM All finite nf.'s - up to 7 exceptional cases
are Di ckson nf.'s.
Now we are going to describe these 7 "outsiders" N^ ,. . . ,Ny
(numbering is the one of Zassenhaus):
All of them are of order ρ with ρ = 5,7,11 (two cases),
23, 29 or 59.
Since all Νε{Ν.,...,Ν,} can be considered as vector spaces
over lb of dimension 2 and since for each ηεΝ* the map
χ -* xn is an element of Aut., (N) it suffices to describe
Nd
(N*,·) via 2*2-matrices over Nrf = GF(p) (N is a vector snace
0 -1,
over
i
1
2
3
4
5
6
7
Nd):
Ρ
5
11
7
23
11
29
59
order of N.
52
ll2
72
232
ll2
2
29^
592
(Nf ,·
and
, 1
. 1
l-5
, 1
l-l
, 1
42
, 2
1 1
, 1
l-12
, 9
l-10
(N^,·) is generated by A:= ({ ~q)
_2) and (0 4)
3)
-2>
-6
.2) and <0
4,
2 0,
■3
■7.
-13 0
-2} and ( 0 -13'
.10) and ί0 4}
The smallest Dickson nf. which is not a field is given by
if χ is a square in
(GF(3 ),+,o) with x0y: =
Γ
(SF(3'),+,·)
■y" otherwise
Its multiplicative group is the quaternion π roup (of order 8).

258
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
8.35 THEOREM (Li gh-Neal (1)). Let N be a finite nf. such that
к i
\/ ηεΝ: η = η where к is of the form к = pJ+l
(ρεΡ\{2), jelN ).
Then N is a field.
5.) NEAR-FIELDS AND DOUBLY TRANSITIVE GROUPS
Near-fields (and some similar structures) can be used to
describe "sharply transitive" permutation qroups, The
followino discussion follows Kerby (7).
8.36 NOTATION If Μ is an arbitrary set, let S,. be the
symmetric group on И (i.e. the ciroup of all 1-1-maps Μ
If к ε IN , (m.,...,m.)cM- is called a proper k-tupl e
M)
if all
nys
are distinct.
8.37 DEFINITION Г < SM is called (sharply) k-transitive (on M)
if for all proper k-tuples (m,,...,m.), (ml,....m/)εΜ
there is (exactly) one γεΓ with y(m·) = m'. for all
ϊε{1 ,... ,k} (cf. 4.26).
1-transitive groups are simply called tran si tive, the
sharply 1-transitive ones are just the re quiar permutation
groups.
8.38 NOTATION Let (nij ,. . . ,mk )cMk be proper and Γ < SM.
Then г : = {γεΓ| \/ i ε{ 1,. . . ,k): γ(η.) = m,}
nu , . . . ,m. ι ι
denotes the stabilizer (subgroup) of (m,,...,m.).
Near-fields are primarily applicable to sharply k-transitive
groups. There is no need to consider large k's:
8.39 THEOREM (C. Jordan (1872), M. Hall (1954) and others - see
e.g. Kerby (7)):
If к > 4 then all sharply k-transitive permutation nroups
are finite and isomorphic either to S (n > 4), Ap (n > 6)
or to the "atrisu groups of degree 11 or 12,

8a Near-fields
259
Since regular permutation groups are wel1-studied (see. e.g.
(Wielandt) or (Passman)), we turn our attention to sharply
2- and 3-transitive permutation groups.
We start with the sharply 2-transitive ones. Our interest stems
from
8.40 EXAMPLE Let N be a nf.. Then the group T2(N) of all "affine
transformations" (cf. §9c)) χ ■* xa + b (a.beN, a =f 0) is
sharply 2-transitive on FJ.
However, not all sharply 2-transitive groups seem to arise in
this way. We have to consider a new alnebraic system which is
a "little bit" more general than a near-field.
8.41 DEFINITION A near-domain is a set FJ with two binary
operations "+" and "·" subject to the followinn conditions:
(a) (N,+) is a loop (with zero 0)
(b) V η,η'εΝ: n+n' = 0 => η'+n = 0.
(c) (N*,·) is a group.
(d) V ηεΝ: nO = 0.
(e) V η,η',η"εΝ: (η+η')η" = nn"+n'n".
(f) V η,η'εΝ 3 d ,εΜ* V η"εΝ: n+(n'+n") = (n+n')+dr , n?
Near-domains can be viewed as "additively non-associative near-
fields" (cf. 8.75):
8.42 REMARK A near-domain with associative addition is a nf..
It is not known if there exist near-domains which are no near-
fields. Anyhow, those ones must be infinite:
8.43 THEOREM A finite near-domain is a near-field.
We define for a near-domain N T?('J) as in 3.40 and get

260
8 NEAR-FIELDSAND PLANAR NEAR-RINGS
8.44 THEOREM
(a) For each near-domain N, Tp(N) is sharply 2-transitive,
(b) Conversely, for each sharply 2-transitive permutation
group Γ on a set Μ, Μ can be made into a near-domain
such that Γ = T2(M).
8.45 COROLLARY All finite sharply 2-transitive permutation
groups are exactly the T2(N)'s, where N is a finite nf. .
So by 8.31, 8.32 and 8.34, all finite sharply 2-transitive
permutation qroups are determined.
There exist many conditions under which a near-domain is forced
to be a near-field. They are excellently presented in Kerby (9).
We mention only one:
8.46 NOTATION If Γ is a group then I : = ίγεΓ|γ'
the subset of the "involutions" of Γ.
Let (Ir) : = ίγ1γ2|γ1,γ2εΙΓ}.
= 1} denotes
8.47 THEOREM Let Γ be a sharply 2-transitive permutation oroup
on Μ and (M,+,·) "it's" near-domain (8.44(b)). Then Μ
is a near-field <=> (I„) < Γ.
8.48 REMARK Sharply 3-transitive groups can be characterized
in a similar, but more complicated way by qroups of things
like "fractional affine transformations" on certain near-
domains (so-called "Karzel-Tits-fields "). See Kerby (7).
See also all S"-labeled items in the blblionraphy.
6.) NORMAL NEAR-FIELDS AMD INCIDENCE GROUPS
In order to be able to formulate the connections between nf.'s
and geometry we drive in another country and recall some
geometry. For a detailed account see Andre (4). Cf. also the
appendix to Thomsen (1).

8a Near-fields
261
8.49 DEFINITION Let Ρ be a set and /«ξ2Ρ . The pair (P,£)
is called an incidence structure. (P,^) is an i nci dence
space provided that
(a) V ρ,ςεΡ, ρ + q 3 lzt\ pEL Λ qEL.
(b) V Lzt: |L| > 2.
The elements of Ρ are then called "poi nts" and those of
/ "lines". L of (a) is called the "line determined by p,q"
and denoted by p~q. If ί,ΜεΖ, set L//M: <=> (L = M) ν
ν (LnM = 0). Call (P,/) degenerated if every set of 3
points is on a common line.
8.50 DEFINITION Two incidence spaces (P,£) and (P',£') are
called i somorphic if 3 h:P -+ P' with h bijective and
\/ MsP: h(M)cJt,' <=> Mcit. h is then called an i somorphism
or (if Ρ = P' and L = L') an automorphism.
8.51 DEFINITION A subset S of an incidence space (P,£) is
called subspace if it is "convex", i.e. if V s,tcS,
s + t : ItcS.
8.52 REMARK. The subspaces of an incidence space (P.j£) form
an inductive Moore-system. Hence it makes sense to speak
about the "subspace oenerated by a subset of P".
8.53 DEFINITION A non-degenerated incidence space (P,j6) is
called an
(a) affine plane if \j lc£ \j ρεΡ 33 Με& ρεΜ Λ L//Ч.
(b) projective plane if V Lzt>: |L| > 3 and \/ L,Mc£:
: L л Μ + 0.
Each affine plane can be extended to a projective plane by
adding some points. Conversely, one gets an affine plane from
a projective one by taking out one line.

262
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
8.54 DEFINITION A subspace of an incidence space {P ,£)
generable by 3 points (not on a common line) is called
a plane in (Ρ,£) .
8.55 DEFINITION An incidence space (P,j£) is called a projecti ve
space if each plane in (PX) is a projective plane.
8.56 DEFINITION Let (P,£) be a projective space. BSP is
called a base of (P,Jt) if В is a minimal generatino set
for (Р,£).
8.57 THEOREM Each projective space has a (non-empty) base and
all bases are equipotent.
8.58 DEFINITION If В is a base for the projective space
P: = (P.«6) then dim P: = (В | -1 is called the dimension
of P.
8.59 PROPOSITION The automorphisms of a projective space Ρ
(the "с о 11i η e a t i ο η s ") form a group Coll (P).
8.60 DEFINITION A projective space (P ,£) is called
Desarguesi an if, whenever two "triangles" {ajja-.a^}
and {b1,b2,b,} (ajjagiaj .bjjbgibjcP) are "perspective
w.r.t. a center οεΡ" (that means that
3 L1,L2,L3cZ \j ΐε{1,2,3}: ocLi Λ a^L., Λ b^L^
then гГГаТ η F7FT, a ,al0 b, b., and TTaT ft БТБТ are in
some common line L:

8a Near-fields
263
/
/
8.61 REMARK Each projective space of dimension > 3 is
Desarguesian.
8.62 NOTATION If V is a vector space over some field К then
V*/K»: = {K*v|vcV*} and
£: = {L | L is a subspace of V of (vector-space-)
dimension 2} .
8.63 THEOREM (Karzel (1)). In the notation of 8.62, (V*/K»,jC)
is a Desarguesian projective space of dimension dim V-l.
Conversely, one gets all Desarguesian projective spaces
of dimension > 2 in this way.
8.64 DEFINITION The triple (P,£, ·) is called a (projective)
incidence group i f
(a) (P,£) is a projective space,
(b ) (P , ·) is a group.
(c) V ρεΡ: с : Ρ - Ρ cCol1 (Ρ).
. Ρ q - ρο
(Ρ ,Χ, ■ ) is called ΰ с- s a r n u e s i a n if (Ρ,ί£) happens to be
the same.

264
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
8.65 REMARK It seems to be clear how isomorphic projective
incidence groups and the dimension dim Ρ of a projective
incidence group are defined.
What has all of that to do with near-fields ?
8.66 DEFINITION Let N be a nf. and F a proper subfield of N.
N is said to be normal over F provided that
(a) (F*,.) <l (Ν*,·)·
(b) V f,f'EF V ηεΝ: n(f + f) = nf+nf.
(N,F) is then called a normal near-field. N can be
considered as a vector space over F.
8.67 REMARK If (N,F) is normal and N is a field then by
Cartan-Brauer-Hua's theorem (see (Jacobson), p. 186)
F = C(N). This does not hold for general nf.'s (see
Wan ling (6), p. 76).
The basic fact is in
8.68 THEOREM (Karzel (1), pp. 69 - 73).
(a) Let (N,F) be a normal nf. . Then (Ν*/ρ.,£·)
(as in 8.63) is a Desarguesian projective incidence
group.
(b) Conversely, all (up to isomorphic copies) Desarguesian
projective planes Ρ arise in this way from some normal
nf., which (if dim Ρ > 2) is unique up to
isomorphism.
8.69 THEOREM (Karzel (1), pp. 76 and 78). If (N,F) is normal,
N*/p.» commutative and dinip(N) > 3 then N is a
commutative field.
If dimp(N) = 2 and N is finite then either N is a field
2
GF(p ) or a Dickson nf. of order 9 (see the remarks
preceding 8.35) or of order 64.

8a Near-fields
265
For generalizations ("normal local near-modules") see e.g.
Piener (1), Andre (4), Kuz'min (1), Maxson (15) and Theobald,
(1),(2). For ordered nf.'s see e.g. Kerby (3), Grb'ger (2).
Now we turn to affine planes.
7.) PLANAR NEAR-FIELDS
First again a little bit of geometry.
8.70 DEFINITION An automorphism α of an affine plane (P,£)
is called di latation if \/ Lc«£: ct(L)|| L. A dilatation α
is a transl ata ti on if α = id or α is f i xeci-poi η t-f ree .
8.71 DEFINITION An affine plane (P,£) is called a translation
pi ane if the set (it is a group!) of all translations in
(p>£) works transitively on P.
Consider, for a nf. N, .in N2 = : Ρ the "lines"
{(x,xa+b)ΙχεΝ} =: L, . (a,beN). Two of such, L. h and
α ) D α ) D
L , ., can be considered as "parallel" if a = a'. In order
3 ) D
to get something like an affine plane, we want two "non-
parallel" "lines" to have exactly one conmon point. This is the
case iff every equation xa = xa'+c with a + a' nas exactly
one solution.
8.72 DEFINITION A nf. N is called planar (or projective) if
each equation
xa = xb + c (a 4" b)
has exactly one solution.
Evidently, every field is planar.
8.73 REMARK (Wahling (6)). It suffices to want xa = x+1 to
have a unique solution for each a ± 1.
In тасс we get without too much work:

266
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
8.74 THEOREM Let N be a near-field, L, , as in the motiva-
tional considerations preceding 8.72 and L := {(c,x ) ]χεΝ}
(с ε Ν) the "vertical lines".
Let ίί: = U . |a,bcN} υ Η IccNb Then (N2,^) is an
α t D С
affine plane <-> N is planar.
о
In this case, (Μ ,jC) is a translation plane.
8.7 5 REMARK Not all translation planes arise in this way from
a planar nf. . One has to use "multipi ιcatively non-
associative planar nf.'s" (cf. B.41 - 8.42), so called
"Veblen-Wedderburn-systems" Μ to net all translation
planes as sone (11 ,;£) (definition of £ as above) ("Each
translation plane can be coordinatised by a Veblen-
Wedderburn system".) See e.g. Hall (1), p. 362.
For the more general question, which oeometric structures
can be coordinatized by which types of alaebraic structures,
see all G-labeled items in the bibliography.
We look a little bit around to find some planar nf.'s.
8.76 THEOREM (Maxson (10) et al.) A nf. П with dimN (N) finite
is planar.
8.77 COROLLARY (e. α. Zemmer (1)). A finite nf. is planar.
So our search is turned around: does there exist non-planar
nf.'s at all ?
8.78 EXAMPLES
(a) (Zemmer (1)). The Dickson nf. (F(x),+,o ) of 8.29
(char F = 0) is not planar.
(b) There exist planar nf.'s Η with dim.. (N) infinite
л
(Maxson (10)). Hence the converse of 8.76 does not
hold.
The following concept is usually only defined for finite groups.
See e.g. (passnan) or Kerby (9).

8a Near-fields
267
8.79 DEFINITION Γ < SM is called a Frobenlus group if
(a) Each γεΓ, γ 4= i d has at most one fixed-point.
(b) K_: = {γεΓ|γ is fixed-poiηt-free} и {id} is a transitive
proper normal subgroup of Г.
8.80 REMARKS
(a) К is called the Frobenius-kernel of Г.
(b) Other characterizations of finite Frobenius oroups are
e.g. :
(а) Г < S,, is a Frobenius-group iff Г is transitive,
but not sharply 1-transitive (= regular) and
V (ρι,,Ρΐ^εΜ2, m, + m, : Гт т = {id} (8.38).
(8) The Frobenius oroups are exactly the semidirect
products of a group Δ with a fixed-point-free
automorphism group 4· {id} of Δ.
Anyhow, if Γ is a finite Frobenius group, К is
characteristic, regular and nilpotent.
(c) The finite sharply 2-transitive permutation nroups of
degree > 3 are exactly the 2-transitive Frobenius
groups .
The connection to planar nf.'s is given by the followino two
theorems.
8.81 THEOREM (e.g. Andre (3)). Let N be a planar nf. with
|N| > 2. Then T2(N) =: Г (8.40) is a Frobenius nroup
and К is the set of all mappinqs χ -* x + b. Moreover, if
char N = 2 then I K, = К (8.46) (*)
char N + 2 then \j ηεΝ 3 γεΙ„ : γ(η) = η (**).
14 Γ
Conversely, all Frobenius groups with (*) or (**) can be
obtained in this way from a planar nf.

268
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
Of course one can define К for every permutation group as
in 8.79(b). Then one gets one more characterization of a
pianar nf.:
8.82 THEOREM (Kerby(7)). Let N be a nf. and Г: = T2(N).
N is planar <=*> (If)2 = К .
We close our round-up of near-field theory with
8.83 REMARK Let N be a nf. and к ε IN .
к »
Consider the N-group N and define SC by
£: = {(aj,. . . ,ak) + M(b1,. . . .Ь^Ца^. . ..ак),(Ь1.. ..,bk)eNk}.
ι.
Then (N ,jfc) is a "nearly affine space".
For the representation of "affine incidence groups" see
Pieper (2). Cf. Grbger (1) and Theobald (1),(2). Wefelscheid (10)
showed that every nf. can be embedded in a minimal planar one.
b) PLANAR NEAR-RINGS
1.) THE STRUCTURE OF PLANAR NEAR-RINGS
The "planarity property" 8.72 can of course also be formulated
for near-rings. But it is not very wise to do so: a near-ring
with this property is a near-field (see 8.91), so we wouldn't
get anything new. Therefore we oeneralize this concept
(Anshel-Clay (1)):
8.84 DEFINITION Let N be a nr. and a.bcN.
a ξ b: <=>> V ηεΝ : na = nb.
In this case, a and b are called (right) equivalent
multip!iers.

8b Planar near-rings
269
Of course, this is an equivalence relation on N.
8.85 DEFINITION A nr. N is said to be a planar near-rino if
|Ν/ξ|>3 and if every equation
xa = xb+c (a i b)
has a unique solution (in N).
8.86 NOTATION If Nc77, let A: = {ηεΝ]η = 0); denote N\ A
by N*.
By 8.85, N has at least two elements.
8.87 PROPOSITION (Anshel-Clay (1)). Every planar nr. is zero-
symmetri с.
Proof. Take ηεΝ. Let a be εΝ. Then 0 and nO are
both solutions of xa = xO+0, hence equal.
8.88 PROPOSITION (Anshel-Clay (1)). Let N be planar.
(a) acN is a riqht zero divisor <=> a ξ 0 <=> агА.
(b)\/ ηεΝ* V ριεΝ 3 xcN: xn = m.
Proof, (a) lie only have to show that na = 0 (n =f 0)
implies a ξ 0. In fact, a i 0 implies that 0 and
η are solutions of xa = xO+0, a contradiction.
(b) If ηεΝ*, xn = χθ+m has a unique solution.
The last result gives rise to the followinn definition.
8.89 NOTATION For acN* let 1. be the unique solution of
w a
xa = a. Let В : = {χεΝ 1. χ = χ} .
α α
Evidently, βεΒ., Ι.εΝ* and Ν* = U Β .
aeN* a
These Β 's help to clarify the structure of a planar near-
ri ng.

270
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
8.90 THEOREM (Anshel-Clay (1)). Let N be a planar nr. . Then
(a) Each (Ba>') is a group with identity 1 .
(b) A and the В 's (acN*) form a partition of N.
(c) V acN*: Β Ν* = Ba.
α α
(d) If a, be Ν , then φ: Β ■> В. is a (qroup-) isomorphism.
χ - lbx
(e) Each 1 (acN ) is a right identity for (N,+,·).
(f) If S«=N* and SN*sS then S = (J В .
acS a
Proof, (с),: Let a be cH , ηεΝ and bcB. Then
ι a
l.(bn) «-' (l.b)n = bn, whence bncB,.
α α α
(a) By (с), , {В .* ) is closed w.r.t. multiplication,
i a
Now 1.1. and 1 are both solutions of xa = xO + a ·
a a a
s° 1,1, = Ι,εΒ, is a left identity in (B,,·)·
a a a a a
If bcB , let F be the unique element of N with
БЪ = la (8.88(b)). Then b and laF solve xb = la,
whence l.F = FcB,.
a a
(c)-'· Conversely, every bcB can be written as
b = KbcB J*.
a a
(b): It is enouoh to show that \/ a,bcN either
0 or B.
В.л В.
a b
Jb*
If ηεΒ r\ В., In = η = 1. η, hence la and lb
are solutions of xn = xO+n; so 1 = 1. and
B, = B. .
a b
(e): For each ηεΝ, nl and η solve xl = xO+nl.
(d): By (с), ф really noes from Ba into Bb. If
a',a"EBa, ф(а'а") = lb(a'a") = ((lba')lb)a" =
= (lba')(lba") = ф(а')ф(а"), since lba'EBb, where
lb acts as identity by (a).
If ф(а') = Ф(а") then lba' = lba". By (e),
a' = la' = 1 La1 = 131. a" = a".
а а о а ь

8b Planar near-rings
271
If b'eB. , then lb'eB, is mapped onto b'. Hence φ is
D σα
an isomorphism.
(f) is clear.
8.91 COROLLARY (Anshel-Clay (1)). If N is planar and ξ is
discrete (i.e. = is the identity) (N then fulfills 8.72)
then N is a nf..
Proof. Μ a.bcN*: la = lb (by 8.90(e)), so la = lb·
Hence (8.90(b)) (N*,·) = (N*,·) = (Ba,·) is a qroup.
Hence ΙεΝ, Ν planar =*· N nf.. So planar nr. ' s avoid 1 as Duke
Drakula the sunlight !
8.92 COROLLARY A planar nr. is intenral <-> N* = N* <=> A - {0}
8.93 THEOREM (Anshel-Clay (1)). Let N be an intenral planar nr.
such that each 1",
υ{0} (acN*) is a normal subnroup
of (N,+). Suppose that no Β = Ν, but for all a + b,
В +В. = N.
a b
Then
(a) (N,+) = (Ea.+)+(Eb>+) for all a,bEN», a + b.
(b) Each (Έ^ ,+,·) is a nf.
α
(c) N is abelian.
(d) If j£: = {n+F |ηεΝ, acN*}, (N ,jt) is an affine plane
(8.53).
Proof, (a) is clear from 8.90(b).
(b) follows from 8.90(a).
(c) is a consequence of (a), (b) and 8.11.
(d) : If ρ,ςεΝ, ρ + q.
Then 3 ηεΝ: ρεη+"Β"
take acN* with ρ-ςεΐΓ. .
α
. . L and qcL. If L' = n'+B.
also has the property that p,qcL' then p-qcB3 r\
and
Since ρ + q, ~B я = "B~K
!a
a ■ "a·
L = L* . Since 0,1аг"В"а,
a b
each |n + Ba|>2. Hence (N ,£) is an incidence space
(8.49). |N*| = |N*/=I>2 implies that (N ,£) is not
deqenerated.
Now take L = η+ΤΓε«6 and ρεΝ. If M: = p+E. then
a α
ρεΜ and M|| L. If Μ'ε;£ has the same Drooerty then

272
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
M' = P+^k for some bcN*.
If M' = L then P+^k = n+^a> s0 a = b, whence
Μ = M' .
If M' + L then M'n L = 0. If a + b, Ν = Β +Β. .
α D
Hence 3 xcBa 3 УЕ^к: n"P = *+У· So -x+n =
= у + рс(п+1"а)л (p + B'b) = 0, a contradiction.
Consequently again a = b and Η = Μ'.
8.94 REMARKS It can be shown that the affine plane (N,£) in
8.93(c) can be coordinatized by a skew-field. A similar
result can be obtained if the 1" ' s are alternately
α
defined as "B : = B3 a {0} и -3,. There is also a close
α a a
connection to "Φ(I , IV)-nroups ".
For all of that see Anshel-Clay (2). As Clay points out,
there is also some relation to "inverse planes" (cf.
Ferrero (12)). If N is an integral olanar nr. with identity
then N is a skew-field or isomorphic to the near-field
{f : N -+ N|g m, η ε N: f(mn) =mf(n)}.
8.95 EXAMPLES (see Anshel-Clay (2) or Clay (10)).
(a) Every planar nf. with more than 2 elements is a planar
nr..
(b) Let V be a normed vector space over IR . Define v*w: =
= || w|| v. Then (V, + ,«) is an intearal planar non-
r i η π .
(c) Let V be a vector space over IR and *: V -+ IR have
the property that 3 ctcIR* \f tcIR, t > о \/ vcV:
φ(ΐν) = t%(„).
Define v«w. = | Φ (w) [ '%. Then (V,+,*) is a planar
near-ring.
See Anshel-Clay (2) for the aeoraetric interpretations
of the В 's as lines, rays, hyperbolas etc. .
a
(d) No Μ (Γ) or Μ(Γ) is planar: ξ is discrete, so
planarity would imply that ^(0\(r) 1S a pf· with
more than 3 elements, which is certainly not the case.
So in contrast to near-field-theory, a finite nr. is
not planar in oeneral (cf. 3.77).

8b Planar near-rings
273
8.96 THEOREM (Ferrero (5), Betsch-Clay (1)).
(a) Let Γ be a group and G + {id} be a fixed-point-free
automorphism group of Γ. If г is finite then each
N: = (Г , + , ·B ) of 1.4(b) is a planar near-rinn. N is
integral iff {Β·|ι'εΠ is the complete set of all
nonzero orbits (notation as in 1.4).
(b) Conversely, let N be a planar near-rinn. Consider for
aeN: q ·. N
■ a
η
N
na
Then G: = {g.UcN } is a fixed-
a '
point-free automorphism group =f {id} of (N,+).
For each bcN
R ^ B.
Proof. (a): Consider again the situation of 1.4(b)
|N/ξ[ = |G w{o}|>3, since γ ξ δ <=> a = η&
So it remains to show the "planar property":
9γ(ζ) = 9δ(ξ)+Π or (-ηδ+ηγ)(ξ) = η with g^ f q&
or (with n: = g"1Q6 + id): (-q + id) (ζ) = -д"^(п)
Assume that ξ·γ = ζ·δ+η, γ ^ δ. This means that
4.
1
γ 'б ' ' " -' " · '* · - γ
But -q+id is bijective, so this equation has exactly
one solution:
Suppose that (-n+id)(a) = (-g+id)(B) then -g(ct)+« =
= -q(8)+8 and g(a-S) = q(a)-g(B) = α-β. Since g
is fixed-point-free and g ^ id, а = 6.
Since Г is finite, -o+id is bijective.
(b): If acN , V ccN 3 χεΝ: ga(x) = xa = с by
8.88(b). So g cAut(N,+) and G = {g JacN*} is
α а
a group.
Consider the map ψ: (Β,·) ■+ G , where bcN .
a ► ga
Evidently, ψ is a homo morphism.
If ф(а1) = Ф(а2). then \/ χεΝ: xaj
1, a, = la-, so a, = a?» and φ is shown to be a
monomorphi sm.
Now take some g , ceN . Since lb ccSk by 8.90(c),
xa-, whence
Ф(1Ь с) =■ 9
luC
= g and ψ is an isomorphism.

274
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
G is fιxed-point-free: let
for some ηεΝ, η 4= 0, then 0 and η fulfill
α,εδ
" α
fulfill π (η) = η
■ α
xa = χ·la+0 (8.90(e)).
that V xcN:
deduce that g
ga(x)
xa
So
= xl.
1.
which means
= x, from which we
= id.
8.97 REMARK (Betsch-Clay (1)). This shows that (similar to the
situation in planar near-fields) every finite planar near-
ring can be characterized by some pair (r,G) of groups,
where G 4* (id) < Aut Γ is fi xed-po i nt-free . So every
finite planar near-rinn determines a Frobenius aroup
(8.79) and conversely (cf. also Ferrero (5)), and the
construction of a planar near-rina on a given additive
group Г is nothing else than the construction of a non-
trivial fixed-point-free automorphism nroup on Г.
Cf. 8.124, Heather!v-01ivier (3) and Adler (1).
8.98 COROLLARY (Betsch-Clay (1)) Let N be a finite planar near-
ring and let G be as in 8.96. Then
(a) |G| divides |N| - 1.
(b) (N , +) is nilpotent,but not necessarily abelian.
Proof, (a) is clear from 8.96(b) and 8.90(b), and
(b) follows from (Thompson) (cf. 6.33(b) ^> (d)).
See also 8.124.
The last result is in some other way remarkable: planar near-
rings are "not far away from being near-fields" (cf. 8.83(b)).
But they are far enough to have non-abeiian members in contrast
to 8.11. We need
8.99 DEFINITION (Ferrero (5), Szeto (3)). A nr. N is called
strongly uniform if \j ηεΝ: (0 : η ) = {0} or (0 : η ) = Ν ,
but 3 "ΐεΝ: (0:m) = {0}.
For the following result, cf. Ferrero (5), Heatherly-01ivier (3'
and Olivier (3).

8b Planar near-rings
275
8.100 THEOREM (Ferrero (5), Clay (11), Szeto (3)).
(a) Let N be a planar nr.. Then N is strongly uniform,
the multiplication is not trivial (1.4(b)) and all
non-zero orbits of G (see 8.96(b)) are princi pal
(that means that for all x,y in the same non-zero
orbit there is exactly one qcG with g(x) = у).
(b) Conversely, if N is a finite nr. which is strongly
uniform, has non-trivial multiplication and the
property that every non-zero orbit under Π (defined
as in 8.96(b)) is principal, then N is planar.
Pjro_o_f. (a) If асЛ, па = б and (0:a) = N. If aeN*\
gacAut N and (0:a) = {0}, hence N is stronnly
uniform. Since |N/=j>3, the multiplication cannot
be trivial.
G is fixed-point-free (8.96), so all orbits are
principal (cf. 4.28).
(b) Since N is finite and stronaly uniform, all
g : χ ■+ xa are either = б or automorphisms,
a
(Observe that Ker q = (0:a)). Let G be the group
of all those automorphisns. Since all orbits ηΊΟ]
are principal, G is fixed-point-free. Since · is not
trivial, G + {id}. Now apply 8.96(a).
8.101 REMARK (Szeto (3)). 8.100(b) does not hold in the infinite
case: Take (2,+) & (2,+) and define (n,m)*(n',m'): =
= n'(n,m). Then N: = (^*?, + ,*) is an infinite stronaly
uniform nr.. *is not trivial and all non-zero orbits dre
principal .
On the other hand, N is not planar, for (2,0) t (0,0),
but x(2,0) = x(0,0)+(l.l) has no solution. Cf. also
П. Betsch's report in the "Zentralblatt fur Mathematik".
8.102 REMARK In Ferrero-Cotti - Pellegrini (1) it is shown that
if N is planar then N2=N. For I < Ν, Ν planar, N/I is
not necessarily planar, but it is if N is finite. In this
case (if I+{0}), each ax= bx+i has exactly one solution
in I for a, b e N, a^b, i ε I .

276
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
Finally, we describe a construction method for finite inteqral
planar near-rings for future use (in the next number).
8.103 THEOREM (Clay (11)). Let F be the field of order pn
(ρεΡ) and t a non-trivial divisor of pn-l. (F*,·)
is cyclic of order ρ -1. There is a cyclic subgroup В
of F* of order t. Choose representatives
Uj = l,u2.--..um for the cosets Bu,,...,Bu of В in F*.
fO if g = 0
f-b if q -f 0, gcBu- , d = b u, .
Then (F,+,*t) is an integral planar near-rino.
Proof: straightforward and hence omitted.
2.) PLANAR NEAR-RINGS AND BALANCED INCOMPLETE BLOCK DESIGNS
8.104 DEFINITION Let N be a nr., acN* and bcN. Then the set
aN + b
is called the block determined by a,b. Blocks of the
form aN (a -j» 0) are called basic blocks.
8.105 REMARKS If G = {g |χεΝ*} is as in 8.96(b) and if
G°: = G u(o} then aN+b = G°a+b.
The Ga = aN* = aN\{0} form a partitition of N*, for
they are exactly the non-zero orbits of (N,+) under G.
For applications in the (near) future we renark
8.106 PROPOSITION Let N be planar and acN*, bcN. Then
[ aN + b[ = |G°| = ]N/=|>3.
Proof. V η,η'εΝ: an+b = an'+b <«> an = an' <=> χ = a and
χ = 0 fulfill xn = xn' <"> η ξ η' <=> g = g ,.
Observe 3.85.

8b Planar near-rings
277
In order to be able to formulate our principal results we need
8.107 DEFINITION acN is called an "element of the first
category" (Ferrero) if aN = (-a)N+a = -(aN)+a =: -aN+a.
C^(N): = {acN|a is of the first category}.
Clearly OcC^N).
8.108 THEOREM (Ferrero (8), (19)). Let N be planar and aEN.
Then acC^N) <-> aN < (N,+ ).
In this case and if aN is finite, aN is elementary
abeli an.
Proof. "=>": Let a be cC^N). Let bcaN™, b'caN.
We want to show that b-b'caN. Nov/ 3 ηεΝ : b = an.
Hence using 8.90(e) and 8.88(b), bN = aN = aNn =
= (-aN+a)n = -aNn+an = -aN+b = -bN+b.
Now 3 η'εΝ: b' = bn'. Thus we net
b'-b=bn-b ε bN-b = -bN + b-b = -bN = -aN. We claim
that aN = -aN. We may assume that a =f 0·
By 8.106, 3 c.dcaN : с 4= d. From above we get
Q+c-d ε -aN*. Hence 3 η"εΝ*; c-d = -an".
Tnerefore an" = d-cε(aN )e\(-aN ) since we have
shown above that the difference of any two elements
of aN* is in -aN*. So (aN*) л (-a N*) 4=0, whence
aN* = -aN* by 8.105, from which aN = -aN.
Now our considerations imply that aN < (N,+).
"<=" is obvious, since a = a-1 εβΝ (8.90).
The remark follows from the observation that the
automorphisms of Π, restricted to aN, form an
automorphism group on aN which acts transitively
on aN\{0} = aN and from theorem 11.1 of
(Wielandt).
From the first lines cf the preceding proof we can deduce

278
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
8.109 COROLLARIES (Ferrero (19)). Let N be planar and acN.
Then
(a) V bcaN*: h>N = aN if acC^N).
(b) All bcaN are of the same cateqory.
(c) Cj(N) is a union of - say u - orbits of (N,+)
under G.
(d) If 3 xcCj(N): xN = N then Cj(N) = N.
8.110 REMARK See Ferrero (12), (19) for the connection to
"difference sets".
Which blocks coincide ?
8.111 THEOREM (Ferrero (19), cf. Ferrero (8) and Clay (11))
Let N be planar, a , a ' ε Ν * and b , b'ε Ν.
Then aN + b » a'N + b'<=-> (a) or (b), where
(a) b
and aN = a ' Ν ,
(b) b + b\ -aN = a'N , b'caN +b and a.a'cC^M).
Proof. =■>: First let aN + b = a'N+b'. If b = b' then
aN = a'N and we are in case (a).
So suppose that b -f b' . From aN = a'N-t-(b'-b)
we get some ncN with 0 = a'n+(b'-b). b =f b'
implies that ncN . So a'n = b-b'. Similarly,
3 η'εΝ*: an' = b'-b.
Hence 0 + Ь-Ь'с(а'гГ)п (-aN*), whence a ' N* =
= -aN* by 8.105. So aN+b = -aN+b'. Consequently
3 η"εΝ*: b' = an"+b, so aN+b = -aN+b' = -aN+an"+b,
whence aN = -aN + an" = -aN + q „(a). Applyinq q ,.
gives aN = -aN+a, so acC,(N). By symmetry,
a'cCj(N) and (b) is shown.
<=■: (a) trivially implies aN + b = a'N + b'.
So assume (b). Let xca'N+b'. We have to show that
xcaN+b. If χ = b'caN*+b, xcaN+b. If χ + b',
3 η,η' ,η"r N' : χ - a'η + Ь' = - a n'+ b ' = -an'+an"+b.
Since aN < (N,+), by 8.108, -an'+an"caN, whence
xsa'l-b. T^e converse inclusion ": s shown sinilarly.

8b Planar near-rings
279
These blocks prove useful for constructinn block desinns.
First we define these items.
8.112 DEFINITION An incidence structure (P,B) (B=2P - 8.49)
is said to be a tactical confinuration with parameters
(v,b,r,k)cIN 4 if
(a) |P| = v.
(b) IB! = b.
(c) Each pr.P is in exactly r elements of i?.
(d) Each Βε3 contains exactly к elements of P, i.e.
V ΒεΒ·· |B| - k.
Λ tactical configuration is a balanced block desion if
(e) Each pair (ρ,ς)εΡ , P+q , is in exactly λ elements
of Ъ
and complete if
(П ь = (*),
otherwise i ncomplete.
The elements of £ are called Ы ocks. "Balanced incomplete
block design" is abbreviated by " В_ШГ; (v,b,r,k,X) are
the parameters of the BIBD and Ε: = -^ (<1) is called
its efficiency.
8.113 EXAMPLES
(a) Let Ρ be a set with ν elements, keIN , k<v. Let
B: = (B-P| |B| = k}.
Then (P.B) is a tactical confiauration with
parameters (v, (j[), {k[[), k) and (if k>2) г
complete balanced block desion with λ
(V"2)
4-2'
(b) Consider the field 2 (ρεΡ\{2}) and the affine
plane (ZJ;,£) as in 8.74. Then (zl.t.) is a BIBD
2 2
with parameters (ρ , ρ +p , p+1 ,p , 1). (For ρ = 2
we get a complete balanced block design.)

280
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
For the theory of block designs we refer the reader to (Hall)
or (Dembowski), where also 8.113 and 8.114 can be found.
The parameters of a BIBD are not independent at all:
8.114 PROPOSITION Let (P,3) be a BIBD. Then
(a) kb = vr = |{(ρ,Β)|ρεΒ, ΒεΒ)| (this holds for every
tactical confiquration!).
(b) r(k-l) = λ(ν-1) = |{peP!ρ + η Λ 3 Βεβ: ρεΒ Λ ςεΒ}),
where q is arbitrary in P.
(c) If b>l and k<v-l then b>v ("Fisher's inequality")
and r>k.
BIBD's are an essential tool in experimental designs. The
following example shall illustrate this and provide enounh
motivation for the reader to endure also the next panes.
8.115 APPLICATION Suppose you have b kinds of fertilizers and
want to test some combinations of r fertilizers always
on the same number к of experimental fields.
Take some BIBD (P.B) with parameters (v,b,r,k,X),
and divide the whole experimental area into ν parts.
Since |3| = b * number of fertilizers, Ъ can be
written as 3 = {В. ,B2 ,. . . , Bb}. Rive the fertilizer
number i on every field of the block B^ . Then:
(a) every field contains exactly r different fertilizers,
(b) every fertilizer is applied on exactly к different
fields, and
(c) every pair of different fields has exactly λ kinds
of fertilizers in common.
8.116 REMARKS Of course, given b,r,k, it is a non-trivial
problem how to get a BIBD with suitable parameters.
In general, it is an open question whether for every
quintuple (v,b,r,k,X) of natural numbers which fulfill
the conditions of 3.114 there exists a BIBD with these
parameters. We will now apply planar near-rings to qet
new classes of В I 3D ' s .

8b Planar near-rings
281
The efficiency of a BIBD can be interpreted economically
in the example above. BlBD's of efficiency >0,85 are
usually considered to be "good". Many of them are listed
i η (Coch ran-Cox).
Balanced complete block desiqns are usually "rather
inefficient". This is the reason for looking at the
i ncomplete ones.
8.117 THEOREM (Ferrero (12)). Let N be planar with |N| =: νεΙΝ
Denote by 3 the set of all blocks (8.104). Let α γ (α2)
be the number of non-zero orbits of (N,+) under R
consist i no of elements of C,(N) (not of C,(N),
respectively) (cf. 8.110). Then (N,3) is a tactical
configuration with parameters
(v,
α. ν
+do·ν, a,+И2|R
)
Proof. The first parameter is clear.
We compute the number of different blocks and apply
8.111: The number of blocks aN+b with aeCj(N) is
alv
the one of those with a sj: С ι (N) (case (b)) is
Now apply 8.106 to get к = |B°| in 8.112(d).
Next observe that the number rn of blocks containinn
an element ηεΝ is the same for each ηεΝ, since
it equals the number of blocks containing 0. Now we
know that (N,5) is tactical and we can apply 8.114(a)
to get r = —
k_b
ν
alv
+a?v)
aita»j G
(of
course, this could be accomplished directly, too)
Observe that ν = (α,+α2) | R|+1.
Nothing is more natural now than to ask, under which conditions
(N ,Ъ) is a 315 D. The ne,u theorem answers tins question, thus
bringing joy and happiness into our life.

282
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
8.118 THEOREM (Ferrero (9) - (12)). Let (N,B) be as above.
(N.fc) is a BIBD <=■> C^N) = Μ (then λ = 1) or
Cj(N) = {0} (then λ = |Γ,°| ).
Proof. =>: It does not seem to be possible to deduce
this from the fact that —^^~γ^- εΖ (8.114(b)).
So we have to work.
Call (for a,bcN) a,b equivalent if aN = bN
(a and b are then in the same orbit under 0) and
denote this by a^b. We need a lemma.
Lemma: Let N be planar and η',η" be ε ΓΙ, η'+π".
Let λ: =■ λ ρ be the number of blocks В with
η',η"εΒ ("blocks throuqh n' and n" ").
Let и be the number of different representations
of n: = n'-n" as a difference of two equivalent
elements not contained in C,(N). Then:
If ncC^N) then λ = μ+1.
If n^C χ (ΓΙ) then A = u + 2.
Proof of the Lemma: First observe that if the
block aN+Ь contains 0 and η (= n'-n")
then aN+(b+n") contains n' and n". Hence
λ is the number of blocks throunh 0 and
η (4= о).
How many different blocks with {0,n}«=afWb
exist? Let аП+b be such a block.
Case (1): If b = 0, ncaN*, whence nN =
= aN by 8.109(a) and there is only
one possibility to have {O.nJ^a'N
for some a ' εΝ* .
Case (2) : b = n. Then 0 ε a ΓΙ + η , η ε -a ΓΙ,
whence a ΓΙ = -ηΠ. So there is aaain
just one block throunh 0 and n.
an,+b.
Case (3): 0 + b + n. 3 n^N : 0
Hence a'I = -b!l, and afl + b - -bN + b.
So if η is a difference as stated in
the lemma, the blocks in co'is i de >-5 * ι с '

8b Planar near-rings
283
have the form -cN+c.
Conversely, for the block -bN+b
we qet, since ncaN+b, 3 Ποε^ ' n =
= -bn2+b, which is a representation
of η as a difference of two equivalent
elements of bN.
If bcC^N), bN<(N, + ) implies nebN.
Since also Ocbfl, we are in case (1),
a contradiction to 0 j· b =j» η.
So let b be φ С г(N ) . Then -bN + b is
neither in case (1) nor (2) nor equal to
some other -b'N-i-b1 containinn 0 and
η, but with b=fb' by 8.111.
So in case (3) are just as many blocks
not in (1) and (2) as there are
representations of η of the described kind,
namely u.
So the result follows if one observes that
the two blocks in (1) and (2) coincide iff
ncC^N).
Proof of the theorem. By the planar property,
\/ ηεΝ* V η',η"εΝ*, η' ϊ η" Д χεΝ: η = χη'-χη".
So n has |(5!·(|γ, 1-1) such representations (when
varyi ng n' ,n") .
Now take some arbitrary qcG.
Then η
пп.(х)-оп„(х) = (V0^)^"1^))-
"(9 и ° Q) (ο λ(χ)), providino all other ways to write
η as a difference of equivalent elements. So there
xn ' -xn"
■1,
are just
mi -(|ni-i;
π!
1 different ways to
write η as such a difference.
(a) If n^C,(N) and η = a-b (a^b) then a,b
are both φ С j (N ) . For if e.o. acC^N) then
bcC^N), whence a-beC^N) by B.10B and B.110.
By our ler-ma, λ = μ + 2 = ( | G | -1} +2 = i^+1.

284
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
(b) If ncCj(N) then the [G|-1 ways to write η
as η = a-b with a^b are exactly π = a-(a-n)
with acnN\{n}. For nN is (8.108) an abelian
group of order | G° | ; so \j acnN*\{n}; a-n =
= -n+a ^ a. Observe that a and (a-n) are in C,(N)
So none of the |G|— 1 differences of equivalent
elements giving η are as described in the lemma,
whence μ = 0, and λ = 1.
It may happen that (in 8.117 and 8.118) neither C,(M) = N
nor Сj(Γ4) = {0} (see Betsch-Clay (1)).
One can even say more (see Ferrero (12)):
8.119 REMARK Let
(N,B) be the BIBD of 8.117/8.118.
N then (N,+) is elementary abelian (S.108)
If C^N)
and there is some finite field F such that (N,b) =
= (F >ί) °f 8.74; (N,B) can be considered as affine
space, and the blocks are just the lines of this space.
Looking at the other case (which brinns up possibly new desicins)
yields first
8.120 COROLLARY (Ferrero (12)). Let N be a finite planar nr.
Let |G°| have not the form pa, where ρεΡ and
pa/|N|. Then (PI,J) of 8.117 is a BIBD with к = λ =
Proof. Assume that 3 ηεΝ*: ncC^N). Then nN is
elementary abelian, so [ η Ν [ = ρα with ρεΙΡ ,
and ρα/Ι Ν! (8.108), a contradiction. Hence
Cj(N) = {0} and 8.118 gives the result.
See Ferrero (12), Teorema 8 for the connection to finite Mcibius
planes. Cf. also Anshel-Clay (1).
Another way to reach the case C,(N) = {0} is the following.

Bb Planar near-rings
2B5
8.121 COROLLARY (cf. Ferrero (8)). Let N be a finite integral
planar nr. without subnear-fields. Then the same
conclusion as in 8.120 holds.
Proof. Suppose that ηεΝ* is in C.(N). Then nN is
an abelian subgroup of (N, + ) by B.10B. (nN)* = Br
(пП)*= В is cloar from B.89, while every bcB can
_ 1 ^
be written as b = lb = nn~ b ε nil by B.90 (a).
Consequently, ((nil)*,·) is a group and η Μ is a
subnear-field of N, a contradiction.
Hence C.(N) = {0} and the result follows from
8.118.
In (8), Ferrero constructs BIBD's from near-rings N with
(|N|,6) = 1, having parameters (v, - ^" » ^r^> 3, 3) (where
|N| = v).
Both cases in 8.118 can be obtained by the following near-rinns:
8.122 COROLLARY (Clay (11)). Consider the planar nr. (F,+,*t)
of 8.103.
(a) If t = pm-l for some m<n then (F,J>) - as in
8.117 - is a BIBD with parameters
П / П . . П ·,
/Dn Ρ (Ρ -1) Ρ -1 Dm n
ρ (ρ -1) .ρ -1
(b) If t is not of the form pm-l then (F,b) is a
n P"(P"-1) (t+l)(pn-l)
t ' t
BIBD with parameters (ρ ,
t+1, t+1).
Proof. First observe that t = |G| = JB| (of 8.103),
so t+1 = Ifi°|.
(a) Let t be - pm-l (m<n). Set Ъ: = B„{0}.
Take acF*. Then a*F = aF has t+1 = pm elements
and is a subgroup of (F,+): f consists of all xcF
m
with xp = x, hence being a sub π roup of (F,+).
This is easily transferred to a¥. Now apply 8.118.
(b) follews from 8. 120.

286
§8 NEAR-FIELDS AND PLANAR NEAR-RINGS
8.123 REMARK Observe that one can get BIBD's out of 8.122 with
η n-m
efficiency Ε = ~ (in (a)) and Ε
Ρ -1
(in (b)), which is close to 1 for large n.
Pn-t
pn-l)(t+r
8. 12 4 R.EMARK BIBD's can also be constructed from non-abelian
finite planar near-rinqs (see Clay (11)),
Define on Ζ^χΣ^χΈγ an addition "®" by
(a.b.c) © (a'.b'.c1): = (a+a\ b + b', c + c'+a'-b).
Let g:N ■* N be defined via g(a,b,c): = (2a,2b,4c).
Then (B.H. Neumann (2)) (Ν,Φ) is a non-abelian nroup
2
and G: = {id,g,g } is a fixed-point-free automorphism
group of (N ,©).
8.96(a) gives some planar near-rinn (N,®,«R). Clay noes
on to prove that (N,J>) is a BIBD, of course with
к = λ and CjfN) = {0} (this follows from 8.120).
Clay also generalizes this example.
See Betsch-Clay (1) for an excellent summary of the theory
of planar near-rings tonether with new results (e.q. connections
to partially balanced incomplete block designs) and hints for
further research. See also Clay (17),(13).

2B7
9 MORE CLASSES OF NEAR-RINGS
a) contains commutativity theorems similar to the "n(x)-theorem"
of Jacobson and the "n(x,y)-theorem" of Herstein in rinq theory.
Our discussion is done in the world of IFP-near-rinqs (that
are nr.'s N where ab = 0 implies anb = 0 for all ηεΜ).
Λ dqnr. with the "n(x,y)- property" is a commutative rinq.
p-near-rinqs and Boolean nr.'s are also considered (as special
cases ).
Next, we study nr.'s without nilpotent elements. They are
(if in Ύ) ) subdirect products of intenral nr.'s which are
studied in part 2) of b). The finite integral near-rinns are
planar iff they are not "trivial". Special inteqral nr.'s are
called "near-integral domains". Their characteristic is zero
or a prime.
c) contains a discussion of affine nr.'s (i.e. a neneralization
of nr.'s of type Μ ff(V)). We examine the ideal structure,
the radicals and nr.'s constructed out of affine nr.'s.
Fundamental for these nr.'s is the fact that N is a ring
and N an ideal of N.
d) brings (for certain classes of a roups) answers to the
questions, which nr.'s (nr.'s with identity, ...) are
definable on a qiven additive aroup. For instance, every nr.
with identity on a cyclic aroup is a commutative rinn. Several
groups are explored which cannot be the additive nroup of a
nr. with identity.
We go on by discussing ordered nr.'s in e) (and discover that
very few nr.'s can be fully ordered). Regular nr.'sare studied in
f), tame nr.'s in g), while h) contains information on М<-(Г),
where S is not a fixed-point-free automorphism group.
We close with the connections between nr.'s and automata in i)
and a survey on other topics in j).

288
§9 MORE CLASSES OF NEAR-RINGS
a) IFP - MEAR-RINfiS
In ring theory, the following two theorems are certainly
among the most famous commutativity theorems (see e.g. (Procesi)):
THEOREM 1: Let R be a ri no with V xcR 3 n(x)eINVl}: χΠ'Χ'= χ.
Then Π is commutative.
THEOREM 2: Let R be a ring with
V x,yeR 3 n(x,y) ε 1М\Ш: (xy-yx ) n ( х 'у ■ = ху - vx .
Then R is commutative.
(The first one was obtained by N. Jacobson; the second one is
due to I.N. Herstei η.)
We will generalize these results to certain classes of near-
rings (includina the dgnr.'s) usinn subdirect decompositions.
In order to get a satisfactory treatment we start with a more
general class of near-rings:
1.) IFP-NEAR-RINRS
9.1 DEFINITION A nr. N is said to fulfill the insertion-of-
factors-property (IFP ) provided that
У a.b.ncN: (ab = 0 => anb =■ 0).
Η has the stronn IFP if every homomorphic imaoe of N has
the IFP.
The next is an intrinistic characterization of the strong IFP:

9a IFP-near-rings
289
9.2 PROPOSITION (Plasser (1)). N has the strona IFP: <=->
<=> V IsN \j a.b.ncN: label => anbcl).
The proof is strainhtforward and hence omitted.
We will soon get examples of iFP-near-rinos. But before we
characterize these near-rinns.
9.3 PROPOSITION (Bell (1), Plasser (1)). The following
assertions are equivalent:
(a) N has the IFP-property.
(b) V ηεΝ: (0:η) <t N.
(c) V S«eN: (0:S) й N.
Again, the proof is obvious.
Observe that every IFP-near-rinq N with left identity e is in
fL , for eO = 0 implies that enO = 0, whence nO = 0 for
all ηεΝ.
9.4 DEFINITION Consider the followino properties:
(P ): \f χεΝ 3 η(χ)>1: xn^x^ = x.
(Pj): (P0) and N is ε^.
(P2): V χ,γεΝ 3 n(x,y)>l: (χy-yx ) n(x 'y^ = xy-yx and
(Pg): \j χ,γ,ζεΝ: xyz = xzy ("weak commutati vi ty") .
(P4)'· V х.угН V IaN: xycl ~> yxcl.
9.5 REMARKS
(a) The "χπ(χ) = x"-property does not imply that Νε7)0,
for every Νε?? fulfills it. Nr.'s with (Pj) are
called "L-near-rinns" in Ligh (11). See Szeto (6), (8)
for a characterization via sheaf representations.
(b) Abelian nr.'s N with \/ χεΝ: χ2 = χ and (P3) were
studied by Rati i f f (1) and Subrahnanyarn (1)

290
§9 MORE CLASSES OF NEAR-RINGS
("Boolean semirinns"). Abelian nr.'s with (P3) are
called "semirinqs" there. The nr.'s U with
•и
V xcN: χ = χ and (P3) are the "6-near-rinns" of
Ligh (14).
(c) (P4) was considered by Bell (1), (2) and Plasser (1).
Every nr. with (P.) is in fl0· But, on the other
hand, every constant nr. has (p3)·
9.6 PROPOSITION (Bell (2), Linh (16)).
(a) (Ρχ) -> (P2) -> (P4).
(b) Each one of (P.) to (P4) implies the strong IFP-
property.
Proof, (a): (P,) =■> (P2) is immediate. Assume (P2) and
xycl. Then yx-xy ξ ух (mod I) and 3 ncIN\{l}:
yx-xy = (yx-xy)" - (yx)" = yxyx...yx - 0 (mod I).
Hence yxcI.
(b): Since (P,) - (P3) art inherited to homomorphic
images it suffices to show the iFP-property in this
case. By (a), we only have to look at (P3) and (P/)·
(P3): If ab = 0 and ηεΝ then anb = abn = On = 0.
{Рл): If abcl and ηεΝ then bacl, hence Ь(ап)г1,
whence anbcI by (P.).
See e.g. Ligh (16) and (Thirrin) for the connection to "d_u£
ri ηos" (i.e. rings, in which every one-sided ideal is two-
sided). Clearly each duo ring is a stronn IFP-nr. (but not
conversely). For a detailed study of "duo-near-rinos" see
Choudhari (1), ch. VIII, Choudhari-Goval (1) and P.amakota iah-Rao (>;
For easy reference, it rewards to define for this chapter
9.7 DEFINITION Let a nr. N be of
type I if Ne7L> N simple and strongly uniform (8.99).
_^y_p e H_ if Ί ε |} is not simple, but the intersection n
of all non-zero ideals contains no non-zero
i decipoten t.

9a IFP-near-rings
291
type III if NinQ)
Ν φ 7) and if Ρ (as above) has a
nonzero idempotent then Ρ = N .
о
type IV i f Hc7lc ■
type V if \/ χ , у ε Ν : xy = 0.
The structure of strong IFP-near-rinqs is niven by
9.8 THEOREM (Lioh (16)). Every strong IFP-near-rinq N is a
subdirect product of subdirectly irreducible IFP-near-rinns
of type I ,11 ,HI ,IV or V.
Proof. Let N be the subdirect product of some subdirectly
irreducible near-rinns !J · (iesome index set I)
(1.62(a)). The N^'s have the IFP-property by 9.1.
(a) If N^cTJy and N^· is simple, use 9.3 to net
Nj into type I or type V.
(b) Now let tl-εΤ^ be not simple and Ρ be as in 9.7.
By 1.60(c), Ρ + {0}. Assume that Ρ contains the
idempotent e =)= 0.
If 3 χεΝ·: xe j- χ then 0 =f xe-xc(O.-e) <! N.,
2
so P=(0:e), ec(o:e) and e = e =0, a
contradiction. Hence e is a right identity, contained in
Ρ, whence Ρ
N · , a contradiction.
(c) If N.j is neither г7^ nor c??c then
fO} + (N^o = (0:0) <| U-, so P«=(0:0). As in (b),
every idempotent e f 0 is a rioht identity in N^· .
If χε(0:0), χ = xecP, hence Ρ = (0:0) = (rji)0·
In special cases one aets more out of 9.8:
9.9 COROLLARY (Rati ιff (1), Linh (16)). A nr. with (PQ) and
the strong IFP is a subdirect product of subdirectly
irreducible near-rinqs
Ni
{0} with right identity of
type I (in which case N· is simple and intenral),
type III (in which case the annihilator ideals are exactly
{0} and (N.) ) or type IV.

292
§9 MORE CLASSES OF NEAR-RINGS
Proof. Clearly, type V cannot occur. Suppose that N.
has type II. By the subdirect irreducibi'lity, Ρ =f {0}.
!f χεΡ*, xn^x'~ is a non-zero idempotent in P,
a contradiction. By the same amument, we net Ρ = N
J о
in the case that N. is of type III. If then
(0:n) + {0}, we deduce from 9.3 that NQe(0:n).
If m = m +m ε(0:η), m is zero. Hence (0: π)
is either {0} or = (N-) and the same follows
for all (0:S) (SsfK ).
Now let H- be of type I. (P ) forces N to be
integral. Now pick up some e + 0. en^e'~ =: r
is a right identity, for У ζεΝ.: (zr-z)e = zre-ze =
= zen^e'-ze = ze-ze = 0, whence ζ г = ζ.
9.10 COROLLARY If N has (Ρχ) then N is isomorphic to a
subdirect product of simple integral near-rinas ε??0
with a right identity.
Corollary 9.9 cries for
9. 11 DEFINITION (Ratliff (1)). A nr. N is called almost small
if N has at most 2 different annihilator ideals.
9.12 COROLLARY (Ratliff (1), Li qh (11), (16)). Every strong
IFP-nr. with (P ) is representable as suboirect product
of almost small near-rings.
See also Ligh (11) and Szeto (1) for more detailed versions
of 9.8 for near-rings with (P ) or (P3)·
Now we turn to (P2) . First we need (cf. Pamakotaiah-Pao (2))
9.13 PROPOSITION Let N = N be subdirectly irreducible.
(a) N has the IFP, but no nilpotent elements beside 0 ->
—> N i s integral .
(b) If N has (p2). "is intenral and has Nd + {0} then
N fulfills both cancellation laws, is abelian and
either commutative or '-^\ ■

9a IFP-near-rings
293
Proof. We may assume that N j= {0>.
(a) Consider any χεΝ*. The semigroup ({x |kcIN},·)
does not contain 0 and is contained (Zorn !) in a
multiplicative senigroup Μ
containing 0. Consider I :
maximal for not
U (0:m). The IFP
mc'l.
implies that I d N. Since χ έ I (П
' χ * χ v χ
η,Λ - ί0}·
ζεΝ
is closed
w.r.t. multiplication!),
ζεΝ* "
Since N is subdirectly irreducible, 3 γεΝ*: I
-- {0}.
If
by
ηεΝ
is not in Μ , the subseminroup aenerated
Μ and η contains 0. So some product containing
У
at least one tines η (and possibly eleiients of Μ
must be zero. Such a product has one of the follow inn
forms :
m.nmj = 0, nm' = 0, m"n = 0, η = 0 (m. ,m2,m' ,πΓεΜ )
An application of (P.) yields
3 πεΜ: nm = {0}, in which case
again η = 0.
Thus M„ = N* and N is integral,
η = 0
or
ncly «= {0},
so
My = N·
(b) 1) If 3 x.ycN: xy-yx + 0, take kcIN\{l}
к к-1
with (xy-yx) = xy-yx. (xy-yx) =: e is a
nonzero idempotent. Let dcNj be =f 0 and let η be
arbitrary εΝ. Then (ne-n)e = 0, so ne=n and
d(en-n) = den-dn = dn-dn = 0, whence en = n. So
N has an identity 1 and each non-zero idempotent = 1.
2) Now let a,b,ccN with ab = ac, a ^ 0. If a is
central, we net b = c. If a is not central,
3 feN*: af-fa j- 0, hence (af-fa)a + 0. Let
5. ε Ш \ {1} be such that ( a (f a )- ( f a ) a )l = a(fa)-(fa)a.
Then (a(fa)-(fa)a)?'"'1 = 1 by 1) and a has a left
inverse which anain results in b = c.
3) We now show that (N,+) is abelian. Let 1+1 =: 2.
If 2=0, each element of N is of order 2 and N is
abeli an.
If 2 4* 0, but 2 is central then expanding (n+m)(l+l)
in both ways gives n+m = η+η for all η,ηεΝ, so
again N is abelian.

294
§9 MORE CLASSES OF NEAR-RINGS
If 2+0 and 2 is not central, we have to examine
the conditions of 1.109(c). By the considerations
above, N is cV, and n(-l) = η = η·(1) yields
η = 0.
2 has a left inverse u (say). Then u is a Hnht
inverse, too, for (l-2u)2 = 2-2u2 =» 2-2-1 = 2-2 = 0
implies 2u = 1. Let rcN be arbitrary; call
h: = u-r. Then h+h = ur+ur = (u+u)r = (2u)r =
= lr = r.
Finally, r = h+h = h'+h' gives 2h = 2 h ' , whence
h = h' .
ft η application of 1.109(c) shows that (N , + ) is
a b e 1 i a n .
9.14 THEOREM (Bell (1), (2), l.inh (12), Li qh- Luh ( (1)) .
Let N be a dqnr. . N has (Ρ £) <^> N is cornmuta ti ve.
r^roo_f. "'=>": Decompose, as in 9.8, N into subdirectly
irreducible rtr.'s N· =f" (0). Consider some N..
N1- has also (P2) and (N. )d + {0}.
(a) We first show that each nilpotent element is
central (i.e. in C(N) - B.16). We will accomplish
this by induction on the denree к cf nilpotence.
к = 1 is trivial, but we also need к = 2:
2
Suppose that η =0. Then \j χεΝ: (xn-nx)xn =
= xnxn-nxxn = 0-0 = 0, since nn = 0 implies
nxn = 0 and nxxn =0 by the IFP. Similarly,
2
(xn-nx)nx = 0, so (xn-nx) = 0, whence
xn-nx=0 by (P,).
к -1
Now assume that У ηεΝ: η' = 0 => ηεΟ(Ν). and
к к- 1 2
take m ε N with πι =0. Then (m ) =0, so
mk_1cC(N); hence V ΧεΝ: 0 = mkx-xmk = mxmK_1-
-xmm " = (mx-xm)m " = (IFP!) = (mx-xm)m(xm-xn)m . .
k- 1 k- 1
(mx-xm)m = ( (mx-xm)m) = (m(xm)-(xm)m)
Applyinq (P,) ana in yields m(xm)-(xm)m = 0.
2
As above, it t u >· r, 3 out that ( r. χ - a r·.) - 0 , w h e η с ;
mx-xm
0.

9a IFP-near-rings
295
(b) From (a) and the IFP-property one nets (as for
rinqs) that the set Npt(N-) of all nilpotent elements
of N. forms an i deal .
(c) If Npt(N.) = Nit (P2) instantly results the
commutativity of (N,·)·
(d) If Npt(N.)={0} and NeKj, N. is integral by 9.13(a)
hence abelian by 9.13(b), consequently a rinq (6.9(c)
and 6.6(c)) and therefore a cownutative one
(Theorem 2). If Ν έ4χ , N is commutative by 9.13(b).
(e) If {0} + Г J ρ t ( N ^ ) =f Г^ , consider ТТ.: = N
Ν· has no non-zero nilpotent elements, but is anain
ig. with (P2). By (d),
1 / Ν η t ( N n- )
anai г
is a commutative rinq.
So for all η',η"εΝι η'η"-η"η'cNpt(N·), from which
we qet n'n" = n"n' by (Po)·
(f) Since all N. are commutative near-rinqs, the
same applies to N.
(q) "<=" is trivial.
9.15 REMARK (Liph-Luh (1)). The assumption in 9.14 that
"N is dq." can be relaxed by "M is a D-nr■" which means
that each homomorphic imaqe ΤΪ of ΝεΤ? has TT. =f {0}
OQ
and is either non-abelian or a rinn. Clearly each dqnr.
is a D-nr., but there exist others, too (see Appendix,
number 6 of the nr.'s on S,).
9.16 COROLLARY (Bell (2), Liqh (12), (16)). Let N be a dqnr.
with N = N. Then N has (P2) iff N is a commutative
ring.
Proof: apply 9.14 and 1.107(c).
9.17 COROLLARY (Bell (1), Linn (7), (11)). Let Μ have IFP, (PQ)
and non-zero distributive elements in every non-zero
homomorphic imaqe. Then Μ is
of hf.'s (hence abelian).
:?l
and a subdirect product

296
§9 MORE CLASSESOF NEAR-RINGS
Proof. NQ = (0:0) <1 N by the IFP. Now N/NQ is constant,
so (N/NQ)d = {0}, whence N/NQ = {0} and N = Ν0ε7?0-
Thus we have (Pj) and hence (P-) and the stronq
IFP avai Table.
By 9.10, N is the subdirect product of simple inteoral
near-rinqs N. with riqht identities and no nilpotent
elements. By 9.13(b), every N^ is abelian and either
commutative (then a simple commutative rino with
identity, hence a commutative field) or zfl, (then
(P) implies that N ^ is a n f.).
Observe from 2.52(b) that the DCCI in N will turn П into a
finite direct sum of nf.'s.
9.18 COROLLARY (Bell (1), Liqh (7), (11), (16)). Every dqnr
W1'th (P ) is a subdirect product of commutative fie!
(by 9.17 and 9.14) and hence a commutative rino.
9.19 COROLLARY (Ratliff (1)). A nr. with (PQ) and (P3) is
a subdirect product of nr.'s N· =f {0} of type I, HI or
IV. If
N. is of type I then
N1
is a commutative field
or has more than one riqht identity.
Proof. Accordinq to 9.9, N cannot be of type II. If N'^
is of type I and has just one riqht identity e then
V χεΝ? : χΓ'(χ)-1 = e (as in the proof of 9.13(b)).
Hence \/ χ,γεΝ* : xy = χ ^x' xy = exy = eyx =
= У '^' ух = ух. So N· is commutative, e is an
identity and N. is (by 9.13(b)) a simple intenral
domain, hence a field.
9.20 COROLLARY (Ratliff (1)). Let N have (P ), (P3) and
nonzero distributive elements in every non-zero homomorphic
image (this happens e.g. if Nc*W, or if N is dq.).
Then N is a subdirect product of commutative fields and
hence a commutative ring.

9a IFP-near-rings
297
Proof. By 9.17, ΝεΤ) . so in the subdirect decomposition
of 9.9 al1 N. are of type I.
By 9.13(b), ΝεΤ^. Due to 9.19, N. is a commutative
field.
9.21 REMARKS Now we mention (without proof, but with reference)
other commutativity theorems.
(a) (Bell (6),(7)). Let Neflj be a dqnr. with
V χεΝ 3 η(χ)εΙΝ : x-xn (x)eC (N) (8.16). Then N is a
commutative rinq.
(b) (Liqh-Luh (1)). A finite D-nr. with identity in which
all nilpotent elements are central is a commutative
ring.
(c) (Ligh (11)). A finite dqnr. without nilpotent elements
is a commutative rinq.
(d) (Ligh (12)). A finite dqnr. is commutative iff all
zero divisors are central.
(e) (Plasser (1)). If N has (P ) and a left identity e.
Then ΝεΤ? <■=■> V χεΝ : e-χθ is idempotent <=-> all
idempotents are distributive <=> all idempotents are
central <=> N has (P,) -=> N is subdirect product of
nf.'s. Anyhow, (N,+) is torsion and each element
has a square-free order in (N,+).
(f) (Ligh (8)). Call Νε?? an g-nr. if deMd implies
-deNj. Every α-nr. without nilpotent elements is а
ring. Each n-nr. with (P ) is a commutative rino.
(g) (Ligh (15)). Each nr. N with (P3) fulfills
\f ηεΙΝ \/ x.yeN : (xy)" = хПуП. Every α-nr. Ν with
this property (or with \/ η ε IN \/ x.ycN: (x+y)
= хП+УП) has only nilpotent commutators of (N,·)·
A nr. ΗεΎΙ-, without nilpotent elements and with
V χ,γεΝ: (xy) = x у is abelian.
(h) (Ligh-Utumi (1)). N is a C^-nr. (C,-nr.) if
\j ηεΝ: nN = nNn (Nn = nNn, respectively). Neither
one implies the other:

298
§9 MORE CLASSES OF NEAR-RINGS
If F is a field then Maff(H (1-4) is a Cj-, but
not C^-nr. .
Λ finite integral nr. has C,, but not C„,
Every C~-nr. (but not every C,-nr.) has the IFP.
N is C,- and C^-nr. iff Η is C,-nr. and every idem-
potent i s central.
See this paper for decomposition theorems for C,- and
Cg-nr.'s with finiteness conditions.
(i) A ring R is called a P,-ri nn if for all rnR there is
a central idempotent r with rr = r and
\/ e2 = eER : (er = re --> r°e = r°) (this Pj has
not hi nn to do with our (P-j)).
See Plasser (1) for a similar concept for near-rinns.
(j) For more results see Bell (9), Ligh (IS), Marin {?.) ,
Ramakotaiah-Rao (2),(5) and Kim-Park (1).
2.) p-NEAR-RINftS
9.22 DEFINITION Let ρ be a prime. A nr. N is called a p-near-
ri ηρ provided that \/ χεΝ: xp = χ Λ px = 0.
Evidently, every p-nr. has property (p0)·
9.23 PROPOSITION (Plasser (1)). A p-nr. with left identity is
zero-symmetri с.
Proof. Let e be the left identity. Then it is easily shown
by induction that \/ xeN \/ keW : (e + xO)k = e + k(xO).
Hence e+χθ = (e+xO)p = e+p(xO) - e, whence xO = 0.
9.24 REMARK (Plasser (1)). 9.23 does not hold for neneral nr.'s
with (P0).
9.25 COROLLARY (Ratliff (1)). A p-nr. with (P3) and non-zero
distributive elements in every homcmorph i с inane :'s
isomorphic to a subdirect product of copies of the field Ζ ,
hence a o-ring.

9a IFP-near-rings
299
Proof. By 9.20, N is a subdirect product of simple
commutative p-rinqs N. with identity. Rinn theory
tel Is us that N. = TL .
9.26 THEOREM (Plasser (1)). A finite p-nf. N is isomorphic to
the field Ж .
Proof. N is a Dickson nf., for N cannot be one of the 7
exceptional cases (8.34 and the subsequent discussion),
5 5
since in each one of these cases A = A, but В 4s B ·
ΙΊον/ 3 ςεΡ 3 ηεΙΝ : \U\ = qn by 8. 13.
Since (N,+ ) is a finite p-qroup, |N| is some
power of p, consequently q = p.
Now (N*,·) has qenerators a,b with b" ab = aq =
= ap (8.33). Thus ab = bap = ba, N is commutative
and hence has (P,).
Now the result follows from 9.25.
9.27 REMARKS
(a) Cf. also 8.35.
(b) The finiteness condition in 9.26 is indispensable, for
there exist infinite p-fields.
An application of 2.52(b) nives with 9.17 and 9.26 the followinn
9.28 COROLLARY (Plasser (1)). Let П be a finite p-nr. with
IFP and with non-zero distributive elements in every
nonzero homomorphic imane. Then ΓΙ is isomorphic to a (finite)
direct sum of copies of Ж , hence a finite p-rinn.
9.29 REMARK Ratliff (1) studied p-nr.'s Μ (especially for p=3
and p=5), which can be derived from a p-rino R in a way
that (N,+) = (R,+) and the product in N is defined via
a fixed polynomial function over R. The nr.'s considered
in this dissertation fulfill (P ) and (P3)·

300
§9 MORE CLASSESOF NEAR-RINGS
3.) BOOLEAN NEAR-RINGS
It does not seem to be quite clear how to define a Boolean near-
ring. So we take what seems to be the most qeneral possible
definition.
9.30 DEFINITION A nr. N is Boolean: <=> \/ xcN : x2 = x.
Hence a Boolean nr. is a (P )-near-rinq with n(x) = 2 for
all x.
9.31 REMARKS
(a) Every constant nr. is a Boolean nr. with (P ) and
(P3), but not a 2-nr. in general.
(b) A Boolean nr. with (P3) (a B-nr.) and non-zero
distributive elements in every non-zero homomorphic
image is a subdirect product of copies of Έ.?. This
result of Ligh (14) follows from 9.25.
(c) (Ligh (5), (14), (8), (10), Heatherly (7)). The same
assertion holds for dg. Boolean nr.'s. Of course, this
follows from 9.18, but there is also a direct elementary
proof in Ligh (10).
(d) See p. 418/419 for a list of all Boolean nr.'s definable
on the two non-abelian aroups of order 8.
(e) Ferrero-Cotti (2), (3) considered nr.'s with the
identities abc = acbc = abac. These are those ones
which contain an ideal I with Г = {0} and N/I is
a Boolean ring.
If) A Boolean nr. with left identity is a Boolean ring with
identity (Ligh (5)).
(g) More results are contained in Heatherly-Stone (1) and
Ramakotai ah-Rao (2) .

9b Near-rings without
301
9.32 COROLLARY (Heatherly (7)). A Boolean nr. ε7?0 with DCCI
is a finite direct sum of ideals which are intenral simple
f x У + 0
nr.'s with the trivial nultinlication xy = j
L о у = о
Proof. Apply 9.10, 2.52(b) and the fact that every
nonzero element is (as an idempotent - see the proof
of 9. 13(b)) a right identity.
9.33 EXAMPLES
(a) (Clay-Lawver (1)). Let (В,+,л) be a Boolean rfno
with identity 1. Let a': = a+1 and avb: = (а'лЬ')'.
If χεΒ, define for a,beB a *χ b: = aA(bvx). Then
(B,t,t ) is a Boolean nr. with (P,) which is a rinq
iff χ = 0.
Nr.'s derived from Boolean rings are called "special
Boolean n_e_ar-_ri "."j." ^n this paper. Their ideal
structure is considered.
(b) Subrahmanyam (1) called an abelian Boolean nr. with
(P3) "Boolean semirinn". Every Paring (9.21(f))
(B,+,·) gives rise to a Boolean semiring (B,+,·),
where a*b: = ab .
Every constant abelian nr. is a Boolean semiring.
A Boolean semiring can be represented as a disjoint
union of "nearly distributive" lattices.
See this paper for more details.
b) riEAR-RHins WITHOUT
1.) NEAR-RINHS WITHOUT HILPOTENT ELEMENTS
Nr.'s without non-zero nilpotent elements came up
different places in our discussion of near-rinqs.
some of the results concerning these near-rinqs.
at several
We collect

302
§9 MORE CLASSESOF NEAR-RINGS
9.34 REMARKS Let N be a nr. without non-zero nilpotent elements.
Then
(a) N has no nil(potent) subsets (2.96).
(b) If Nzfl has DCCN then every non-zero Μ-subgroup
contains a non-zero idenpotent (3.51). Moreover,
ί1/2(Ν) = J0(N) = 7)(N) =3?(N) = {0} in this case
(5.40).
(c) In any case, 71(N) = #(N) = Ш.
9.35 EXAMPLES
(a) Every constant nr. has no non-zero nilpotent elements.
(b) Every integral nr. (hence every nf.) has this property,
too.
The connection to the previous chapter is given by
9.36 THEOREM (Bell (1), Marin (1), Ramakotaiah-Dao (2)).
Let N be zero-symmetric. Equivalent are:
(a) N has no non-zero nilpotent element.
(b) N is a subdirect product of inteoral nt.'s.
Proof, (a) —> (b) is nothinn else tnan in the proof of
9.13(a): N has a family of ideals Ι (χεΝ*) with
zero intersection and each Ν/Ιχ is integral.
(b) => (a): If xn = 0, in each component rr ^ (N)
of the subdirect representation of ΓΙ we aet π^(χ)=0>
whence χ = 0.
Hence we will devote the next number to intenral near-rinos.
But before, some more results minht be appropriate.
9.37 PROPOSITION (Bell (1), Heatherly (7), Marin (1), Ramakotaiah-
Rao (2)). A nr. Ν εΊΐ without non-zero nilpotent elements
is an IFP-nr .

9b Near-rings without
303
Proof. If xy = 0 (x.yeN) then yxyx = yOx = 0, whence
ρ
(yx) = 0, so yx = 0. Now \/ ηεΝ: xny = (ny)x =
= n(yx) = nO = 0, so N has the IFP.
9.38 COROLLARY (Heatherly (7)). Every subdirectly irreducible
nr. ^^fL without non-zero nilpotents is integral. Every
non-zero idempotent is a rinht identity.
Proof. The first assertion holds by 9.13(a).
If e 4" 0 is idempotent, \/ χεΝ: (xe-x)e = 0,
whence xe = x.
To get more, we have to impose some finiteness conditions on N.
9.39 PROPOSITION (Heatherly (7)). Let Uc7l0 be a subdirectly
irreducible nr. 4* С0> with DCCN and without non-zero
nilpotent elements. Then
(a) N is inteqral and 2-primitive on N.
(b) N has a right identity.
(c) Nd + 0 => N is a nf. .
(d) If N is dg. then N is a field.
2 3
Proof, (a) Consider, for χεΝ*, the chain Nx?Nx ?Nx ?. . .
There is some ηεΙΝ with Nxn = Nx = .... N is
integral by 9.36, so Nxn = (Nx)xn implies Η = Nx.
Therefore N is 2-primitive on N.
(b) holds by 4.46 or by 9.38.
(c) By the same argument as in the proof of 9.13(b),
N contains an identity. Now apply 4.47(a) and 9.17.
(d) is obvious.
9.40 REMARK (Heatherly (7)). There exist even finite simple
abelian nr.'s N with (Pj) and without non-zero nilpotent
elements, which are not nf.'s.
We can reduce the theory of near-rings with DCCN and no
nonzero nilpotent elements to that of 9.39:

304
§9 MORE CLASSES OF NEAR-RINGS
9.41 THEOREM (Heatherly (7)). Let Νε?70 have DCCN and no
nonzero nilpotent elements. Then N has a riqht identity, is
2-semisimple and the finite direct sum of nr.'s which
fulfill all conditions of 9.39. If every non-zero homo-
morphic imaqe of N has non-zero distributive elements then
N is a finite direct sum of nf.'s; if N is dq. then N is a
fi'nite direct sum of fields.
Proof. Decompose N into subdirectly irreducible integral
nr.'s N.j (9.36). In fact, N is a finite direct sum
of these ones (2.52(b)). Now apply 9.39(b), 9.39(a)
and 5.49.
9.42 COROLLARY (Heatherly (7)). If Νε7)0 is a finite nr.
without non-zero nilpotent elements. Then N has (pi)·
This is clear by 9.38 and 9.41 (Heatherly noes on to show
that n(x) (9.4) can be chosen to be constant for all χεΝ.
Cf. also Ligh (11)) .
Moreover, we have some information concerning the near-rinns
in discussion, which guys belong to the center C(N) of N:
9.43 PROPOSITION (Bell (1)). Let Νε7?0 have no non-zero nil-
potent elements. Then
(a) Every distributive idenpotent is central.
(b) If UcTlp all idenpotents are in C(N).
Proof. First we show that for each idempotent e,
\/ χεΝ: ex = exe. Now (ex-exe)e = 0, so (9.37)
e(ex-exe) = 0 and ex(ex-exe) = 0 (IFP). Hence
(-exe) · (ex-exe) = (-ex)O = 0. Therefore (ex-exe) -
= ex(ex-exe)+(-exe)(ex-exe) = 0+0 = 0, whence
ex-exe = 0.
(a) If ecNrf, γ χεΝ: e(xe-exe) = exe+e(-exe) =
= exe-exe = 0, hence (xe-exe)e = 0, whence
xe = exe = ex.

9b Near-rings without
305
(b) If N has an identity 1, consider again some
idempotent e. (l-e)e = 0, so \f χεΝ: (l-e)xe = 0.
Also, (xe-exe)e = xe-exe and (l-e)xe = xe-exe,
2
therefore (xe-exe) = (xe-exe )e (xe-exe) =
= (xe-exe)(1-е)xe = 0, so xe = exe = ex for all
χεΝ.
9.44 REMARKS
(a) See Marin (1) for characterizations of those near-
rings without non-zero nilpotent elements which are
(finitely or not) completely reducible into certain
other near-rings. See also Szeto-Wong (1).
(b) Recall 9.21(f).
(c) Again, let NcT? have no non-zero nilpotent elements.
Then (Bel 1-Li gh (1)):
a) If N is dg. with finitely many subnear-rings, N is
a finite commutative ring.
B) If N has at most 2 idernpotents and no proper
(finitely many) subnear-rinqs, N is a finite field
(a near-field, respectively).
(d) Don't forget to observe 9.54.
2.) NEAR-RINRS WITHOUT ZERO DIVISORS (INTEGRAL NEAR-RINSS)
9.45 EXAMPLES
(a) Every constant nr. is intearal.
γ ... δ+0
0 . . . δ = О
defines an integral nr. (Г.+.») (cf. 1.4(b)).
(b) If (Г.+) is any group, also γ»δ:

306
§9 MORE CLASSES OF NEAR-RINGS
So one can say nothing about the additive qroup of an integral
near-ring. To overcome this we will give the following
9.46 DEFINITION An integral nr. N is non-trivial if its
multiplication is not one of 9.45(a) or (b).
These гтоп-trivial integral nr.'s are sometimes called "near-
integral domains" (see Ligh (13), Heatherly-01ivier (1), (2),
Adams (1), (2)). But they are not always embeddable into a
near-field, so we reserve this distinguishing name to a more
special class of non-trivial integral near-rings (see 9.52
and 9.65). See also Olivier (?.).
9.47 PROPOSITION (Clay (8), Heatherly-01ivier (1), Plasser (1)).
If N is integral then ΝεΤ^ or ΝεΊ^..
Proof. Suppose that 3 χεΝ: χΟ + 0. Then for all ηεΝ we
have (nxO-n)xO = nxO-nxO = 0, whence nxO = n.
Hence nO = nxOO = nxO = n, and N is constant.
Thus every non-trivial integral near-ring is zero-symmetric.
Integral near-rings also appear in previous chapters. In order
to present a good aerial view on this topic we compile these
facts :
9.48 REMARKS Recall that an integral near-ring N has the
following properties:
(a) N has the right cancellation law (1.111(a)).
(b) If N is finite and non-abelian then each element of
N has a unique square-root (1.112).
(c) N is a prime near-ring (2.66).
(d) If N is non-trivial in the sense of 9.46 and has the
DCCI then N is subdirectly irreducible (2.107).
Applying 9.39 we get: If N has moreover the DCCN then
there exists a right identity, N, « {0} implies that
N is a nf. (Ligh-Malone (1)), N is dg. implies that N
is a field (cf. also 6.14(b)), N is 2-primitive on N

9b Near-rings without
307
and so N is simple (Heatherly (7)). See also Graves-
Malone (1).
(e) On the whole, 9b) 1) is applicable, for N has no
nonzero nilpotent elements. So if N is non-trivial, it
has the IFP.
9.49 REMARK to 9.48(d). Without chain conditions one cannot
conclude that an inteqral nr. N with N, j· {0} is a near-
field: take a field F and form N: = FQ [x] (7.78). N is
inteqral (7.68(c), 1.111(a)), each ax (aeN) is in Hrf,
but N is no nf. (7.68(b)). In fact, for every kcIN,
к к+1
*k: = ^akx +ak + lX +. . .|ak,ak + 1, ...cF} is an ideal and
I i=I2=>I3='· · · 1S a strictly descending chain (Heatherly (7)).
Cf. also Graves-Hal one (1).
9.50 THEOREM (Ferrero (8)). Let Μ be a finite integral near-ring.
N is non-trivial <=> N is planar.
Proof. ==>: Consider 6 of 8.96(b). Since \j ηεΝ*: Nn = N
(9.48(d)). Each g (acN*) is a monomorphism since
N is inteqral. N is finite, so R < Aut (N,+).
N is non-trivial, so G + fid}.
G is also fixed-point-free (Heatberly-11ivier (1)):
Let ga (acN*) have a fixed-point nQ 4> 0. Let χ
be arbitrary in N. Since NnQ = N, 3 УхеМ: х = Ухпо·
Hence ga(x) = xa = yxnQa = УхРа(п0) - yxnQ = x,
so g = id.
Since N is trivially stronqly uniform we may apply
8.100 and are through.
<=-: is immediate, since j N / = | > 3 (8.85).
We apply 8.98 to get (cf. Adams (1),(2), Olivier (2)):
9.51 COROLLARY (Liqh (13)). The additive group of a finite
non-trivial integral nr. is nilpotent, but not necessarily
a b e 1 i a n .

308
§9 MORE CLASSES OF NEAR-RINGS
Mathematics is a crazy job: the additive groups of these nr.'s
without nilpotent elements J_s nilpotent.
Anyhow, we can use 8.11 to get:
9.52 COROLLARY Not every non-trivial integral nr. can be
embedded into a near-field.
9.53 REMARKS (Betsch). If one recalls (Γ,+,·Β) of 1.4(b),
1.15 and 8.97, the following results are simple corollaries
from group theory, since a group G is the additive croup of
a non-trivial integral near-rina iff G has a non-tivial
group of 'Fixed-point-free automorphisms.
(a) (Adams (1), (2)). For each ксШ there exist both
finite and infinite non-trivial integral nr.'s N such
that (N,+) is nilpotent of degree к (see (Huppert),
p. 499). Cf. Blackett (7).
(b) (Ligh (13), Heatherl y-Ol i v.i er (2)). Neither the
commutativity nor the nilpotency of (N,+) force N
to be i ntegral.
(c) (Heatherly-Olivier (2), Adams (1), (2)). If N is
infinite and integral, (N,+) is not necessarily
nilpotent (not even for dgnr.'s with both cancellation
1aws).
The following result is somewhat nostalgic in nature, for it
concerns nr.'s of 9b)l).
9.54 COROLLARY (Heatherly-Olivier (2)). Let NcT)Q be a finite
nr. without non-zero nilpotent elements, such that no
homomorphic image is trivial (9.46). Then (N,+) is
ni1 potent.
Proof. 9.36 and 9.51.
9.50, 9.51 and 9.54 exclude nany aroups from being additive
groups of non-trivial intenral nr.'s:

9b Near-rings without
309
9.55 REMARKS
(a) (Ligh-Malone (1)). Complete groups, the dihedral qroup
of order 8 and the quaternion group cannot be the
additive group of a non-trivial integral nr..
(b) (Betsch, Heatherly-01ivier (1)). The same applies to
all finite groups Γ of order 2n, wher.e η is an odd
integer > 1 . For Γ is nilpotent in this case, having
an element of order 2 which is fixed by all automorphisms
of Γ. Hence there is no non-trivial fixed-point-free
automorphism group on Γ.
(c) If |(Γ, + )| = p+1 (ρεΡ\{2}) then either there is no
non-trivial integral nr. definable on (Γ.+) or the
only one is GF(2n), where 2n = p+1.
9.56 EXAMPLES For non-trivial integral nr.'s on Zr, Z, see
Clay (6), for ones on ^31· ^ . consult Ferrero (8), for
some on 2„, Z*®^ and TL see Heatherly (5), for all
Of them Heatherly-01ivier (2). Whittington (1) gives a
computer-aided description of all non-trivial intearal
nr.'s on groups of "low" order. Additional information can
be found in Adams (1), (2). See also p. 346 and p. 348,
Lawver (3) and Olivier (2).
Clay (7) raised the question, if every non-trivial intearal nr.
has as characteristic zero or a prime (cf. 8.9). Some of the
examples above answer this negatively. So we give a condition
where this is the case:
9.57 THEOREM (Heatherly-01ivier (1)). Let N be an integral near-
ring with N, =f 0. Then the characteristic of N is either
0 or a prime.
Proof. Take dcN*i. Let χεΝ have a finite order, say
p*j, where ρεΒ* , jcIN. Then 0 = dO = d(pjx) =
* (pd)(jx), whence pd = 0. So d(px) = (pd)x =
= Ox = 0 and hence px = 0.

310
§9 MORE CLASSES OF NEAR-RINGS
9.58 REMARK (Heatherly-01ivier (1)). The same conclusion holds
if the non-trivial integral nr. N has no non-trivial riqht
i deals .
Confer also 9.17.
9.59 REMARKS
(a) (Heatherly-01ivier (2)). If a non-trivial inteqral nr.
has the property that 3 ηεΝ* V mcN: m(-n) = -(inn).
or if N is finite with |N| = Ρι·-·Ρη qf · ··q^ p3 2k
(Pj,...,pn, q^,...»qm» Ρ distinct odd primes, where
ρ is of the form 2r + l and 2-1 is a prime or
к = 0) then N is abelian.
The same follows for integral nr.'s N with a non-zerc
idempotent e such that 3 hcN: h+h = e (B.H. Neumann
(1)). С f . Olivier (2).
(b) (Ferrero (8)). Given a prime power ρ , there exists
a non-trivial integral nr. on Ж . if there is some
kcIN such that the smallest of all numbers kx-l which
are divisible by ρ is also divisible by ρ .
Ferrero called N Z-distri buti ve if V a.bcN V zt£:
a(zb) = z(ab). A finite non-trivial inteoral Z-
distributive nr. has an elementary abelian additive
group.
(c) See Heatherly-01ivier (2), Szeto (2) and Ramakotaiah-
Reddy (1) for a description of the multiplicative
semigroup (N*,·) of a non-trivial integral near-ring N.
Much more is in Heatherly-01ivier (2),(3) and Olivier (2).
Having 9.52 in mind we look for integral nr.'s which can be
embedded into a near-field.
9.60 DEFINITION A nr. N is called near-integral domain (nid.)
if N fulfills the left cancellation law and the left Ore
condition (1.64).
Generalization to "Η-monogenic nr.'s" can be found e.g. in Olivier

9b Near-rings without
311
9.61 WARNING Nid.'s are called "near-domains" by Graves-Mai one
(1) - (3). Since this collides heavily with 8.41, we will
not use this name.
Near-integral domains are really integral:
9.62 THEOREM (Graves-Ma 1 one (1)). Let N + {0} be a nid. . Then
(a) N is integral.
(b) Nc970 and if ]N| > 2, N is not trivial.
(c) If N + f0> has the DCCN then N is a near-field.
Proof. Suppose that rtO
= η
η + 0 for some
с '
η η
с с
η 0 we get
О
η ε Ν. Since
by the left
с с с
cancellation lav;, a contradiction. Hence ΝεΤ10·
If nm = 0, but η + 0, we aaain use left
cancellation to get m = 0 out of nm = 0 = nO.
m + 0
So if N is trivial, \/ η,ριεΝ: nm = ■{ (9.45).
С
0
3, take η,ρι,,η^εΝ* with m. -f* m? ■
If |N| >
then nm^
diction m
To show (c), observe that for
But
η = nrig is again resulting in a contra-
Ί = m2'
N| = 2 we oet
N = 22. Otherwise N is non-trivial. 9.48(d) tells
us that N has a right identity e. If e' is another
one and χεΝ* then xe = χ ·-- xe' implies e = e'.
By 1.112,N is a nf. .
9.63 REMARKS
(a) Observe that we did not use the Ore condition in 9.62.
(b) (Graves-Malone (1)). 9.62(c) does not hold without
a chain condition: Let (Γ,+) be the free group on
two generators x,y. For ηεΙΝ let Τ be the map
Γ ->· Γ sendinq a word w(x,y) in χ and у into w(nx,ny).
Let N be the dqnr. nenerated additively by the set
{Τ ΙηεΙΝ }. Then Wzf^ , П has the left cancellation
law but neither the left nor the riaht Ore property.
N is not a near-field.

312
§9 MORE CLASSES OF NEAR-RINGS
9.64 THEOREM (Graves-Mai one (1)). Let N be a nid. and
S: - (N*,·)· Then the nr. H$ of left quotients of N
w.r.t. S is a nf. .
Proof. S fulfills the conditions of 1.55, so N exists,
In Ν , each non-zero element is invertible, hence
Ns is a nf.
Now we get two corollaries due to Graves-Mai one (1) (see 8.9,
8.10 and 8.11).
9.65 COROLLARY Every nid. can be embedded into a near-field.
9.66 COROLLAKY Let N be a nid.. Then
( a) N is abeli an .
(b) у η,ρίεΝ: n(-m) = -nm.
(c) N has as characteristic either zero or a prime.
This is a satisfactory result, which may conclude our
considerations of integral near-rings. We only make some
remarks:
9.67 REMARKS
(a) (Graves-Malone (1)). If I is an ideal of the nid. N
then the near-fields of quotients of I and N coincide.
(b) Berman-Si1verman (1) defined a nr. N to be a D-ri η η
if N is integral and V ηεΝ 3 ηθ·ηΓεΝ*: nencC(N) and
n n ε С (N ) . (C(N) is again the center of N).
Graves-Malone showed in (1) that each D-rinq is a nid.
and in (3) that the converse does not hold by lookina
at the nr. N of formal power series over IR with
a2n = " ^or a^ ηε^ο' ^ 1S a "id·» but no D-rina.
(c) See Graves-Malone (2) for a discussion of nid.'s which
have an Euclidean alaorithm ("Euclidean near-domains").
Each such nid. N has the ACCN, an identity, only
monogenic [.'-suboroups and a uniauc factorisation into units
and prines for every non-zero element of N. Confer 7.72,

9c Affine near-rings
313
c) AFFINE NEAR-RINGS
Now we study a class of near-rings which are in a certain sense
the "most elementary non-zero-symmetric near-rings". The
dominant property is that the constants form an ideal.
Let F be a field, V a vector space over F and Μ --(V) the
near-ring of affine transformations on V as in 1.4(c).
9.68 PROPOSITION (Blackett (2)). Μ ff(V) = : N has the
following properties:
(a) N is abeli an.
(b) NQ = Nd.
(c) Nc <l N.
(d) NQ = N/N is a ring.
The proof is straightforward and omitted. Observe that all sub-
near-rings of Μ ff(V) also fulfill these properties.
Now we consider "affine transformations" over groups:
9.69 NOTATION Let Γ be an abelian group.
Ма(Г): = Нот(Г,Г)+Мс(Г).
9.70 PROPOSITION Μ,(Γ ) < Ч(Г).
Again, the proof is obvious.
Μ (Γ) also enjoys the properties (a) - (d) of 9.68. This gives
α
motivation for an axiomatic treatment:
9.71 DEFINITION (Ronshor (1)). A nr. N is called an abstract
affine near-ring ( = : a .a.η.r. ) provided that
(a) N is abelian.
(b) NQ - Nd.

314
§9 MORE CLASSES OF NEAR-RINGS
Evidently, Μ.(Г), М --(V), every ring and every nr. of linear
polynomials (linear polynomial functions) are examples of
a.a.n.r.'s. See also 9.81.
There is no need to postulate (c) and (d) of 9.68 since
9.72 PROPOSITION Let N be an a.a.n.r.
(a) (N. + ) = (N0,+)+(Nc,+).
Then
(b) Nc S3 N.
(c) NQ - N/Nc is a ring.
The proof of (a) is done by rememberinq 1.13, the rest
is established by straiahtforward computations (for (c
one can use 2.8).
The main types of substructures of an a.a.n.r. are easily
character!zed:
9.73 PROPOSITION (Gonshor (1)). Let N be an a.a.n.r.. Then
(a) All N-subgroups S of N are of the form S = S +N
with S <N NQ.
о
(b) All right ideals R of N are R = R0+Rc where
Ro ^r No · (Rc + ) s ('V + ) and R0NcsRc-
(c) АП ideals I of N are given by I = I0+Ic with
I 53 Ν. Ν Ι Ε I , IN
о о' о с сое
I
(d) All two-sided invariant subgroups Τ of N are
Τ = T0 + Nc, where TQ <3 NQ.
Again, the proof consists only of standard arquments (observe
2.18). The formulation of (a) - (d) is meant in this way that
every S +N with S <N N is an N-suboroup of N, and so on.
о
9.74 COROLLARY Every two-sided invariant subgroup of an a.a.n.r.
i s an i deal , and ever)
conversely, of course;
is an ideal, and every N -subqroup is a le^t ideal (and

9c Affine near-rings
315
In the ca?e of Maff(v) one can improve 9.73 (see 9.76).
9.75 REMARK Not every left ideal of an a.a.n.r. can be directly
decomposed similar to 9.73: take for instance N
Μ
aff
(ПП
and
((0,0): (0,1)). UN, defined by fc(x,y): =
(x+y-l,x+y-l) is in L, but
and
are not in L,
for *с(х,у) = (-1,-1) and fcQ(x,y) = (х+У,х+у).
9.76 PROPOSITION (Wolfson (1)). For each ordinal λ > 0 let
{hcHomF(V,V) | dim Im h <·Κχ}. Let Τ_χ: = {ό},
Then
(a) All ideals of Maff(V) are given by TX+MC(V) (λ>-1);
hence the ideals are MC(V)=T0+MC(VJ^Tj+M (V)=...c
=M
aff
(V).
(b) For v>-l, Maff(V)/T +M (V) =- HomF(V,V)/T , so every
А С А
homomorphic image + Μ ^^(V) is a ring.
(c) In particular, dim-V < » implies that ΜC(V) is the
unique ideal of Μ ^(V).
Essentially, this follows from 9.73(c) and the fact that every
ideal in Homp(V.V) is some Τλ (see e.g. (Baer)).
9.77 THEOREM Let N be an a.a.n.r., ^(NQ ) the Jacobson-radical
of the ring N and ^(tJ N ) the radical of the N -module
N (= the intersection of all maximal N -subaroups of Nc).
Then 70(Ν) = ···=72(ΓΙ)=^(Ν0) ♦ }(NoNc).
Proof. By 9.74, ^0(fl) = ... = ^2(N). If ηε?2(Ν)η NQ,
take some N -group Γ of type 2. Since Ν ^ Ν/Ν , Γ
is an N-qroup of type 2 by 3.14 (b). Hence ηε(ο:Γ),
so ηε92(Ν0) = ^(NQ). Therefore ?2(N)nN0 = ^(NQ)
by 5.25. The rest follows from 5.32 and the
observations in 5.67 (t).
9.78 REMARK It is not known if one can obtain similar results
for the prime and nil radicals of an a.a.n.r..
Concerning prime ideals one can say that if PQ^ flQ
then Ρ is prime in N iff Ρ +N„ is prime in N.
о о oc

316
§9 MORE CLASSES OF NEAR-RINGS
To get a theorem on constructions of a.a.n.r.'s it will be
convenient to have (cf. Dasic (1), Natarajan (4))
9.79 DEFINITION For any nr. N and η,η',η"εΝ let
D(n;n' ,n"): = n(η'+n")-nn'-nn"
denote the "distributor of η w.r.t. n' and n" ".
9.80 THEOREM Let N be a nr. Then the followinq statements are
equivalent:
(a) N is an a.a.n.r. .
(b) N is abelian and (N, + ) = (Nd. + )+(Nc, +).
(c) (N ,+) is an N (ring-) module (where n0nc is
defined as in N).
(d) There exists an abelian group Γ with Nc»M (r).
(e) N is abelian and У η,η',η"εΝ: D(n;n',n") = -η = -ηΟ.
Proof. (a ) => (b ) ■=> (с ) is obvious.
(c) => (d) (Gonshor (1)): Extend M N to a faithful
% c
NQ-modu1e Г (this can always be done). Consider
h: N + Ма(Г), where h(n) = h(nQ+nc) = : f : г* Г
Y-noY+nc
It is easy to see that V neN: f„ε Μ„(Γ) and that h
Π α
is a nr.-homomorphism. h is injective since f = fm
implies \/ γεΓ: f_(Y) = fm(Y); taking γ = о yields
η = m and this in turn that η = m , since Γ is
faithful .
(d) => (e) Suppose that N < Ma(r). Then N is abelian
and D(n;n',n") = η(n*+n")-nn'-nn" = nQ (n ' +n" ) + n -
-non'-nc-n0n"-nc = n0n,+n0n"-n0n'-%n"-nc - -nc.
(e) -> (a): It suffices to show that NQ e Nrf.
Take η0εΝ0· Then for all η',η"εΝ we net
D(n0;n',n") = -nQ0 = 0, so nQcNd.
(c) in 9.80 has some sort of a converse:

9c Affine near-rings
317
9.81 PROPOSITION Let R be a ring and RM an R-module. Then
there is exactly one way to extend the multiplication
·: RxM * Μ to a multiplication "o" in (N,+): =
= (R,+) ® (M,+) such that (N,+,o) is a nr. with
Nd = N = R®{0} and Nc = {0}βΜ, namely
(r.m)o(r',m') = (rr'.rm'+m) (cf. Clay (1) and 9.78).
Moreover, (N,+,o) is then an a.a.n.r. and all a.a.n.r.'s
arise in this way.
Proof. It is a routine check to see that (N,+,o) is an
a.a.n.r. with the indicated properties.
Suppose that ( N , + ,») is a nr. with (N,+) =
= (R,+)®(M,+), Nd = NQ = RSiO} and N = (0}SM.
Then tf (r,m),(r* ,m»)eN: (r ,m)»(r',m') =
= ((г,0)+(0,т))»((г',0)+(0,т')) = (r,0)·(r',0)+
+ (r,0)*(0,m') + (0,m)«(r',m') = (rr' ,0) + (O.rm')+ (0,m) =
= (r.m)o(r' ,m' ).
Since Nj = Ν, Ν is an a.a.n.r.
The rest follows from 9.80(c).
9.82 COROLLARY
(a) The class of all a.a.n.r.'s is a variety.
(b) Subnear-rinos, homomorphic images and direct products
of a.a.n.r.'s are again a.a.n.r.'s.
(c) Every a.a.n.r. is a subdirect product of subdirectly
irreducible a.a.n.r.'s.
Proof. 9.80(e) shows that the class of all a.a.n.r.'s
is equationally definable, hence a variety, (b) and
(c) are well-known consequences.
The last assertion of the preceding result cries for a
description of subdirectly irreducible a.a.n.r.'s:
9.83 THEOREM Let N be an a.a.n.r. with Η ± NQ. N is subdirectly
irreducible <=> N has a smallest under all rjQ-suboroups
+ {0>.

318 §9 MORE CLASSES OF NEAR-RINGS
Proof. By 1.60, N is subdirectly irreducible iff N has a
smallest non-zero ideal I. By 9.73(c) I has the form
I = Γ+Ι with I <1 Nn, N I„ S N and I N «ξ I .
ос о о о с с ос с
Since Ν ^ Ν, Is Ν , whence I = {0}. So I is
just an N -subgroup of N and of course the smallest
one.
Conversely, let N have a smallest non-zero N
-subgroup M, and let J be an ideal + {°) of N. Again,
J = J0+Jc* wnence M - Jc - J· ВУ 9.73, Μ is an
ideal of N and hence the smallest under all non-zero
ones.
9.84 REMARKS
(a) Neither "Nc ^ N" nor "N0 is a ring" alone imply
that a nr. N is an abstract affine one (take e.g.
any non-abelian constant near-ring as counterexample).
(b) Also, not every a.a.n.r. N can be embedded into some
Μ -f(V) (each element of any Μ ,f(V) has as
characteristic 0 or a prime; this does not necessarily
hold in the M,(r)'s. ).
α
We will discuss these problems now.
9.85 THEOREM Let N be a nr. with rjQNc = Nc and Nc a base.
Then:
N is an a.a.n.r. <=>> N <l N.
с
Proof. =-> holds because of 9.72(b).
To prove "<=", let N be an ideal.
(a) First we show that N is abelian. By N <1 N and
1.13, the elements of NQ and Nc commute.
By NQNC - Nc, V nccNc 3 n^Nc 3 η0εΝ0: nc - n^.
So V nc,n^cNc: nc + n^ = nQn4n^ = (n0 + n^)n^ =
= (n£ + nQ)n(l = n£ + nQn(l = n^ + nc, proving that
(Nc>+) is ahelian.
Since N is a base, we get N «4 Μ (N,.). Hence N is
abelian.

9c Affine near-rings
319
(b) Now we show that N s N. (N.
N is always
true). Take η εΝ .
' о о
V η^εΝ„ V η^εΝ^: χ: = η«(nI + n^)-n«n«EfL» since
о о с с оч о с о о с
Nc <l N.
Hence χ = хО = η (η'0+η 0)-η η'Ο = η η . Therefore
ον о с ' οο ос
Then
оv о с' oo ос
Furthermore, let nc»nc be arbitrary εΝ <
3 η;εΝο 3 n»eNc: n^ - n^. Thus n0(nc+n;) -
- no(nc + nonc) = no(nc + noK - KWiK -
= nonc + nononc -" nonc + nonc·
Consequently, for all ηό>η0'εΝ0 and ncENc we oet
(noK + no>>nc " Vnonc + nonc> = nononc + nononc =
= (nono+nono)nc·
Since N forms a base, η0(η0+ηο) = nnno+nono'
Plugging these results together yields
V η',η'ΈΝ V η0εΝ0: n0(n'+n") = πο((η4η;)ι(η4η»))
= nn((n>n:) + (nl + n")) = ηΛ(η!+η")+η (η'+η") = ηΛη' +
0ЧЧ0 О ' ν С С' ' О v 0 О' О v С С ' 00
+ n,X + n,X + n,n:i = ηΛ(η'+η')+ηΛ(η"+η'') = η η'+η η",
о о о с о с ον о с' ον о с' ο ο
Hence η. is distributive and H,
*d-
If instead of Ν Ν = N more is postulated,one nets much more
information out of 9.85 concerning 9.84(b):
9.86 THEOREM (cf. Heatherly (2)). Let Nc be a base and ideal
of a nr. N, such that V ncEfJc: Nonc
N . We may assume
that NeM(N ) (1.96). Then there exists a field D making
N into a vector space over D, N is 2-primitive and dense
in Maff(Nc) and N0 is a primitive rino, dense in
Homn(Nc,Nc).
Proof. 9.85 tells us that N is an a.a.n.r., so NQ is a
ring. The assumptions imply that N is a primitive
ring, hence a dense subring of Homn(N ,N ) , where
D is the centralizer Horn., (N ,N ). An application
No
of 4.27(a) finishes the proof, since Ν = Μ (N,)?N.

320
§9 MORE CLASSES OF NEAR-RINGS
We close this section with theorem 9.88 for which we need
9.87 DEFINITION A set N together with two binary operations
+,· is called a generalized ring (Beaumont (1)) if
(a) (N,+) is an abelian group.
(b)(N,·) isasemigroup.
(c) 3 r,SElN\{l} \f n,,...,n ,η{,...,η:εΝ: ( J n.) ( \ ni) =
r s i=l j-l J
i=l j-l Ί 3
9.88 THEOREM (Beaumont (1), Ferrero (3)). Let N be a nr. with
bounded order of the elements of (N ,+). Then:
N is an a.a.n.r. <=*> N is a generalized ring.
Proof. —>: Let N be an a.a.n.r. . Then, by 9.80(e),
\j η,η',η"εΝ: η(η'+η") = nn'+nn"-nO. Hence
к к
V kcIN V η,,...,η.εΝ: η( ? η.) = У nn.-(k-l) nO.
1 K i=l 1 i=l 1
Let s' be the l.c.m. of the orders of the elements
of (N ,+), and set s'+l = : s. Then s > 2 and
s s
V η,η.,...,η εΝ: n( l л,) = I nni .
1 s i=l Ί i=l Ί
From this one aets (c) in 9.87 for arbitrary rcIN\{l).
<—: From 9.87(c) we can conclude that
\/ η,η',η"εΝ: n(n'+n") = (n+0+...+O) (n'+n"+0+...+0) =
r-summands s-summands
= nn'+nn" + (s-2) (nO) + (r-l) (0 (n'+n''+O+...+Ο)) =
> nn'+nn"+(s-2) (nO).
But nO = (n+0+...+O) (0+0+...+0) = nO+(s-l) (nO).
Thus (s-l)(nO) = 0 and (s-2)(n0) = -nO.
Consequently we get for all η,η',η"εΝ:
D(n-,n',n") = nn'+nn"-nO-nn'-nn" = -nO.
By 9.80, N is an abstract affine near-rino.

9d Near-rings on given groups
321
9.89 REMARKS
(a) (Maxson (1)). Maff(V) has a unique maximal ideal.
(b) (Heatherly (3)). HomF(V,V) is a maximal subnear-
ring of Maff(V).
(c) Malone (5) describes how automorphisms of N and N
(where N is an a.a.n.r.) can be "mated" to give an
automorphism of N (cf. 1.114).
(d) See Blackett (3), (4) for matrix representations of
affine transformations over a finite-dimensional vector
space V.
(e) Observe the connections to near-fields and doubly
transitive groups (see e.q. 8.40).
(f) By 9.82(a), for every set A there exists the free a.a.n.r.
A~ over X. For later use (see §9 i ) we describe, how A~
looks like: A" consists of all finite sums of elements
±α^, where each a^ is an element of the free monoid
over А и {0} (see 9 . 245).
(g) If R is a ring and Μ ε Л, the variety of all R-modules
then Μ [χ] = {rx + m|reR, ριεΜ} is an a.a.n.r.. In fact,
each a.a.n.r. N is isomorphic to such an Μ [χ] (the
underlying ring R can be chosen as R=N ). See 7.149(e).
(h) See Clay (18) for an excellent survey on a.a.n.r.'s.
d) NEAR-RINGS ON GIVEN GROUPS
1.) MULTIPLICATIONS ON A GROUP
9.90 DEFINITION Similar to 8.24 - 8.28, we can define couplino
maps on a nr. N as maps φ: N -»■ End (N, + ,·) with
Ф„°Фт = Φα /m\ „ f°r all η,ρίεΝ.
Tn rm τφ (m)·η

322
§9 MORE CLASSES OF NEAR-RINGS
As in 8.26/8.27 one gets a new nr. (N,+,o ) =: Ν , aaain
called the φ-deri vation of N, where η ο m: = <J>m(n)-m.
If for a nr. N there exists a ring R and a couplino map φ
on R such that R* = N, we call N a Dickson near-ring.
More on Dickson near-rings can be found in 9.153. We only need
9.91 PROPOSITION Let (Γ,+) be a group and define the "constant
multiplication" * as in 1.4(b). Then (Г, + . *) eTJc .
Every nr. (Г,+,·) on (",+) can be coupled to this
(Г.+,*) by φ: Γ + End (Γ,+,«), given through
γ - φ : Γ - Γ
Ύ δ + δ·γ
Since End (Γ,+,*) = End (Г,+), we get
9.92 THEOREM (Clay (2), (4), (6), (8)). There is a 1-1-correspon-
dence between all near-ring multiplications on г (that are
binary operations "·" making (N,+, ) into a near-ring) and
all maps φ: Γ ■* End г with φ οψ = ψ .
γ - Φγ Υ 6 \(6)
9.93 NOTATION
(a) If φ is defined by · we write φ" .
(b) If · is defined via φ we write · (as in 8.26).
With straightforward proofs one nets
9.94 PROPOSITION (Clay (4)). Let Г,ф Ьр as before.
(a) ·φ gives a distributive nr. <=> \/ γ,δεΓ : φ = φ +<b&.
(b) ·, is commutative <=> \/ γ,θεΓ : φ (δ) = φ,(γ).
(c) ·. produces a zero-symmetric nr. <=> φ = о.
We have to look,which multiplications yield isomorphic nr.'s:
9.95 DEFINITION Two multiplications «^ and «2 are similar
if there is some axAut Г with γ,δεΓ; α(γ· ,f> ί=α(γ) ·2«(ί ) .

9d Near-rings on given groups
323
9.96 PROPOSITION (Clay (4)). The followinn conditions are
equi valent:
(a)
1
and
are similar.
(b) (Γ,+,.,) = (Γ,*,·-)
9.97 PROPOSITION (e.g. Clay (8)).
(a) γ is a zero divisor in (r,+,·)<=> φ is not injective
(b) (Γ, + ,· ) is intenral <=--> all φ. are injective (γεΓ*;.
or γ ε Ker 3 for some 3.
(c) If Γ is finite, (Γ,+,· ) is intenral iff
V γεΓ*: фуеАи1 (Г,+).
At this place it might be appropriate to remark the construction
method of 1.4(b) of a nr. on (Г. + ) via a fixed-point-free
automorphism group, (cf. also Theorem G of Clay (8)).
We also mention without the evident proof
9.98 PROPOSITION (Clay (4)). The multiplication · on (Г.+)
is trivial (in the sense of 1.4(b)) <—> \j γεΓ: φ*ε{δ,1(0.
9.99 THEOREM (Clay (4)). (Γ.+,·) is constant iff φ^εΑυί(Γ).
This holds by 9.91. In this case, φ* = id.
9.100 REMARKS
(a) See the appendix for all nr.'s on groups of order
< 7 and of many ones of order 8 and 12.
Examples of nr.'s on non-abelian groups of order 12
and 18 are in Malone (1). Yearby (1) contains
many more examples.
(b) There are groups (Z~ for instance), on which no
rings except the zeroring is definable. Abelian
groups with this property are called "nil groups".

324
§9 MORE CLASSES OF NEAR-RINGS
Lawver (1),(2) has shown that there might exist non-
trivial near-rings on nil groups. Ligh-Malone (1) have
shown that near-rings without zero divisors on complete
groups are constant or fulfill ab = j ?. .T0
(a group is said to be complete if its center = {o}
and if all automorphisms are inner· all S (n 4 2,6)
are complete).
(c) Lawver (3) studied nr.'s on free groups and on direct
sums of groups. Cf. 1.22(a).
(d) Clay (1),(6) studied the multiplications on an abelian
group by giving them a group structure.
(e) "Multiplications" turning a ring into a composition
ring are studied in Adler (1).
See also the lines on lamineted near-rings in 9.277.

9d Near-rings on given groups 325
2.) NEAR-RINGS ON SIMPLE AMD ON CYCLIC GROUPS
9.101 PROPOSITION (e.g. Heatherly ( 2 )). Let (Γ. + ) be a simple
group and (Γ,+,·) =:Γ a near-ring on Г. Then
(a) Γε9?0 ν rcT?c.
(b) V γεΓ : (φ* = δ) ν (φ* is a monomorphism).
(c) Г finite -> V γεΓ : (φ* = δ) ν (φ'εΑιιί (Γ, + )) ·
Proof, (a) follows from the fact that (rQ,+) <l (Γ.+).
(b) and (c) result from considerino Ker φ*.
9.102 THEOREM (Heatherly (1),(2)). Let Γ = (Γ,+,·) be a nr.
on the finite simple qroup (Г.+). Then Г falls into one
of the following disjoint classes:
(a) \/ γεΓ: φ* = б (in this case, · is the "zero
multiplication") .
(b) rd = {o} and Г has a right identity.
(c) Г, -f {°} and r nas an identity.
Proof. Suppose that · is not the zero multiplication. By
9.101(c), 3 δεΓ : φ^: Γ + Γ eAut(r,+). Now
γ ·* γα
3 ксШ : (фа)к = idr . V γεΓ : γ = id(Y) = (Фа)к(у) -
к к
= γα , and α is a right identity.
If Γ, + {o}, take δεΓί. Consider δΦ:Γ+Γ εEnd(Γ,+)
γ+δγ
Since · is not the zero multiplication, (ο:Γ) = {о}
for (о:Г) ^ (Г,+). Hence .ψ + б and (as above)
δψεΑυί(Γ,+) and some power of S is a left identity,
hence the identity.
Observe that 9.109 implies that in case (c) of 9.102,Γ has to
be a finite prime field.
We now turn to cyclic groups.

326
§9 MORE CLASSES OF NEAR-RINGS
9.103 THEOREM (Heatherly (1) and others). Let N be a nr. on
η
ring
or Ζ with a generator gcN.· then N is a commutative
Proof. In this case, N is an abelian d g η r. , hence a ring.
Every ring on a cyclic group is commutative (see
(Beaumont)).
9.104 COROLLARY (Heatherly (1)). If N is a nr. on Ж (ρεΡ)
or TL with N. \ {0} then N is a commutative ring and
there is some χεΝ with ]j η,η'εΝ : nn' = η·η'·χ
(usual product in Ж or Ζ ).
Proof. If dr.N, is 4* 0, a short calculation (cancel d!)
shows that 1 is also εΝ,; now we may apply 9.103
to get the first assertion.
Let 1·1 =: χ and η,η'εΝ. Then
nn' = (1+...+l)(l+...+l) = η·η'·(1·1) = η-η'.χ.
n-summands n'-summands
9.105 REMARK The same result as in 9.104 holds in every TL^
if ΙεΝ.. On the other hand, Heatherly (1) gives an
example of a nr. on Z4 which is not a ring (in fact,
1 and 3 are not distributive).
For the next result, let C(k,j) be the number of combinations
of к elements to the class j). Without proof we state
9.106 THEOREM (R. Jacobson (1)). The number of different nr.'s
definable on (2p.+ ) (ΡεΙΡ) is given by
2+ I ( I C(k,j)(ulA)J).
k/p-1 J-l K
More informations can be found in Adler (1), Feigelstock (2) and
Heatherly (2).

9d Near-rings on given groups 327
3.) NEAR-RINGS WITH IDENTITIES ON GIVEN GROUPS
We start with
9. 107 PROPOSITION (Clay (4)). Let Ν = (Γ.+ ,·. ) be a nr. on Г..
φ
Then ΙεΝ is an identity of N iff φι -■ id and
V γεΓ: φγ(1) = γ.
The proof is obvious.
Out of 9.103 and 9.104 we qet (observe that under the piven
assumptions, χ of 9.104 is invertible in fl) :
9.108 COROLLARY (Clay-Malone (1)). If N is a nr. ε^ on the
cyclic group (N, + ) then N is a commutative rinci. All
nr.'s on (N,+) are isomorphic. There are Φ(η) ( Φ the
Euler function) ones on (Zn,+) and 2 on 2.
9.109 COROLLARY There are exactly p-1 nr.'s with identity
definable on (Z ,+); all of them are isomorphic to the
field Ж and hence all are finite prime fields.
This result was obtained by Malone, Clay, Maxson and Heatherly
under different circumstances.
Observe that if in (Γ, + ,·)εΤ7, (Γ, + ) is abelian with exactly
one proper subgroup then (Γ,+) = Ϊ 2 and (Г.+,·) is a
commutative rinq by 9.108 (Liqh (9)). But there do exist non-
rinqs with identity on qroups of order ρ (cf. also 9.115(c)):
9.110 PROPOSITION (Maxson (1)). For each ρε IP there exists a
2
group Γ of order ρ and a non-rinq with identity on Γ.
The proof is established by defining a multiplication on
(Γ,+): = (Z ,+)®(2 ,+) in an appropriate manner (see
Maxson (1) for details.)

328
§9 MORE CLASSES OF NEAR-RINGS
Now we study nr.'s of square-free order. First we need
9.111 PROPOSITION (Clay-Malone (1), Maxson (1)). Let (Γ,+,>)
be a nr. with identity 1 on the finite group Г. Let
ord(y) be the order of γεΓ. Then
ord(l) = 1.c.m.{ord(y)|γεΓ} =: I.
Proof. If γεΓ, ο = ογ = (ord(l).l)y = ord(l)-y, so
ord(y)/ord( 1) . Hence £./ord(l). But ΙεΓ, hence
ord(l)/2. whence ord(l) = I.
9.112 THEOREM (Maxson (1),(2)). Let (Γ,+,·)ε?71 have finite
square-free order. Then (Γ.+) is cyclic, and (Γ.+,·)
is a commutative rinq.
,Pr are
Proof. Let ]Γ[ = ρjp2·- -Pr» where p^,,
distinct primes. Using the Sylow theory we get for
each ic{l,...,r} some Υ,-εΓ of order p^. Hence
|G| > ord(l) = 1 .c.m.{ord(y)|γεΓ} >
& l.c.m.{ord(Y1),...,ord(Yr)} = |G|.
So ord(l) = |G| and P. is cyclic. Now use 9.108.
Several groups cannot bear a nr. with identity (call a subset
Ρ of a partially ordered set an anti chai η if no distinct elements
are comparable ):
9.113 THEOREM (Krimmel (1),(2))· Let (Γ,+) be a group havino
elements Y^>...,Yr of distinct prime orders Pj»...»Pr
(r > 2). If every antichain in the lattice of normal
subgroups of Γ has cardinality < r then (Γ.+) cannot
be the additive group of a nr. with identity.
Proof. Suppose that (Γ.+,·) is a nr. with identity 1.
If there are 1,je{1 r) with (ο:γ.|) ? (o:Yj)
then h: (Γγ·,+) ■* (Γγ·, + ) is a well-defined aroup-
YYi * YY·,·
homomorphism. But п(у^) = h(lY.j) = 1γ.= = γ·, whence

9d Near-rings on given groups
329
p. = ord(Yj)/ord(yi) = pi , so i = j. Hence
{(ο:γ,),...,(ο:γ)} is an antichain with r elements,
a contradiction.
Observe that we didn't use associativity of ·; 1 could have been
only a left identity.
9.114 COROLLARY (Clay-Malone (1)). Λ nr. with identity on a
finite simple group Γ is a finite prime field.
Proof. Γ cannot have a composite order by 9.113. Hence
Γ is a simple p-group, thus coinciding with it's
center. So г is isomorphic to J and we can apply
9.109.
9.115 COROLLARIES (Krimmel (2), Clay-Malone (1), Clay-Doi (1),
Ligh (9)).
The following groups Г cannot be the additive qrotips of
near-rings with identity:
Sn (^3>'
(a) groups of composite order in which the lattice of
normal subgroups is linearly ordered (e.g
(b) simple groups of composite order (e.g. A (n>4)),
(c) finite ηοπ-abelian nroups with exactly one proper
normal non-zero subgroup,
(d) non-cyclic groups of square-free order.
Proof. Evidently 9.114 => (a) => (b) and 9.112 => (d).
In (с), Г must be of composite order since otherwise
Г is a non-abelian p-group, hence of order ρ with
k>3. In this case, Γ has at least two non-trivial
normal subgroups (see e.g. (Rotman), Cor. 5.5 and
Ex. 5.2) .
We now mention without proof some more results on this subject,
If Ν ε ?7j is finite such that the invariant subgroups of (N, + )
form a chain then Μ is isomorphic to a ring Ж n.

330
§9 MORE CLASSES OF NEAR-RINGS
9.116 THEOREM
(a) (Ligh (9)). There is no nr. with identity definable
on a torsion divisible group.
(b) (Clay-Doi (1)). The same holds for S : = (J S
and A.
U Ar
ηεΙΝ r
ηεΙΝ
(c) (Clay-Maxson (1)). There are also no nr.'s ε?ϊ,
definable on generalized quaternion groups.
(d) (Ligh (13)). There do exist nr.'s with identity on
perfect groups (that are groups coincidino with its
commutator subgroup) (cf. Ligh (9)).
(e) See Johnson (4) for the nr.'s on the dihedral nroups
D- of order 2n. There are no nr.'s εΤ), on D„
Zn 1 2n
for odd η (this follows from 9.111), for the only
ones exist on D. (ρεΡ ). They are zero-symmetric
and normal N-subgroups and left ideals coincide (and
all left ideals are annihilator left ideals).
There are (up to isomorphism) 7 nr.'s with identity
on Dg (p.418) and (again up to isomorphism)
just one on D. (ρεΙΡ\{2>). There is just one such
nr. on the infinite dihedral group (Lockhart (1),(3))
(f) (Clay-Maxson (1)). All nr.'s with identity definable
on p-groups with exactly one subgroup of order ρ are
commutative rinns.
(This follows from 9.108 and (c) since a group as
described above is either cyclic or a generalized
quaternion group.)
4.) NEAR-RINGS WITH OTHER PROPERTIES ON GIVEN GROUPS
Now we briefly study nr.'s with special properties (other than
having an identity) en some nroup (Γ,+).
Vie will only cite the results or even only the napers which
are concerned with these topics. See also the chapters
concerning the types or near-rings in discussion. For example,

9d Near rings on given groups
331
there are no near-fields definable on non-abelian nroups (3.11),
and so on.
We start with nr.'s with chain conditions. Heneralizinn 9.102
one qets
9.117 THEOREM (Linh (3)). Let N be a nr. with DCC on mononenic
N-subnroups on the simple nroup (N,+) such that
N. 4" {0}. Then N is either the zero-nr. or a field.
9.113 REMARK For a detailed study of nr.'s N on a nroup which
fulfill the DCC on mononenic N-subnroups and the "ЛСС on
principal annihilate- left ideals" (i.e. each (0:x)s
2 3
s(0:x )e(o:x )=... terminates) see Linh-Ramakotaiah-
Reddy (1).
9.119 THEOREM (Timm (3)). (Γ.+) is the additive nroup of a
(not necessarily associative (!)) near-rirtn in which every
non-zero element has a rinht inverse iff Γ is invariantly
simple and every γ^Γ has (the same) prime order.
The question concerninn the additive nroup of near-fields is
settled by the followinn theorem.
9.120 THEOREM (Timm (3)). The following conditions on a group
(Γ , + ) are equi valent:
(a) Г is the additive nroup of a near-field.
(b) Γ is abelian and the additive nroup of a nr. with
rinht cancellation law.
(c) Г is the additive nroup of a vector space over some
field.
(d) Γ is the additive nroup of a commutative field.
(e) Γ is the additive nroup of an alternative field.
(f) There is some ρεΡ such that Γ is the direct sum of
the nroups (Έ +) or Γ is a direct sum or copies of
(rj) Γ is abelian and either each elenent has the sane
prime order or Γ is torsion'ree divisible.

332
§9 MORE CLASSES OF NEAR-RINGS
Finally, we consider the additive group of dgnr.'s and of
integral nr. ' s .
9.121 REMARKS
(a) (Ligh (10)). There are just 3 non-isomorphiс dgnr.'s
on S, and at least 3 on S (ni5, n=f6).
There are precisely 3 non-isomorphic dgnr.'s definable
on D? (ρ ε IP) , but none on the infinite dihedral group
0Ю (Lockhart (1),(3)).
(b) (Ligh (13)). The additive group of a simple dgnr. is
perfect.
(c) Dgnr.'s with identity on free groups are extensively
studied in Zeamer (2) .
(d) Dgnr.'s on groups Г, in which the index of the derived
group Г' is prime, are considered in Chandy (3).
(e) (Malone (7)). There are exactly 16 dgnr.'s on a
generalized quaternion group. All of them are distributive.
(f) More on distributive near-rings on given groups can be
found in Jones (1) and Willhite (1).
(g) From 9.51 we know that integral near-rings on finite
groups Г force г to be nilpotent. If Г is non-abelian
of order ρ (ρεΙΡ) with ρ =2η+1 then there are no
integral nr.'s N definable on Γ, such that N has at
least one right identity ^ 0.
If |r| = p+1, ρεΙΡ , ρ 4 2 then either again no such N
exists or p+1 is a power p+1 = 2n of 2 and N is a
Galois-field (Olivier (2), Heatherly-01ivier (2)).
(h) (Lawver (3)). All near-rings on 2°° are planar. There
are no integral planar nr.'s definable on Zi, but
there are some on iT (with characteristic j 5 !).
(i) "H-monogenic" near-rings (see 9.275) are generalizations
of integral near-rings. Additive groups of H-monogenic
near-rings are studied in Olivier (2) and Heatherly-
01 i ν i e r ( 3 ) .

9e Ordered near-rings
333
If a group Γ is given by a presentation, it is a highly non-
trivial matter to characterize all near-rings on Γ. First
studies in this directions (including "pre-near-rings"
{- multipiicativelу non-associative near-rings)) can be found
in Lockhart (1) and Laxton-Lockhart (1).
e) ORDERED NEAR-PINGS
9.122 DEFINITION A nr. N is called partially (fully) ordered
by < if
(a) < makes (N,+) into a partially (fully) ordered nrouD.
(b) \j η,η'εΜ: (η>0 Λ n'>0 => nn'>0).
"Ordered" means "partially ordered".
9.123 REMARKS
(a) Thus an ordered near-rinn is a nr. where (N,<) is
an ordered set such that n>0, η'>0 implies n+n'>0
and nn'>0.
(b) The standard work on ordered aloebraic systems
(semigroups, nroups, rinns and fields) is (Fuchs).
(c) For an ordered near-ring we will write (N,+,·,<) or
simply (N,<).
(d) Parts of our discussion is implicit in (Gabovich).
Of course, п. й n2 and Oin implies n^nSr^n in an ordered nr. N,
Some authors (K.B.P. Rao (1), for instance) require that also
nnj<nn2 follows. Cf. 9.152(b),(d) .

334
§9 MORE CLASSES OF NEAR-RINGS
9.124 NOTATION We adopt the usual conventions to write n<n',
n>n ' , η>η', η!' n' (n and n' are i ncomparable, i.e.
neither n<n' nor n'<n holds).
"Partially ordered" will be abbreviated by "p.o.",
"fully ordered" by "f.o.".
Just as in the theory of ordered nroups or rinns, it is more
convenient to work with the set of "positive" elements instead
of the order relation itself:
9.125 THEOREM
(a) Let (PI, +,·,<) be a p.o. nr. ; then the "positive
cone" P<: = Ρ: = {ηεΝ|n>0} fulfills
(α) Ρ+Ρ = P.
(β) Pn(-P) = {0}, where, as usual, -P: = {njn<0}.
(γ) V ηεΝ: η+Ρ = P+n.
(6) P-P = P.
(b) Conversely, for every subset Ρ of a nr. N fulfil 11m
(a) - (6) we net an ordered nr. (N , < p ) via
η <ρ η': <=> η '-ηεΡ.
(c) This correspondence between order relations and
subsets with (α) - (δ) is 1-1, that means that
<0 = s and Ρ = Ρ' .
= Ρ'
The proof of 9.125 is easy and left to the reader.
9.125 enables us to say that "the nr. N is ordered by P".
The followinn result is obvious.
9.126 PROPOSITION Let N be ordered by P.
(a) <p is a full order <=> Ρ w(-P) = N.
(b) <p is trivial (i.e. η <D η'<=> η = η') <=> ρ = {0}.
There is no place for finite near-rinns in this section:

9e Ordered near-rings
335
9.127 PROPOSITION Every non-trivially ordered near-rino is
infinite.
Proof. 3 nEf': n>0· But then n<n + n = 2n<3n<... .
9.123 DEFINITION Let N,N' be nr.'s ordered by P,P',
respectively. f: N -*■ N' is order-preservi nn: <™>
<-> f(P)ep'.
If there is an order-preservino mono-(iso-)morphism
f : N ->· N' we write N c» N' (N =. N', respective! ν)
(for isomorphisms f we also want f" to be order-
preservinq since the catenory of ordered near-rinas and
order-preservinn homomorphisns is not balanced).
9.129 DEFINITION Λ subset Τ of an ordered nr. N is called
convex i f
V tl5t2cT \/ ηεΝ : (t1<n<t2 -> ηεΤ).
Similar to rinn theory (see (Fuchs)) one can easily prove the
followinn two results. Anain, they are corollaries of theorems
of (Γ5 a b ο ν i с h ) .
9.130 THEOREM A subset I of an ordered nr. N is the kernel of
an order-preservinn nr.-homomorphism from N to some
ordered nr. M' iff I is a convex ideal.
9.131 THEOREM Let N,N' be nr.'s, ordered by P,P'. Let
h: Μ ■**■ Ν' be an order-preservi nn epimorphism (i.e.
h(N) = N' and h(P) = P' ) .
If I' <l N' is convex then h" (I') =: I is a convex
ideal of N and N/j = N'/j, .
Convexity is quitt trivial, so we won't prove it.
9.132 PROPOSITION Every order Ρ in N can be extended to a
maximal order F.
The proof is accomplished by an application of 7orn's Lemmi.

336
§9 MORE CLASSES OF NEAR-RINGS
9.133 PROPOSITION To every abelian ordered nr. N there exists
an (abelian) ordered nr. N with identity such that
Μ <=^ N.
Proof. By 1.86, we can find a nroup Γ with НцЧ(г)
(say by h). If N is ordered by P, take P: = h(P).
Then α), β), δ) and (since N is abelian) also γ)
of 9.125 are clearly fulfilled, hence Ν: = Μ(Γ) is
ordered by β and h is an order-preservinn monomorphisn.
9.134 REMARKS
(a) Hot every ordered nr. can be enbedded into a f.o.
nr. with identity (this follows from 9.137).
(b) If N contains an identity 1 then 1>0 or 1|| 0,
for 1<0 implies (-1)>0, hence (-1)(-1)50,
whence 1>0, a contradiction to the assumption I<0.
In some instances we can describe π of 9.132 explicitly:
9.135 PROPOSITION Let N be a nr. such that N is ordered by
Pc and such that PQ forms a base (1.91).
Then P: = {η ε Ν | \/ρεΡ : np>0} is the unique maximal
extension of Ρ to an order of N.
Uniqueness is clear for Ρ = (pc:Pch ancl «) " fi) °f 9.125 are
easily verified. So the proof is easy. Nevertheless, this
proposition has heavy conseauences, e.g. that in neneral P(R)
cannot be fully ordered: each ρ cP(R) would have to have only
positive or only nenative values at {rcR'r>0}!
If N is complete (i.e. if \/ ηεΝ \/ kcNc 3 "neN \j ccM : Wc =
= n(c+k)) then 9.135 implies that one cannot get full orders
except N = N0 or N = Nc (see Pi 1 ζ (1) , (3) , (б )) .
Fundamental for the followinn is
9.136 THEOREM Let N be f.o. (by P) and Nc = {0>. Then
V ηεΝ V crP :=PnN : nc = nO.

9e Ordered near-rings
337
Proof. Suppose there are some ηεΝ and οεΡ with
nc 4" ηΓ* · W.l.o.g. we may assume that n0>0
(otherwise chanqe to -n). Now η с = (n-nO)c = nc-ηθ =f 0.
Hence η =f 0.
If n„ would be >0 then 0<n„c + 0, whence n c>0
О О · О
Consider I: - η - η с + η Q ; I fulfills i. с = η с > 0,
whence £>0. On the other hand, P.O = ηθ-nc =
= -nQc<0 implies iUO and we arrive at a
contradiction.
The assumption n0<r1 leads to the same disaster.
9.137 COROLLARY If Μ has a left identity and is fully ordered
then ΝεΤ) .
9.138 REMARK If under the circumstances of 9.136 N forms a
с
base and N is considered as subnear-rino of Μ (N )
4 с'
(1.96) then 9.137 tells us that each ηεΝ is constant
on all positive elements of Μ We will see that in
fact f.o. nr.'s are closely related to constant near-
rinos (9.141(a)).
9.139 DEFINITION An ordered nr. N is called archimedian ordered
if (N,+) happens to be this (see e.g. (Fuchs)).
If a f.o. n.r. N is archimedian, (N,+) is abelian and
(N, + )C>Q (IR, + ). If N is not archimedian then there exist
pairs (a,b) of N2 with к · | a | = |a I + I a| + ... + I a I<\ Ы for all
kcIN (where |aj has the usual meaning - see (Fuchs)). In
this case we call a "small w. r. t. b" and write a«b. If
AsN, "A«b" and similar notations are then clear.
9.140 DEFINITION Let N be ordered and ηεΝ.
η is nearly constant: <=> n„ « N*.
i О С
9. 141 THEOREM Let N be fully ordered and Nc J- {0}. Then
(a) Every ηεΝ is nearly constant.
(b) N 4= [0} => the orde>- is non- a^chimed ian.

338
§9 MORE CLASSES OF NEAR-RINGS
Proof, (a) If N ± {0}, take any ηεΝ with nc>0.
For kcIN, let ak: = k-(-n +n)-n ,
bk: = (k-l)-n0+n.
Then a.O = -n <0, hence a.<0 and
t>k0 = η >0, whence bk>0, so \/ кгТО : ak<0<bk>
Thus -n„+n-n +...+n-n <0<n-n +n-n + . ..-n +n. So
с с с ее с
-η +к · η <0<(к-1 )n +n. From the first inequality
we qet k-n -n <0. Hence k·n.-n <0<(k-1 )n+n,
* ОС О С v ' 0
so -n <-k-n <-n +n = η and k-ln |<n for all
kcIN .
Now let хсЧ and ccN* be arbitrary, but c>1.
Let m: = xQ + c. Since m = c>n, we can apply our
considerations above to m and net ν kcIN: k-'n l<n,,
о' с
Since mQ = xQ, we see that xQ « N*.
(b) follows at once from (a).
It is hiqh time for examples,
9.142 EXAMPLES
(a) Non-zero near-rinns of the kind f4r)> 'Vont^
(1.4(a)), R [x] and f1aff(v) cannot be fully ordered
since they contain an identity and have non-zero
constants.
(b) Let R be a fully ordered commutative nr. with identity.
к ,
Then R^Txl with Ρ: = { У а.χ !au>0} υ {ή} ("lexico-
graphic order") and RQCCX]]: = (R[[x]])0 w1 th
P: = { I a .x1 I at>0} υ{δ} ("antilexiconraphic order")
i>k 1 K
supply non-trivial examples of ordered near-rinns
In R TxJ, for example, we net the followino
"archimedian classes" {pldeg ρ = 1} « {p'deg ρ = 2} «
« ... .

9e Ordered near-rings
339
(you may multiply a linear polynomial by any natural
number you want and you will never arrive at a
quadratic one).
(c) Let R be a rinn, f.o. by P. Take an arbitrary, but
fixed subset 0^-P*, and form the near-rinn
of Q).
the indicator function
Orderinn NQ 1exiconraphical ly nives a f.o. n.r.
with {0} + (Nr))c x Νη> in which all elements are
nearly constant.
9.143 Ρ Ε MAR К 9.141 shows that one cannot expect for "real"
near-rinos to qet any "neat" full order.
This is not very surprisinn: one has some subnear-rinn
of some М(Г) in hand, whose cardinality is, in oeneral,
"much biqqer" than those of Г. Anybody, who ever had to
brinn order in a larne storehouse knows: the larner the
set, the harder is it to aet a full order.
Also, one minht expect that well-ordered near-rinns are
qui te special:
9.144 ТНЕОРЕИ (Maxson (11)). Let ΝεΤ^ be well-ordered. Then
N = (2,+,·,<) ( < the usual order in J) .
Proof. Anain, let Ρ be the positive cone of N. Let a be
the smallest element of p* and A the cyclic
subgroup of (N,+) nenerated by a.
Suppose that A J= N. Let b be the smallest positive
element of U: = fl\A.
Consider a-b. If a-b>0 then a-b = a, a nonsense,
Hence b-a>0. b-acLH> b-a = b, which is a
contradiction, too.
Hence U = 0 and N is cyclic and infinite (9.127).
The map h: N = A ■* Ж i s an order-isomorphism
za ■* ζ
between the additive groups.

340
§9 MORE CLASSES OF NEAR-RINGS
By 9.137, N is isomorphic to the rinn Ζ = (Z,+,·)
(usual multiplication) or !': = (2,+,») with
z*z': = -ζ·ζ'. Zand 2' are order-isomorphiс via
ζ -+ -ζ. Hence in any case ΓΙ = Σ .
In an ordered nr. N one can ask, how !nn'| and 'η[ | η ' !
might be related. In neneral there is no direct relationship,
but for R[x] of 9.142(b) we net the followinn result which
we state without proof.
9.145 THEOREM (Pilz (4)). In (R(Y],+,o) we have for all
P>qsR[x] :
(a) IpoqI = |p!ojql <"> (q5Q) ν (ρ contains only even or
only odd deqrees).
(b) | ρ о q | < I p | о ! q ! <=> (q<0) Л (the coefficients of the
qreatest even and nreatest odd
denree of ρ have the same sinn).
(c) !poq| 5 | ρ! о j q[ <=> (q<0) Л (the coefficients of the
greatest even and nreatest odd
denree of ρ have opposite sinn).
J. Zemmer has shown that a direct sum of f.o. rinns can be
f. ordered iff all but at most one of the summands are zerorinns
We now obtain a similar result for nr.'s implyinn some
statements on the structure of f.o. nr's.
s
9.146 THEOREM If N = ® N- is f.o. then in all but at most
i.-.l 1
one of the N.'s all positive elements (in the order
induced by N via the projection maps) annihilate N.
from the riqht.
Proof. Assume that 3 i,je{l,...,s), i Α j
3 0<η.εΝ. 3 0<η.εΓ1.: Ν.η. J= {0} Λ Ν.η. J* {0}.
Then one can choose η'. ε PJ ^ and η^εΝ·, both
positive, such that η ί η . > 0 and ηlπ·>0.
If ni<nj then 1<nin.<nin. = 0 (2.27 and 2.6(b)),
a contradiction. nj>ri4 yi°Hs the same, so η ι || η j >
and we have no full order.

9e Ordered near-rings 341
S
9.147 COROLLARV Let N = Щ N. be fully ordered and all
i-1 1
Ni + (0).
If N is either strictly ordered (i.e. η > 0, η ' > 0 =>>
=>> ηη'>0 — R Гх] of 9.142(b) is strictly ordered if R is
inteqral) or if N contains a left identity then s = 1.
Proof. For strict orders this is immediate from 9.146.
s
If N contains a left identity e, let e = У е.
1=1 Ί
with e^cil·. As in 3.43, e. is a left identity
in "I· and anain we can employ 9.146 to net s = 1.
9.148 COROLLARV Every 1-semisimple f.o. nr. Ne7?1 with OCCL
i s sinple.
Proof: by 9.137, 5.31 and 9.147.
9.149 REMARK One cannot improve 9.146 to net the exact analogue
of Zemner's result: take for Ν.,Νρ any f.o. constant
non-zero near-rinns and use the 1exiconraphiс order.
Examininn abstract affine near-rinns nives a stranne result
which shows that "nearly no" a.a.n.r. can be fully ordered:
9.150 THEOREM Let N be an a.a.n.r. such that N0,NC are fully
ordered. Then N can be f.o. <~> Ν Ν = {0} л (Ν = {0} ν
ν Ν is a ζ e r ο Η η о ) .
Proof. =>: (a) First we show that (N, + ) = (N , + ) + (NQ,+)
(9.73(a)) must have the lexiconraphiс order.
If n>0 then η = n0>0; likewise n<0 implies
nc<0.
If n„ = 0 then n>0 <=> η = η >0.
с о
So we qet for ηεΝ: n>0 <=■> (пс>0) ν (η = О Л
Л η >0), i.e. the lexicographic order.

342
§9 MORE CLASSES OF NEAR-RINGS
(b) Assume now that NN J- {0}. Since ΝΛ = Ν,,
v ' ос ο α'
we can find n 0 ε Ν 0 a n ^ ^' г ε ^ с v"tn nr>ri anc'
η η < 0. Then η: = 2 η - η η >0 bv (a),
ос о о с .. ν /
But nn = n0nc<^> a contradiction. Hence N Ν = {0}
(c) If П20 А {0} and Nc -f {0} 3 η0·ηόεΝο: ηο>0 Λ
Λ η'<0 Λ η η'<0. Also, 3 η„εΝ : η >0. Let
О 0 0 С С С
η:
nQ>0 and n
n^+n >0. Then nn' = η η'+
о с
= nnn'<^ (use (b)!), a contradiction.
Hence either Pl£ = {0} or Nc = {0}.
<=: The multiplication rule in N is
npi = (no+nc)(n;+nc) = nono+nc·
If N
{0}
we net
N
N.
as a т.о. ring.
If Nc J= {0} then N£
{0} and all
nn
In this case it is easily verified that the
lexicographic order in Nc + N0 = N makes N into a f.o.nr.
9.151 COROLLARY No a.a.n.r. N J= Nc in which Nc forms a
base (this happens e.n. in Μ ,,(V) and Ч (Г)) can
be fully ordered.
Proof. If N is f.o., the same can be said about N and
c*
N.N = {0} implies N = {0}, hence N = N
о с
9.152 REMARKS
(a) See Kerby (1),(3),(5) for a theory of ordered near-
fields with some neometric interpretations.
(b) (Pilz (1),(4)). ηεΝ is called even (odd) if
γ η'εΝ: n(-n') = nn1 (n(-n') = -nn', respectively).
For instance, fcM(IR) is even (odd) iff f is an
even (odd) function. A nf. contains only odd elements
(8.10(b))·, the same applies to rinas.
N is said to be clea yable if each ηεΝ is the sum
of an even and an odd element. R Γχ"1 , ρΓΓχ*!Ί, R Γχ1
and the subnr. N generated by id, sin and cos in
M(IR) are examples of cleavable near-rinns.
9.145 can be extended to f.o. cleavablp near-rings.

9e Ordered near-rings
343
(c) R Γχ] with the antilexicographic order
о ,
(a k χ +...+arx >0: <=> a k > 0 ) . is not a f.o. nr.
(althouah a f.o. rinn when we use multiplication
instead of conposition (Fuchs)), for e.n. in this
ordering -x+l>0, x+2>0, but (-x+l)*(x+2) = -x-l<0.
(d) One can define in an ordered nr. Ν ηεΝ to be
monotone (anti tone) if V η',η"εΝ: η'<η" => nn'<nn"
(nn'>nn").
η is positive definite : <=--> \/ η' εΜ : ηη'>0.
See Pilz (1) for results on these concepts.
(e) (Pilz (8)). Let N be a nr. with (N,+) =(N ,+)+(N ,+)
(cf. 1.13), where Ц and Μ are f.o. n.r.'s
(by VV·
Then the f.o. of N and N can be extended to a
f.o. on N iff V ρ0εΡ0 V Pce»c V %EV Po^Vc^V
In this case the order is the "lexiconraphic" one
determined by n0 + nc^ <c=> (nc>0) v (nc = ^ Λ η >0)
(see 9.150).
(f) It is hard to net full orders in "non-deaenerated"
near-rinas (9.141). But it is very natural to look
for 1atti ce-orders (i.e. such that (N,<) is a
lattice). For instance, M(r), where Γ is a f.o.
group, can be oiven a lattice order by
m<m' : <—> \f γεΓ : т(у)<т'(у).
For details and connections to "F-near-ri nns" N
(these are subnr.'s and sublattices of a direct
product Π N. of f.o. nr,4 ε?) , lattice-ordered
icl 1 c
by ( . . . , η . ,...)<(..., η i , . . . ) : <=> \/ i ε 1 : η ^ <n ! )
and to vector-near-rinns (F-nr.'s, where N is a
subdirect product of the N.'s) see pilz (1), Bhandari-
Radhakrishna (1) and Radhakrishna (1).
(g) Kerby (1),(3),(5) and Groger (1),(2) studied ordered
near-fields. A nf. F is formally real if -1 is not

344
§9 MORE CLASSES OF NEAR-RINGS
the sum of products of squares. F can be fully ordered
iff F is formally real (Grbger (1)).
(h) Extensions of partial orders to full orders are studied
in K.B.P. Rao (1) .
(i) Natarajan (3) and K.B.P. Rao (1),(2) also considered
ordered N-groups.
(j) See also Kusel (1) .

9f Regular near-rings
345
f) REGULAR NEAR-RINGS
Von Neumann regular rings play an important r61e in ring theory.
They generalize some properties of near-fields to a much wider
class of rings. This concept not only transfers to near-rings,
it is also motivated by the fact, that some of the most important
types of near-rings are regular (see 9.154).
9.153 DEFINITION A near-ring is called regular if
\/ η ε Ν 3 x ε Ν : ηχη = η
9.154 EXAMPLES Regular n.r.'s are obviously:
(a) M(r) and Μ (r) (Beidleman (10)).
(b) Constant near-rings.
(c) Direct sums and products of near-fields.
(d) Integral planar near-rings N (since for η ε Ν we can
find xeN with η = xn by 8.88 (b) ; now (n-nxn)n =
).
π if it | 0
2 2
= η -η =0 gives the result (Mason.(5))
(e) (Ν,ι-,*) for any group (N,+ ) and n*m: -{
0 if m = 0
In 9.153, xn can be considered as a "private right identity"
and nx as a "private left identity" for n. If HcV. and η has
an inverse χ then nxn = n, of course.
9. 155 REMARKS
(a) In 9.153, nx and xn are idempotent.
(b) By 9.154, regular near-rings are not necessarily
abelian.
(c) Homomorphic images, direct sums and direct products
of regular near-rings are regular. By 9.154 (a),
7.33, 1.86 and 1.88, every (zero-symmetric) near-
ring can be embedded in a (zero-symmetric) simple
regular near-ring. 9.154 (a) also shows that in
general a regular nr. has neither the IFΡ nor
(P0)-(P4) (see 9.1 and 9.4)

346
§9 MORE CLASSES OF NEAR-RINGS
(d) By 9.154 (a), subnear-rings of regular nr.'s are not
regular in general.
Nevertheless, several connections to IFP-nr.'s and their
properties will show up. We now characterize regular near-rings
and display some of their properties afterwards.
9.156 THEOREM (Beidleman (10), Ligh (7)). Let NcH., N is regular
< = >Vn ε Nje = e cN:Nn = Ne.
Proof. ==*· : Take χ e N with nxn = n. Then Nn = N(xn) does
the job by 9.155 (a). *= : Take η c. N. Then Nn = Ne
for some idempotent e. Since e ε Ne, there is some
χεΝ with xn = e. Since \·\ ;:T\. , η ε Nn = Ne , hence
η = ye for some у ε Ν, and we get η = ye = yee =
= yexn = nxn.
9.157 COROLLARY (Beidleman (10)). A regular near-ring with
identity contains no non-zero nil N-subgroups.
Hence we might look at regular near-rings without nilpotent
elements.
9.153 THEOREM (Ligh (7), Chao (1), Oswald (9)). Let N $ {0}
be a regular near-ring with identity. Equivalent are:
(a) N = N has no non-zero nilpotent elements.
(b) All idempotents of N are central.
(c) N is a subdirect product of near-fields.
Proof. (a)=»(b) holds by 9.43 (6). (b) =»( с) : By 1.62,
N is isomorphic to a subdirect product of subdirectly
irreducible nr.'s. N · (i ε I). These N-'s are regular
by 9.155 (с), гУ1, and fulfill the condition (b),
too. Let A: = [\ (0: e) , where e runs over all
idempotents j= 0 and j= 1 in N^. Since each e is central
(0:e) (and hence A) are ideals. If (0:e) = {0}
then e = 1, a contradiction. By 1.60, A ^ {0}

9f Regular near-rings
347
Take a ε A, a f 0. Now axa = a for some xcN..
2
If e = e ε N. then ae = 0, hence ea = 0, whence
ec(0:a). ax is idempotent by 9.155 (a). If
(0 : a x ) = {0), ax = 1 and e = e(a x ) = (e a ) χ = Ox =
= 0, a contradiction. If (0:ax) ^ {0}, we get
ae(0:ax), hence a = (ax)a = a(ax) = 0, again a
contradiction. This shows that 0 and 1 are the only
idempotents in N,. If л cN, i s j= 0 and η = nxn
then nx and xn are k 0 and hence = 1 (by 9.155 (a)).
Hence N- is a near-field. (c)=»(a) is trivial.
The equivalence (b)<-»(c) is true without the assumption Nel.
This result ( and its proof ) show
9.159 COROLLARIES (Beidleman (10), Ligh (7), Heatherly (8),
Chao ( 1 ) , Marin ( 1 )).
(a) A regular near-ring whose idempotents are central is
abelian, 2-semisimple and an IFP-near-ring (9.37).
(b) A regular dgnr. whose idempotents are central is a
semi simple ring.
(c) A regular near-ring N with identity 1 i 0 is a near-
field iff 0 and 1 are the only idempotents in N.
(d) A regular nr. with DCCI whose idempotents are central
is a finite direct sum of near-fields.
(e) In a regular nr. with identity whose idempotents are
central, every N-subgroup is a left ideal.
(f) A regular nr. with identity is integral iff it is a
near-field.
This gives another characterization of regular near-rings.
9. 160 THEOREM (Chao (1)) Suppose N = NQ e 171 has no non-zero
nilpotent elements. N is regular** Na is a direct summand
of N for each a ε N.
Proof. => : By 9.159 (a) and (e), N is abelian and each
Na a N. But Na - Ne for some idempotent e, whence
(N,+) = Ne + (0:e) by 1.33.

348 §9 MORE CLASSES OF NEAR-RINGS
«= : For ηεΝ, let L be a left ideal of N with (N,+ ) =
= Nn4-L. There are m ε Ν and 1 ε L with 1 = mn+1 , whence
η = n.1 = nmn+nl by 2.29. But nl = -nmn+n ε Nnл L = {0}
by 1.34. So η = nmn .
In Ligh - Utumi (1) it is shown that if Ν εψ) has no nilpotent
elements then N is regular iff nN = nNn holds for all η ε N.
We also mention another result of Ligh (2): Let N be a dgnr.
with |N| > 1. For each ηεΝ there is exactly one χεΝ with nxn = η
iff N is a near-field.
Regular nr.'s with one (and hence all) of the three conditions
studied in 9.158 are obviously of particular importance. They
deserve a special notation.
9.161 DEFINITION A regular near-ring N is called stronglу
regular if {0} ^ Ν ε Ή. and if N fulfills the conditions
in 9. 158.
9.162 THEOREM (Marin (1)). Νε7}„ηΉ, is strongly regular iff
_ о I
\/ η ε Ν ] χ εΝ: η = xn .
Proof. =>: Take η ε N. Then η = nxn for some χ ε N. Hence
xn is idempotent, hence central.
2
So η = nxn = xnn = xn .
2
«= : Let η ε N. Then η = xn for some χ ε N. Hence
2 3
η = xn , and so on. Thus there cannot be some к ε Ν
к к - 1 ι
with η =0, but η f- 0, and N is shown to be a
2 2
nr. without nilpotent elements. Now η = nxn ,
whence (n-nxn)n = 0, hence n(n-nxn) = 0.
2
We get (n-nxn) = n(n-nxn) - nxn(n-nxn) = 0-0 = 0,
consequent!у η = nxn.

9f Regular near-rings
349
We remark that M(r) and Μ (г) form examples of regular, but
not strongly regular near-rings. Integral planar near-rings are
examples of strongly regular near-rings. Many results on
strongly regular near-rings can be found in Mason (5). We
present some of these results:
9.163 THEOREM Let N be strongly regular.
(a) Every prime ideal of N is maximal (cf. 2.72).
2
(b) \/ η ε Ν 3 χ εΝ : η = xn д х is invertible.
(c) Every N-subgroup of N is a (two-sided) ideal.
(d) Every ideal I of N fulfills I = I2.
Proof, (a) Let Ρ be a prime ideal and suppose that
p<= Μ <=^Ν , MSN. If m ε M\P there is some χεΜ with
0 = m-xm = (1-xm)m. By 2.61 and the IFP we get
either re с Ρ (a contradiction) or 1-xre ε Ρ, whence
2
za for some z.
2 2 2
η -zan +an
1 εΜ, again a contradiction.
(b) η = an for some a ε Ν and a
2 2
Let x:= 1-za + a. Then xn = (1 - ζ a + a ) η
2 2 2 2
= η -za(an )n+n = η -η +η = η, and xa = (1-za+a)a =
2 2
= a-a+a = a . If χ is contained in a maximal ideal
2
M, a εΜ, by 2.72 hence a ε Μ. So 1 ε Μ, a contradiction.
Hence χ is a unit.
(c) We know already (9.159 (e)) that every
N-subgroup S of N is a left ideal. If seS and ηεΝ then
s ε Ns = Ne for some idempotent e, hence s = We.
Now e is central. Hence sn = n"en = 7fne ε Ne = NssS.
2
(d) Of course, I si. If i ε I then there is some
χ ε Ν with i = xi
.2
>i )i ε Γ
Information concerning the radicals of a regular nr. was
obtained in Johnson (6), which we state without proof.
9.164 THEOREM Let Net be regular. Then
■)3h/2(N) = {0}

350
§9 MORE CLASSES OF NEAR-RINGS
(b) Every minimal left ideal f {0) in N is a minimal
N-subgroup.
(c) If N has the DCCN then N is regular iff N is 2-semi-
simple with η ε Nn for all ηεΝ.
(d) If N has the DCCN then maximal ideals coincide
with primitive ideals.
Still more information can be found in Choudari-Goyal
(1) and Ramakotaiah (3). We shall consider regular near-rings
of the type Μ (г) in §9 (h).
g) TAME NEAR-RINGS
In this chapter we investigate a class of nr.'s which is closely
related to compatible near-rings as defined in 7.137. Most
of the results in this theory are due to S.D. Scott. For the
following definition cf. 1.34.
9.165 DEFINITION An N-group r is called tame if every N -
subgroup of ,,Γ is an ideal. A near-ring N is tame if N has
a faithful tame N-group Г (then N is called tame on ,,Γ).
Hence in tame N-groups, ideals and N -subgroups coincide.
There are several examples which work for different reasons.
9.166 EXAMPLES
[a) If N is 2-primitive on ,,Γ then N is tame on „Γ
(since ΝΓ has no non-trivial N -subgroups in this case).
[b) If Inn(r)eSSEnd(r), let S(r) be the nr. additively
generated by S. If S = Inn(r) then S(r) = I(r), if
S = End(r) then S(r) = E(r), and so on. S(r) is d.g.,
hence zero-symmetric and S(r) is tame on the S(r) -
group Г, since every S(г )-subgroup Δ is normal
(because Δ is invariant under all inner automorphisms),
hence an ideal by 6.6.

9g Tame near-rings
351
P0(A)'
p(b)eB for all pcPQ(A) and be
(c) Let Ϋ be a variety of Ω-groups and А еУ. Then (A,+ )
is a tame A [x]- and P(A)-group, and P(A) is tame
on A. This holds since Β < η ,„\A implies that
Hence all finite
sums of these elements are in B, whence В зА by 7.123.
But the elements of P(A) are compatible by 7.122;
consequently В is an ideal of P(n\A. (The same
arguments are applicable for Av[x] instead of P(A),
with the only exception that A [x] acts not
necessarily in a faithful manner on A.
(d) More generally, every near-ring N between P(A) and
C(A) is tame on A (cf. also 7.140 and 9.168!).
(e) Every ring-module is tame. Every ring with identity
is tame.
(f) Many more examples will come up by 9.168 and 7.137 !
Scott remarked that S(r) in 9.166 (b) is also (by 9.168 we will
say: "moreover") compatible on Г: If γ ε Γ and if s ε S or
-scS then either s(y+<5) - sy = sy+so-sy = ε(γ+δ-γ) or = s6
cor all ι5 ε Γ. Now Jnn(r)^S, hence there is some л cS(r) with
s(y+6) - sy = ηδ for all Л( Г. This extends to all finite i-
sums of elements of S. Hence S(r) is compatible on r. The
similarity between the concepts "tame" and "compatible" is
revealed by
9.167 PROPOSITION (Scott (17). If Nell, and ΝΓ is unitary then
ΝΓ is tame iff for all γ,δ ε Γ and π ε N there is some
m ε N with η ( γ + с
- ηγ = mi
Proof. If .,Γ is unitary and tame then each Ν δ is an
N о
ideal of ., г containing δ. Hence η (γ + δ ) - η γ ε Ν δ.
Conversely, suppose that Δ <Ν Γ. If γεΓ and
о
δεΔ then γ+δ-γ = 1(γ+δ) - 1γ = my ε Δ (for some
iheN ). Hence Δ is normal. If γεΓ, δεΔ and ηεΝ
then η(γ+c
a ε N J.
ηγ
η (γ+δ) - η γ = aJ ε A (for some

352
§9 MORE CLASSESOF NEAR-RINGS
This is shown by the following picture (cf. the diagram after
7.136 !)
/
/
/
9.168 COROLLARY Every unitary compatible N-group is tame.
In (17), (20) and (21), S.D. Scott goes on to the study of a
type of near-rings between tame and compatible near-rings:
9.169 DEFINITION Let к be a cardinal number. An N-group Г
is k-tame on ,,Γ if for all ned and γ ε Γ there is some
mcN with η(γ+δ.) - η(γ) = m(6·) for any collection
of at most к elements <5 ■ in Г.
Hence we get for unitary N-groups:
compatible => ... => k-tame => ... => 2-tame => 1-tame = tame.
We cite some results on 2-tameness without proof.
9.170 THEOREM (Scott (1), (8), (20)). Let Ne^nff, be 2-tame
on the unitary N-group Г.
(a) If h is an N-endomorphism of Г then id-h is an
N-endomorphism, too.
(b) If h is an N-automorphism of г and id-h is an N-
automorphism, too,then -id is an N-automorphism.

9g Tame near-rings
353
.c) If Aut м(г) contains a fixed-point-free element
then -id is an N-automorphism.
d) If -id is an N-endomorphism then (End(r),+,°) is a
ring and End(r ) = E(Г ).
.e) If -id is an N-endomorphism and if г is faithful
without elements of order 2 then N is a ring and
ΝΓ is an N-module.
!f) If МГ has DCCI and ACCI and if Г = Δ.+...+Δ =
N 1 r
= E1+...+Es, where the Δ-'s and E.'s are indecomposable
ideals of .,Γ then r = s and there is a permutation
ρ of {1.....П} with Δ, ~= Ν Ερ(1),...,ΔΓ = N Ep(r).
("Krull-Schmidt-Theorem").
[g) If N is 2-primitive on г as well then N is a ring or
N is dense in Μ (Γ) (i.e. G = {id} in 4.60).
[h) If no non-zero homomorphic image of N is a ring
(N is then called ri ng-free) and if N has the DCCL
then N is finite.
We now mention some elementary facts about tame near-rings.
9.171 PROPOSITION Let all appearing N-groups be unitary
(a) Let N be tame on г and Δ^,,Γ. Then N is tame on Δ
and on Γ/Δ.
(b) If N is tame on r- (i с I) then N is tame on
r: ■ $rt.
(c) If N. is tame on Г; (i ε I) then Π ц is tame on
ι ε Ι
.©/1·
ΐεΐ
(d) If Ν is 2-semisimple then N is tame.
Proof, (a) follows from 1.30, (b) is straightforward
since Δ £N Г implies (...0,1,0,...)(...,δ^,...) =
о
= (.,,,Ο,δ.,Ο,...) ε Δ, whence Δ = © Δ · for
i εΐ
&■ = { δ i ε Γ i | ( . . . , 0 , δ . , 0 , . . . ) ε Δ }
(c) follows from the fact that if N. is tame on
Г- then N is tame on Г (by (... ,n. ,... )γ^:=η ·γ^ ) , too.
Now apply (b).

354
§9 MORE CLASSES OF NEAR-RINGS
(d) If Γ. (i ε I) represent all non-N-isomorphic
N-groups of type 2 then N is tame on their direct
sum (S.D. Scott).
A splitting of N does not induce a splitting of „Γ in general
But it does for tame nr.'s.
9.172 THEOREM (Scott (17). Let Nr be tame, unitary and faithful
If Nc)J is the direct sum of the ideals I and J then
Γ = Δ+Β with (0:Δ) = J and (0: В) = I.
Proof. In N = I+J, 1 decomposes as 1 = e.+e„, where е.,
e„ are central orthogonal idempotents. Let
Δ: = e^ г and B: = e2r. Now J = (ο:Δ), since JA
= J e. г = {о} and if η ε ( ο ·. Λ) , η = η е . + η е 2 ,
then ne,y = 0 holds for all γ ε Γ, hence
ne. = 0 and η = ne„c J. Similarly, I = (o-.B
Next we show that Δ
"Ν,
.then Δ si
Ν'
ΝΔ
Νθ,|Γ
e ^ ΝΓ <
;е,|Г = Δ.
Since
ε Δ for all
ε Δ we
show that Δ is closed under addition. Let ел.,
e1^2eA· Then e2(e ^+e. y2 ) - e 2e 1Ύ1 = me]"^2 ^or
some ieN. Since e2e.y. = 0 we get еЛе-γ.+e .γ2 ) =
= те.ур. Multiplication by e2 gives e„(e.γ.+e.γ„ ) =
= едел» =е»е,1иу. = 0, whence e.y.te.y.eA.
Consequently Δ -;N r, and similarly Β s., г. Obviously
е1е2у2
If e. γ, = θ?γ2εΔηΒ then e .y.
= 0. Hence Γ = Δ+Β.
е.у
14
Without proof we mention
9. 173 THEOREM (Scott (17). Let NcH^ll, have the ACCL, and
let ΝΓ be unitary, tame and f.g.; then any ideal of „г
is f.g., too.
Next we look, how for SsN the left ideal <S>
looks like.
generated by S

9g Tame near-rings
355
9. 174 PROPOSITION (Scott (17!
<Μ>{γ = Μϊ for all γ ! I
Proof. If γεΓ then Μύ
Let ΝΓ be tame and Μ
-,N 1', hence Μγ < Г. Thus
N. Then
(Μγ:γ) ·ί% Γ Since Μ=(Μγ:γ), <M>£ ί(Μγ:γ), whence
<Μ>£γ δΜγ.
9.175 PROPOSITION (Scott (17). Let N be tame and Μ.,...,Μκ be
N -subgroups of Μ with Μ-Plj...M, = {0). Then
<M1>£<M2>£··•<Mk>£ = ^ aS wel1·
Proof. Let к = 2 (and then proceed by induction). Suppose
that N is tame on .,Γ. Now by 9.174, <Mi>{ <M2> Л =
= <M1>„ Μ2Ύ = Μ1Μ2Ύ = ίο) for all γεΓ. Since ΝΓ
is faithful, <Μ,>, < Μ 2 >,
{0}
9.176 COROLLARY (Scott (17). If Μ
N is ni1 potent and if N
is tame then <M>„ is nilpotent, too.
9.177 THEOREM (Scott (17). The sum L of all nilpotent left
ideals L· of a tame near-ring N
Proof. For η ε Ν we get Ln
group of N and is nilpotent, since NL=L. Hence
N is an i deal of N.
Σ L ■ η. Each L.n is an N-sub-
<L-n> is a nilpotent left ideal by 9.176, whence
< L ■ η > < Σ L · = L. Therefore LnsL.
These arguments also show:
9.178 PROPOSITION (Lyons-Mel drum (2)). Let ^Γ be unitary,
N tame and В = д :ίΝΓ. Then the ideal generated by В in Δ
is the same as the one generated in г and is given by
ΣΝ3.
ΒεΒ
Another interesting fact is the following. Call an N-group Γ
completely non-abelian if each non-zero homomorphic image Δ
of an ideal of Nr is either non-abelian or no η εΝ distributes
over Δ.

356
§9 MORE CLASSES OF NEAR-RINGS
It follows from 2.23 that the ideal lattice of ..Γ is then
distributive. If Γ is non-abelian and simple then г is an
example of a completely non-abelian I(r)-group.
9.179 PROPOSITION (Kaarli (5)). If ΝΓ is tame and completely
non-abelian then ί.γηί2γ = (L,/iL.)y for all Li.Lo ' q^
and γ ε Γ .
Proof. (L,/i ί„)γςί,γηί,γ is trivial. Conversely, if
α,β ε L. γ η L„y and ηεΝ then α+β-α-β and
η(α+β) - η β - ηα are in (L. л ί^ΐγ, since
( L ι л L ?)γ is an N-subgroup, consequently an ideal
of Nr. The left ideal generated by all α+β-α-β
and η(α+β) - ηβ - ηα with α,βεί.γηί,,γ and η ε Ν
coincides with Ι .γ η L„y, since .,Γ is completely
non-abelian. Hence L.γ л L2γ c(L. η L„ )γ.
Now we take a look at the structure of a tame near-ring.
Obviously we get
9.180 PROPOSITION If ΝΓ is tame then ыг is 0-primitive iff it
N
is 2-primitive.
N1
From this it looks quite plausible that ^ (N) = ^/2 (N) holds
for a tame near-ring N. We will return to this question later on.
9.181 PROPOSITION (Scott (17). Let N be tame on ΝΓ, Μ SN N and
N1
'N,
γ ε Г. Then M+(0:γ) й N.
Proof. By 2.15, Μ+(0:γ) йц N. If beH, a ε (0:γ) ,
о
η,η'εΝ, n(my + n 'γ) - ηη'γεΜγ, since Μγ ^ Γ, whence
о
Μγ aN Γ. Therefore n(m+n') - nn1 εΜ+(0:γ), and
n(m+a+n') - nn' = n(m+a+n') - n(m+n') + n(m+n') -
- ηη'ε(0:γ) + Μ + (0:γ) = Μ + (0:γ).
9. 182 THEOREM (Scott (17), Lyons - Meldrum (2), Meldrum (13!
Let N be tame with DCCL. Then J.(N) is nilpotent.
Hence f(H) =?1(N) = ? (N) = ... = ?2(N).

9g Tame near-rings
357
Proof. Suppose X(N) is not nilpotent. Let L s?N,
L ί i„ ( N ) be minimal for being non-ni1 potent. If
2 2 2
<L >f<L then L , hence L, is nilpotent. So <L > = L.
Since LO = {0}, Zorn ' lemma provides us with some
L' 53£N, maximal for having LL' = {0}. If L' = N,
L = L = {0}, a contradiction. The DCCL guarantees
the existence of some L" fl£N with L'< L" , but
without left ideals strictly between L' and L" . Then
L"/L is a simple N -group. Among all pairs L.,L2 -oN
with L2/L1 s L" /L ' , choose one with minimal L„ ·
О
If K<L„, KijN, then К С L . , because otherwise
K+L. = 12 and L2/L. s К/К л L. , a contradiction to
о
the minimality of L„. Suppose that Μ £», N is strictly
between L. and L„. Let Nr be faithful and tame.
Since 0Γ(°:γ) = (ο:Γ) = {0}, there is some γεΓ with
(o:y)(\ L„< L„, whence (о:у)л L2< L. by what we have
just seen before. 9.181 tells us Μ+(ο:γ) a N.
Hence L2 η (Μ+(ο:γ)) = Μ+(ο:γ)Λί2 (see p. 48) is a
left ideal of N. But then (ο :γ) л L2ί L1 < Μ and
Μ -„Ν, a contradiction. Hence L2/L. is N -simple,
thus L" /L' is N-simple, too. We go back to L.
Since Ls;^2(N), L (L " /L ' ) = {0}, whence LL"cL'
and L2L"SLL' = 0 . Consequently L2<=(0:L" ) ,and
L = <L2> c(0:L" ) as well. Hence we get LL" = {0},
in contradiction to the maximality of L', and X(N)
is shown to be nilpotent.
The rest follows from 5.59 and 5.60.
This result generalizes 7.58 (b), for instance, as well as 7.127
and 9.77. As far as the author knows, no tame nr. Μ is known
in which the Jacobson-type radicals Ί do not coincide. A
partial result in this direction is

358
§9 MORE CLASSES OF NEAR-RINGS
9.183 PROPOSITION Let Nr be tame with at least one γεΓ such
that (ο:γ) = {0}. Then every N-group of type 0 is tame,
hence of type 2, whence ^Q(N) = ^(Nb
Proof. If (ο:γ) = {0) and νδ is nf type 0 then consider
h: Νγ ->· Λ, ηγ-+ηδ, where δ is a generator of „Δ.
Since (ο:γ) = {0}, h is well-defined and an
N-endomorphi sm such that ί= Ν /ker h. The 1
N-group is tame by 9.171 (a). Now apply 9.180
Tame near-rings have a very clear situation concerning chain
conditions. We mention without proof a result, which reminds
us of 5.50 (c).
9.184 THEOREM (Scott (17). If ti ε \ η Щ is tame and has the
DCCL, then N has DCCN, ACCN and ACCL as well.
In this case, certain minimal and maximal objects exist. The
first part of the next result is obvious by taking Δ = Νγ
for some aoorooriateγ ε Γ; the second part is mentioned without
proof.
9.185 PROPOSITION (Lyons - Meldrum (2)). Let N be tame on ^Γ
with DCCL.
(a) If ΝΓ f- {0} then there is some minimal N-ideal Δ
in г such that „Δ is of type 2.
(b) If Νεί)0^)ί| then ΝΓ has strictly maximal N-ideals.
This turns our attention to minimal N-ideals.
9. 186
DEFINITION The sum of all minimal ideals of ΝΓ is called
the socle of ^r and denoted by soc („r). The socle of
NN ( = the sum of all minimal left ideals of N) is simply
denoted by soc(N).
It follows from 9.185 (a) that soc(ND \ {0} under these
assumptions. The socle of Nr can sometimes be used as a test
semisimplicity of Ц (see 9.188 (b)). By 2.48 we can conclude

9g Tame near-rings
359
9.187 PROPOSITION If N cV and ЦТ
4
then soc(
is the
direct sum of minimal ideals of Nг.
9.188 THEOREM (Lyons - Meldrum (2), Scott (17;
Let Ν ε \n1f}^
have DCCN and be tame on the unitary N-group Γ.
(a) If Δ£Γ fulfills 32(Ν)δ = {0) then AcsoclJ).
(b) If N Γ is faithful and R(N) any one of the radicals
of N mentioned in 9.182 then R(N)={0}<i=> soc(r) = Г.
Proof, (a) If δεΔ then δ с Ν δ and ^(Ν)Νδ = 72(Ν)δ = {0}.
By 3.14 (a), N6 is a monogenic N/^„(N)-group . Since
N/g2<
ideals of ΝΓ by 5.34
■ s о с ( N г ).
\W) is 2-semisiraple , Νδ is the sum of minimal
a). Hence As," N с
δεΔ
(b) By 9.182, all radicals in N coincide in this
case. In particular, R(N) = ^2(N)- If this is zero,
R(N)r = {0), whence rssoc(r) by (a), hence
Γ = s о с (., г
Ν
Conversely, г = s о с ( N Г ) implies that
ΝΓ is the sum of minimal ideals of „Г.
Since Ц.Г
is tame and faithful, R(N) = ^2(N) = {0}·
S.D. Scott proves a result for the socle of a tame near-ring.
We present this result without proof.
If Hc1fl0nTl] nas DCCL and is tame
9.189 PROPOSITION (Scott (17
a un i tary N
:(N) = (δ:γ). Hence soc(N) 3 N and N/soc(N) is tame on
on a unitary N-group then there exists some д я г with
soc(Ν,
Γ/Δ.
Finally we are going to illustrate these results for the tame
(even compatible) near-rings S(r) of 9.166 (b). The proofs can
be found in full detail in Meldrum (13), ch. 10 (the first
result can be shown as in 7.66).
9.190 THEOREM (Lyons-Meldrum (3)). Let г be a group. Then
ι
i f Г is
a) If г is non-abelian then S(r) is dense in M(r) iff
Г is simple. In particular, S(r) = Μ (г'
finite, non-abelian and simple.

360
§9 MORE CLASSES OF NEAR-RINGS
(b) If Δ s Γ is minimal then either Δ is abelian or
perfect.
(c) If Δ in (b) is abelian then either Δ is an elementary
abelian p-group or Δ is divisible (cf. the first 6
1ines on p. 242 ! ).
(d) If Δ in (b) is perfect then S(r)/(o:A) is isomorphic
to a dense subnear-ring of Μ (Δ).
(e) By (b), soc(,wr\r) is the direct sum of an abelian
ideal A and a perfect ideal Π.
(f) If S(D has the DCCN then Π is finite and there
exists an idempotent e of S(r) with еГ = Π and етг = τ
for all π ε Π.
The reader will have observed that this area has a lot to do
with many other topics in near-ring and group theory, so it is
already for this reason a very interesting subject. Much more
should be available soon in a forthcoming paper of S.D.Scott
on tame near-rings.

9h Bicentralizer near-rings
361
h) BICENTRALIZER NEAR-RINGS
We now return to those near-rings whose consideration started in
1.4 (a), was brought into bloom in 4.52 and matured in §7 (a),
namely near-rings of the M<-(r)-type.
If N is a nr. acting on an N-group г it is customary, and in
accordance with ring theory, to call End^(r) the centralizer (4.10)
of „Г, because it contains those endomorphisms h of Г which
N
"commute" with the action of N via η(ηγ) = ηη(γ). If S<^End..(r)
then M<-(r), as defined in 1.4 (a), consists of those mappings
f on Γ which "commute" with some elements S of EndM(r) which
in turn "commute" with N, i.e. fos = sof for all seS. Hence
Μς(Γ) might be called a "double centralizer" or "bicentralizer"
of ..Γ, and we follow an advice of G. Betsch to call these
creatures "bicentralizer near-rings" instead of the commonly
used "centralizer near-rings".
If we start with some group G of automorphisms of a group г and
form N:=MG(r) then we might form its centralizer Ε := AutM(r)
and compare MQ(r) with M^(r). If Nr is monogenic we get (see
9.226) MQ(r) = M^(r) in this case ("closure property"). This
also motivates the "bi" in the title of this chapter: even if we
start with some GeAut(r) we get MQ(r) as a bicentralizer near-
ring.
The knowledge of М^(г), where S is not a group of fixed-point-
free automorphisms of Nr (see § 7 a) for this case) was pushed
forward in the last few years mainly by papers of Betsch,
Maxson-Smith, Meldrum, Oswald and Ramakotaiah-Rao. Since many
proofs in this area are pretty long, quite technical, but easily
available, we only cite a number of results.
We now start with the most general case, in which S is some
arbitrary subset of EndN(r). Although many results carry over
to the general case, we confine our attention to the situation
where S contains the zero endomorphism 0 for the sake of
brevity. Hence we make the

362
§9 MORE CLASSES OF NEAR-RINGS
CONVENTION All near-rings Ms(r) in this chapter have 5cS,
hence are zero-symmetric.
Since id с МЛг) in any case, M<-(r) is a zero-symmetric near-
ring with identity. Surprisingly, all such near-rings arise as
some МЛг), as С Maxson pointed out:
9.191 PROPOSITI ON If N is a zero-symmetriс near-ring with
identity then there is some Nr and some S^EndN(r) with
Ν ~ Μ Л Г ). Г is even a monogenic Мз(Г)-дгоир.
Proof. Take Г:= (N,+ ) and S:= {s |n cN), where sn: N+N,
x+xn are the right translations. For aed let
f : N + N, η-> an be the corresponding left translation,
a
The map h : N -+ {f I a ε Ν) , given by a->- f , is obviously
a a
a near-ring isomorphism (see 1.86/1.89). Now МЛг) =
= {fcM(r)|f°sn = snof} = {feM(r)|f(nm) = nf(m)
for all η,ιηεΝ}. Let f(1) =:a. Then f(n) = f (η 1 ) =
= πf(1 ) = na for f сМЛг), hence f = f and we get
N HfJacN) = МЛГ).
a o
Hence Μ ς(Г) is as general as it can be, and one cannot expect
very strong results, of course. In (2) Maxson and Oswald considered
conditions under which Ms(r) is regular, simple, primitive and
soon, and studied the connections to congruence relations in fV(r).
if г = Sy for some γεΓ or if S* is a union of groups.
Another paper solves the question as to when МЛг) happens to
be a near-field. For that, let γ-R γ„(γ. ,γ„εΓ) if s(y«) = γ~
for some s ε S and let ^<. be the equivalence closure of R^.
9Л92 THEOREM (Maxson-Mel drum (1
Ms(r) with S = End(r) is a
All elements in Γ* are equivalent w.r.t. %<.,
near-field iff
(a.
(b) Γ has proper subgroups r. (i ε I) such that each
γεΓ* is contained in exactly one Γ. and for each
seS either s ( Γ η·) = {о} or S ( г i
and ker s = {ο},
Г· for some j ε Κ

9h Bicentralizer near-rings
363
[c) If Γ = U Sy (with well-ordered J ), and у^ с Г^
then the set of all &■, where 6k (ke J) fulfills
Г = ^/S6k and s> = s'yn (s,s' £S)=*-s6m = s6n
for all m s: η , is just Γ .
In the paper mentioned in 9.192, the authors give the following
9.193 EXAMPLE Let ρ 23 be a prime and Γ be generated by two
elements γ,δ of order ρ such that ρ[γ,δ] = [γ, [γ,6]] =
= [δ, [γ,δ]] = ο (the brackets denote commutators). Then
Γ is a non-abelian group of order ρ and exponent p.
Let Γ· (i ε I) be the set of all cyclic subgroups
£ {o} of Γ and <γ> the cyclic subgroup generated by γ.
Fix some iQ ε I . Let S: = {id^iseEndd^lker ssZfr) +
+ <γ> and s({) tr. }. Then Μ (Γ) fulfills the conditions
о
in 9.192, hence is a near-field (in fact, М<Л Г) s (Ζ ,+,·)
We turn our attention to the question as to when Μς(Γ)
happens to be (semi)simple. For this we need an extension of the
concept of fixed-point-freeness.
9.194 DEFINITION S <End!
(a)
(b)
(c)
\/s
Л
scS
Μ у
ε S
Ker
ε Γ
: Ker s =
s = {0}.
\/ s 1 , s 2 r
S
is fi xed-point-free if
2 3
Ker s = Ker s = .. .
= s
.(γ) ί
=> s
= s-
Obviously, if S is a group of automorphisms, this concept of
fixed-point-freeness coincides with our well-known concept for
automorphism groups (cf. e.g. 4.52). It can be shown that, if
S is fixed-point-free and finite, S can be written as S =
G,j ι/ . . . L/Gnu{o}, where G.,...,G are groups with identities
e.,...,e and e-e. = δ,, (hence S is a completely regular inverse
semigroup). See also Maxson-Smith (11).

364
§9 MORE CLASSES OF NEAR-RINGS
9.195 THEOREM (Maxson-Smith (8)). Let Ne»j be a finite
near-ring. Then N is semisimple all of its simple summands
being either non-rings or fields iff N is isomorphic to
some M<.(r), where S is a fixed-point free set of endo-
morphisms of Γ. More generally, МЛГ), with SiEndr,
Г finite, is semisimple iff S is a completely regular
inverse semigroup.
More information can be obtained if S is specialized. The first
collection of results which we mention concerns a one-element
set S = {s}, in the second series of results, S is a semigroup
of "linear" maps.Of course, Mr ,(Г) = М_(г), where <s> is the
subsemigroup generated by s.
M<s>(r'
9.195 THEOREM (Maxson-Smith (2;
Let Г be a finite group and
s ε End(r). Then the following assertions hold for N: =
= М(5,оЛЬ
(a) If s is non-nilpotent and not invertible then N is
not 2-semisimple.
(b) Let s be nilpotent of degree η >1, L(γ): =
= {f ε ΝI sn"1(Y) = o} and Δ:= {δ ε ker s |\/γ ε Γ:
m- 1 ,
s (γ
= δ}, but for no γ'εΓ we have s '(γ ' ) =δ}
with maximal m. Then 1?{H) = /1ί(6) = {f ε Ν|f/ = 0}.
δεΔ
and N/22(N) г Μ0(Δ) ; hence N/^2(N) is simple.
If Γ is a vector space and s is linear then N is
simple iff N is 2-semisimple.
9.197 THEOREM (Maxson-Smith
Let R be a finite ring with
identity, DN a finite unitary R-module and S:= (fire R)
к r
with f : V ->· V, v-* rv. Then
(a) MS(V) = {f ε MQ(V) | \/ r ε R \/m ε M: f(rm) = rf(m)}.
(b) If R is simple, so is M<-(V), and Μς(ν) is a near-ring
iff R is a field with dimRV>1.
(c) If R is semisimple, with none of its simple summands
being a field then Μ <- ( V ) is a ring.
(d) If R is not a field but if MS(V) is still simple then
MS(V) is the ring EndR(V).

9h Bicentralizer near-rings
365
Now we turn our attention to the structure of Μ (г) for some
G
G<Aut(r). Even stronger than before one can say that MQ(r) =
= Μ r (Γ) if G«Aut(r) and <G> denotes the group generated by G
in Aut(r). Hence we may confine our consideration recording to
the fol1owi ng
CONVENTION For the rest of this chapter, Мд(г) will always
denote the case where A = Gu{o} with G^Autir).
Due to limitations of space, we only cite most of the results.
We start the discussion with some elementary facts concerning
Мд(г) and proceed by characterizing those cases in which Г is
a monogenic Мд(Г)-дгоир, in which МД(г) is a ring, a near-field,
a g.d. near-ring, or a regular near-ring. In order to compute
the radicals and to study (semi -)simplicity and primitivity, we
take a look at the left ideal structure of МД(г). After that,
we consider the problem, when a near-ring can be written as a
bicentralizer near-ring with various properties. In particular,
it turned out that bicentralizer near-rings are as general as
they can be (9. 191). Finally, we close with some closure
properties and some other interesting questions and results. Since
some results concern more than one of these topics mentioned
above, we'll mention some results twice on different places
and hope that this will provide a better overall look.
We have to adjust a concept, first mentioned on 8.38, to our
situation. For a ε A and усГ we prefer to write a/ instead of
a(y).
9. 198 DEFINITION In МД(г), let for γεΓ st(y):= {a£A|ay = γ}
denote the Α-stabilizer of γ.
Obviously, for all γεΓ εί(γ) is a subsemigroup of A and st(y)*
is a subgroup of A*. The following result turned out to be basic
to the understanding of what's going on in Мд(г).

366
§9 MORE CLASSES OF NEAR-RINGS
9.199 PROPOSITION ("Betsch's Lemma"). In МД(Г), let γ,δεΓ. Then
there is some itcH.(r) with m(y) = δ iff st(y ) <= st( δ ).
Proof. If m (γ) = б then clearly ay = γ implies a δ = δ.
Conversely, let st(y)cst(6). We define f by
f(y):= δ, f(ay) := ao for acA and f(3) = о for
βψΑ*γ, the orbit of γ under A*, f is well-defined,
since a.γ = a ~ γ implies a .a. ε stfylestti), whence
a,|6 = a26. Clearly, f гМд(г) and f(Y) = δ.
The fact that f(a^) = af(6) determines all values of f uniquely
in the orbit of γ gives us the following
9.200 COROLLARY Let (y^^j be a complete set of representatives
of the orbits of Γ under A. If δ- εΓ (i ε I) fulfill
st(yi ) £ st( 6i ) then there is a unique f ε«λ(γ) with
f (γ1· ) = δ- for al 1 i ε I .
The proof of the next result is straightforward.
9.201 PROPOSITION For ΎεΓ and a ε A* we have st(ay) = a(st(Y))a"1.
Hence elements of Γ* in the same orbit have conjugated
stabilizers.
If A is abelian (see e.g. Maxson-Smith (1)) then clearly elements
in the same orbit have equal stabilizers. Example 1 in Maxson-
Smith (5) shows that this does not necessarily hold for non-
abelian A.
Following a suggestion of G. Betsch, we fix a very suggestive
notation for elements in Γ.
9.202 NOTATION In the Мд(Г)-situation we write
(a) For γ,δεΓ we say that γ divides δ and write γ|δ
if δ = fy for some feMA(r), i.e. if st(y) stU).
(b) For γ|δΛδ|γ, i.e. for st(-y) = st(6), we say that
γ and δ are equi valent and write γ~δ.

9h Bicentralizer near-rings
367
(c) If γεΓ, let e be the (by 9.200 uniquely determined)
function in Мд(г) which fixes all elements in the
orbit of γ under A element-wise and sends all other
elements to zero.
Obviously, | is a preorder relation in г and an order relation
in r/^ , where ^ denotes the equivalence of 9.202 (b). The
situation in which | is discrete will prove to be equivalent
to the regularity of М«(Г) (see 9.207). The maps e are clearly
η γ
idempotent. If γ.,,.,,γ are representatives of the non-zero
orbits of Γ then e ,...,e are often referred to as the
V ϊ'η
"usual idempotents". In this case, they are also orthogonal.
Now we draw our attention to the case where Γ is a monogenic
MA(r)-group. From 9.199 we get
9.203 PROPOSITION (Betsch (10), (11)). The following conditions
are equivalent:
(a) Г is a monoqenic МД(г)-дгоир.
(b) There exists 0}=Δ<=Γ with Α*Δ = Δ and st(6) = {id} for
all δ ε Δ .
( с ) 3 γ ε Γ : s t(γ ) = {id}.
(d) Γ* has a smallest element w.r.t. |.
When is Мд(г) a ring, a near-field, d.g. or regular ?
9.204 THEOREM (Maxson-Pettet-Smith (1)) Let г be finite.
M„(r) is a ring iff it is a direct sum of fields.
9.205 THEOREM (Maxson-Smith (5)) Let Г be finite. Equivalent are:
(a) Мд(Г) is a near-field
(b) A acts transitively on г
(c) г* is a single orbit under A*.
9.206 THEOREM (Maxson-Smith (12)) Let Г be finite and solvable.If
Мд(г) is d.g. then r has derived length 2 and Мд(г) is a
ring (go to 9.204).

368
§9 MORE CLASSES OF NEAR-RINGS
9.207 THEOREM (Me 1drum-Oswa1d (1)). Мд(г) is regular iff
st(y) £ st(6) implies st(y)= st (δ) for all γ,δ ε Γ.
The proof of this result can also be found in Meldrum (13'
Obviously, МД(Г) i<
other hand we have
Obviously, Мд(г) is regular if A* is fixed-point-free. On the
9.208 EXAMPLE (Me 1drum-Oswald (1
and A = Inn(Г
Meldrum (13'
Take Г = A,
Then A = Aj- , and the stabilizer of γεΓ
is just its norma 1izer. Since their orders are 3,4 and 5,
no proper contaiη me ηt of stabilizers is possible by
Lagrange's theorem. 9.207 shows that Мд(г) is regular.
In Meldrum-Zeller (1), Meldrum-Oswald (1) and Maxson-Meldrum-
Oswald (1), all two-sided invariant subnear-rings of regular
MA(r)'s (under various conditions) are determined. Basically,
two-sided invariant subnear-rings of Мд(г) consists of all
fеМд(г) whose "rank" is smaller than a given cardinal. This
generalizes 7.34, of course.
For computing the radicals of Μ (г) we need some knowledge about
(stictly) maximal left ideals. A first result is
9.209 THEOREM (Smith (1)). Let Г be finite. Then the lattice
of left ideals of Мд(г) is distributive.
9.210 THEOREM (Max son -Smith (9), cf. 7.22) Let L be a minimal
left ideal in Мд(г). Then there is some γεΓ with Lcflj(r)e
Moreover, N = Mu(r)e <-> st(f(y)) = st(y) holds for some
η γ
f eL <=> L is not contained in the intersection of all
maximal left ideals of Мд(г).
We want to know everything: when is a left ideal strictly minimal
(i.e. also minimal as an N -subgroup ?

9h Bicentralizer near-rings
369
9.211 THEOREM (Maxson-Smith (9)) For γεΓ, Мд(Г)е, is a strictl
minimal left ideal iff the orbit of γ is a maximal one.
Generalizing 7.28 (b) we get
9.212 THEOREM (Maxson-Smith (9)) If L is a maximal left ideal in
Мд(Г) then there is some γεΓ* such that either L = (ο:γ)
or L = (ο:γ)+Κ , where К is a left ideal of Mn(r), maxi-
γ γ Α
mal w.r.t. being contained in M„(r)e .
3 Α γ
In the finite case we get the following result which needs
9.213 NOTATION If Δ is an orbit of Γ* then 1_д is the set of
all feM.(r) such that δ properly divides f(6) for some
δ ε Δ.
9.214 THEOREM (Maxson-Smith (1
"Γ
Let Γ be finite
Then
If L <i?M«(r) is strictly maximal then L = L for some
orbit Δ of Γ*
L, is strictly maximal iff for some δ ε Δ we have
Δ
that r,:= {γεΓ|δ|γ, but γ|δ} is a normal subgroup
of Γδ:= {ύ e Γ|δ | γ} , if {γ ε Δ|γ Ιγ0л Ύ0!Ύ ) is a union
: Γ for al1
ν
[with a ε A) implies ay
of cosets of ry for all Ύ0εΓΝΓδ and if a^ " Ύ ε Γ(
γεΓ, for all such γ .
For finite and regular Мд(г)
the situation is much easier.
^see e.g. 7.1!
7.21 and 7.2i
9.215 THEOREM (Maxson-Smith
Let M„(r) be finite and regular.
All minimal left ideals of Мд(г]
are of the form
M.(r)e and automatically strictly minimal
" Ύ
All maximal left ideals of Мд(г
,ο:Ύ;
are of the form
and automatically strictly maximal.
This provides valuable information from which one can compute
the radicals of Μ«(Γ). Recall that the ^. and the ^? radicals
coincide by 5.3(b).

370
§9 MORE CLASSES OF NEAR-RINGS
9.216 THEOREM (Maxson-Smith (1),(5),(9)) Let Γ be finite, let
γ.,.,.,γ be representatives of all orbits of Γ* and let
N:= ΜΑ(Γ).
(a) ^0(N) = AnB, where A is the intersection of all
(o:Ne ), where Ne. is minimal and В is the inter-
~4
Ύ
section of all (((ο:γ.) + Κ J:N), such that Ne is not
J YJ Yj
maximal and К is defined as in 9.212.
(b) ^i/^(f!) is the sum of all those К such that (ο:γ.)
" ' ι <- Ύ ^ Κ
is not maxi ma 1 .
(c) 1j ~ ( N) is the intersection of all those L which are
described in 9.214 (b).
^„(N) is nilpotent iff each L is strictly maximal.
Concerning semisimplicity we get
9.217 THEOREM (Maxson-Smith (5),(9)) Let Г be finite and Ν:=ΜΑ(Γ]
(a) N is O-semisimple <=> Ne is a maximal left ideal
whenever the orbit of γ is maximal.
(b) N is 2-semisimple <=> all stabilizers in Γ* are
maxi mal.
If all st(y) with γ ε Γ* are normal subgroups of A*
(e.g. if A is abelian or fixed-point-free ) then N is
2-semisimple iff N is simple.
i.u , was investigated in several papers.
The simplicity of Μ„(Γ\
Examples of non-simple M.(r)'s are exhibited in Meldrum-Oswald
.5) and especially in Meldrum-Zel1er (1).
.1
Maxson-Smi th
This can happen even if Г is finite. If M.(r) is finite and
simple then M„(r) is 2-primitive by 4.47 (a).
Up to now no complete characterization of simple M.(r)'s seems
to be possible. But there are important partial results. We
present them without proof.

9h Bicentralizer near-rings
371
9.218 THEOREM
(Mel drum-Oswald (1), Meldrum (13), Maxson-Smith (5))
Let M„(r) be regular such that all stabilizers of
elements in Γ* are conjugated. Then there is some
group Δ and some fixed-point-free automorphism group
F on Δ with МД(Г)
μ°(δ;
If diirvA is finite (see
then Мд(г;
and MpU:
are s i mple.
7.7) or if |F | < | Г
(b) (Maxson-Smith (5))
simple iff all stabilizers in Г* are conjugated
(if A is abelian, the last condition can be replaced
by "all orbits in Г* have the same size").
(c) (Maxson-Smith (5)). If г is finite and Мд(г) is simp
but A not fix-point-free then Г is an elementary
Let г be finite. Then МД(Г) is
abelian p-group, hence a vector space over Ж , and A*
acts irreducibly as a group of linear automorphisms.
(Maxson-Smith (5)). If г is finite and M„(r) simple
and non-abelian then A* is fixed-point-free.
9.219 COROLLARY (Meldrum-Zel1er (1), Maxson-Meldrum-Oswald (1)).
If Г is a vector space over a field F and if A = F (acting
by multiplication) then Мд(г) is simple.
In considering v-primitivity of Мд(г), care should be taken.
Suppose that M«(r) is regular such that all stabilizers in Г*
are conjugated, but such that A* is not fixed-point-free. Then
Мд(г)-Мр(л) by 9.218 for some Δ and some fixed-point-free
F ίAut(u). Hence Мд(г) is 2-primitive on δ, but not on г!
Now we turn to 0-primitivity. If г is finite we can say quite
a lot.
Obviously, we can reformulate 0-primitivity as follows to get
(a) »(b) in the next result.

372
§9 MORE CLASSES OF NEAR-RINGS
9.220 THEOREM Let Γ j= {o} fulfill one (and hence all) conditions
of 9.203. Equivalent are (Betsch (10), Ramakotaiah-Rao(3)):
(a) Мд(г) is 0-primitive on Γ,
(b) If {o} =(= Д or, there are шсМд(г) and η, γ2 ε Γ
with η-γ2 εΑ, but m (γ 1 ) - ΐ4(γ2)ι(.Δ.
(c) There is some δεΔ (Δ as in 9.203 (b)) such that
(ο : ΓΝΑδ) does not contain a non-zero left ideal of
ΜΑ(Γ) with L2 = {0}.
9.221 THEOREM (Maxson-Smith (9)) If Γ is finite then Мд(г) is
0-primitive iff (a) and (b) hold:
(a) All maximal orbits in r* are conjugated
(b) If the orbit of γ ε Γ* is maximal then Mu(r)e is a
maximal left ideal.
An important result on 2-primitivity can be derived from 9.218(a):
9.222 THEOREM (Betsch (9),(10)) If Г is a monogenic Мд(г)-дгоир
(see 9.203) then МД(г) is 2-primitive on Г iff A* is
f i xed-poi nt-free.
As an illustration of these results we give information in an
interesting special case.
2.223 THEOREM (Maxson-Oswald (1)). Let F be a field, г abelian
and A* the general linear group of regular nxn-matrices
over F. Suppose that A*SAut(r). Since A* generates the
simple ring R of all nxn-matrices over F, we might view
Г as an R-module. Then the following results hold:
(a) If y|ay then ay|y for all γεΓ, βεΑ*.
(b) All minimal left ideals of Мд(г) are given by Мд(Г)е .
(c) МД(Г) has no nilpotent left ideals except {o}.
(d) Every non-zero left ideal of Мд(г) contains a non-zero
i dempotent.

9h Bicentralizer near-rings
373
(e) M„(r) is O-primitive.
(f) Μ. (Г) is simple iff it is 2-primitive.
If η = 1 then Мд(г
is simple iff di mRr = 1
(h)
in this case, M.(r) is a field isomorphic to F]
(b), 1
1/2(MA(D) = {0}
This results complement and generalize 9.197. The authors
also obtained "sharp" upper and lower bounds for Χ>(Μ.(Γ)).
Now we collect results which concern the question as to when
A* has to be fixed-point-free.
9.224 THEOREM
a) A* is fixed-point-free iff Мд(г) is regular and г is
a monogenic M„(r)-group (Betsch (10)).
[b) If Г is monogenic then A* is fixed-point-free iff
Мд(г) is 2-primitive on г (see 9.222).
See also 9.21!
and
Also, we might ask, which near-rings have a "representation"
as a bicentralizer near-ring. We recall 9.191 and go further
on.
9.225 THEOREM (Maxson-Smith (1
Maxson-Meldrum (1
Maxson■
Pettet-Smith (1))
(a) If N is a near-ring with identity then N = M<-(r) for
some group Г and some SSEnd(r) (see 9.191).
(b ) If N is a near-field then there is some group Г
and some G^Aut.,(r) with N
м°(г:
[c) If N = N10...©Nk, where N1,...,N|< are finite
simple near-rings such that at least one N. is a
ring, but not a field, then N cannot be isomorphic
to some Mg(r!
Let F1 < F2 < .
some vector space V
transformations on
. < F be finite fields. Then there is
over F. and a group G of linear
V such that F^ . ..0Fn - Mg(V).

374
§9 MORE CLASSES OF NEAR-RINGS
See these papers for more details. For instance, Maxson-Mel drum
gave an example of a near-field N with N=M-(r), where S£End(r)
but S^Aut(r). The third paper mentioned in 9.225 brings the
reader also back to the Frobenius groups which come up in the
studies of planarity. Cf. also 9.218(a).
A type of closure brings us back to the beginning of this chapter;
suppose that N< МД(Г). Then obviously A*4AutN(r) =: 7^Aut(r)
and Ν^Μτγ(γ) «sM.(r). Often, the way from A to J is a "closure":
9.226 THEOREM
Maxson-Smi th (10'
Let N 4 Ml
N
МД(Г) for some A iff N
Г) be si mple. Then
1AutN(r)(r)· 0n the 0ther
hand, there exists some group Δ such that N is not
isomorphic to some Мд(л) with A*^ Aut(r).
(Betsch (
in МА(Г)
A* = Aut.
:9),(ю;
(w.r.t.
Let N be densely (see 4.26) embedded
which is monogenic on Г. Then
hence М-д-(Г\
= мА(г
get A* = AutM (r.
;see 7.13
). In particular , we
and 7.37!) as wel1 as
AutMA(r)(r)
Г) = M,
Isn't that a beautiful formula ? The proof will be published in
a forthcoming paper of G.Betsch. These considerations allow us
to get back the ingredients (Г and А), Мд(г) =:Μ is made of.
If γ ε Γ* then obviously Μγ = „ Me , so we get back the monogenic
M-subgroups ofr. Also, for ηεΜ, the map eje +t with
t„:Me -> Me , me -+■ (me )(e ne ) = me ne turns out to be a semi-
n Ύ γ' ΎΎΎ.Ύ ΎΎ
group isomorphism between (e Me ,') and End„(Me ). If we plug
this together and assume that γ generates all of Γ we get a ve
satisfactory result. See also the proof of 4.56.
9.227 THEOREM (Betsch
group. Then Γ =
Let Γ be (by γ) a monogenic M:= M„(r\
,, Me and End„(
Μ γ Μ
A*sgroup of units of (e Me ,
e Me , in particular
We now illustrate these ideas on Klein's four group.

9h Bicentralizer near-rings
375
9.228 EXAMPLES (Betsch
1 1
Let Γ = Z2x22 = {0 ,a ,b ,с}
Then Aut(r)=S, (see Appendix, near-rings of low order, Ε
(a) If A*:= S3, МД(Г) is isomorphic to the field Z2. The
stabilizers of a,b and с are all equal and consist of
id and the three transpositions. Hence 9.192, 9.207
and 9.2 18 apply.
(b) Let A*:= A
is 2-primitive on Г
group is isomorphic to Г.
(c) Take A*: = {id, (° !j £ £)}. Then Г is a monogenic Мд(г:
group with generator b. A* is not fixed-point-free
and М„(Г) is not regular. M„(r) has 8 elements. The
>,. Then A* is fixed-point-free. Hence M«(r)
Мд(г) has 4 elements; its additive
■radical is the annihilator of a and has order 4.
ь
Moreover, еьМд(Г)еь = {0,eb,f} with f(0) = f(a) = 0,
f(b) = c, f(c) = b. Its group of units is {e.,f} and
in fact isomorphic to A*.
We go on with some topological properties. Recall the notation
S(m,Y) of 4.25.
9.229 THEOREM (Rama кotaiah - Rao
For each Δ^Γ*, Δ f Я,
Μ,
Γ
is closed in Μ (Γ) w.r.t. the topology X generated
by the subbase {S(m , ό)|m ε Μ (Γ), δ ε Δ). Furthermore,
7, is Hausdorff iff Δ = г*.
Of course, X* is just the finite topology У {see 4.26) in
Μ (Γ). With this notation, one can reformulate the density
theorem 4.30 for 0-priraitive non-rings N on ..Γ (see4.14 and the
footnote on p. 116!):
9.230 THEOREM (Ramakotaiah-Rao (3), Betsch (11)). If the non-
ring N = N is 0-primitive on Г and A = AutN(г) и {о} then
N is dense in
= МА(Г) w.r.t. /θ
{f -_ Μη(Γ) ! V a ε Α \/γ e 9i:f(aY) = af(Y)}
In 9.230, it is straightforward to see the equality of the two
near-rings involved.

376
§9 MORE CLASSES OF NEAR-RINGS
Another topic was pursued by M. Holcombe:
9.231 DEFINITION Let N be a near-ring with identity. For aeN
let К (N ) be the set of those η εΝ with Nan = Na and
a
with ]hieN yx ε Nд: xmn = xnm = x. Let X be the set of
all Nb which are maximal for being contained in Na. Then
H(Na):= {f : N -+■ N ,x -+■ xn | f indues a permutation on X }
is called the holonomy group of N.
If N is finite, the semigroup of all f (ηεΝ) can be covered
by a wreath product of holonomy groups by a well-known result
of (Eilenberg). Let Ύι,·..,Ύη be acompletesetofrepresentatives
of Γ* under the action of A*< Aut(r).For Na, Nb put Na~Nb
if there exists χ ,y ε Ν with NaxsNb and NbysNa. Then we get
9.232 THEOREM (Holcombe (9), (10)). Let N = МД(Г) and suppose
that Г* has к orbits under A*.
(a) If a ε Ν* then Na~N(e. + ...+e ) , where e.,...e ε
ε {e ,...,e } and s is the number of orbits in
4 Ys
a(r)*.
(b) H(N(e,!+. . .+e )) = {f еМд(г) j f is a bijective map
on (e^'. . .+es )r }.
(c) The semigroup {ί:Ν->Ν,χ->χη|ηεΝ} can be covered
by a wreath product of к holonomy groups of the type
H(N(e,, + . . - + es)) (with s = 1 ,2,.. . ,k).
CIay-Maxson-Meldrum (1) extended the study of the units of
M.(r) (i.e. the group U.(r) of invertible functions in Мд(г)
mentioned at the end of §7 a)) to the case that A* is not
necessarily fixed-point-free. U.(r) turns out to be (as a group)
isomorphic to a direct product of certain wreath products.
If Мд(г) is regular, these wreath products are of the type
mentioned in §7 c). Also a "determinant-like" function D is
defined on a part Ρ of Мд(г). If there are only finitely
many orbits of Г under A*, P = Мд(г). If one views these orbits
as generalizations of one-dimensional subspaces of a vector
space then the following results show that D really behaves
like a determinant function:

9h Bicentralizer near-rings
377
(a) Under certain conditions (e.g. regularity of МДг) with
finitely many orbits), D(f) + 0 iff f is invertible.
(b) If f eM.(r) maps two distinct orbits to the same orbit
then D(f) = 0.
(c) If g differs from f only in one orbit then D(g) is a
"multiple" of D(f) ("multi1inearity").
Also, the concept of " eigenvalues" is discussed in this paper.
In (16), Maxson shows an interesting connection to geometry. A
so-called translation Sperner space with operators leads in a
natural way to an associated near-ring which consists of mappings
on a group (which "coordinates" this space) which commute with
certain endomor phi sms. Hence this near-ring is of the М<-(Г)-
type. Maxson characterizes the case in which this bicentralizer
near-ring is a near-field.
Bicentralizer near-rings arise in many other situations as well.
In Hue к e1's theory of molecular orbits in quantum mechanics, for
instance, one studies transformations which commute with the
Hamiltomian operator.
As Maxson-Smith (1) pointed out, the study of M.(r) is also
motivated by investigations concerning automorphisms of linear
automata. This brings us straight into our last (proper) chapter.

378
§9 MORE CLASSES OF NEAR-RINGS
i) NEAR-RINGS AND AUTOMATA
In his papers (11)-(14), M. Holcombe has established an intimate
connection between near-rings and linear automata. In fact,
linearity is not necessary for various concepts and results, and
from what we have seen up to now, near-rings might develop its
full power just in the non-linear case (but even from linear
automata one does not get rings, but affine near-rings.
Roughly spoken, automata consist of inputs, states, and outputs,
together with maps which describe how "new" inputs affect the
state and the output. For many considerations, output do not
play any role. Hence we start with "one half of automata": those
which only have inputs and states. For a much more extensive
treatment of these creatures see e.g. (Eilenberg), (Holcombe),
(Kalman ) or Lidl-Pilz (1 ).
We will see shortly that it is advisable now to change our
confession for opportunistic reasons (see the lines after 9.24л);
in this chapter we will write maps from the right (hence we
write xf instead of f(x)); this implies that we are now dealing
with 1 eft near-rings instead of right near-rings (see 1.2).
The author is fully aware of the crime committed here. But to
the subject now!
9.233 DEFINITION A semi automaton is a triple S = (Q,A,F), where
Q and A are sets (called the state set and the input set)
and F is a function from QxA in Q, called the state-
transition function. If Q is a group (we always write it
additively), we call S a group-semi automaton and abbreviate
this by GSA.
For q ε Q and a ε A we interprete F(q,a) as the "new state
obtained from the old state q by means of the input a".
If S = (Q,A,F) is a semi automaton, we get a collection of
mappings f from Q to Q, one for each a ε A, which are given by
qf ■ = F(q,a).
α

9i Near-rings and automata
379
Hence f, describes the effect of the input a on the state set
a
Q of S.
If the input a. ε A is followed by the input a„, the semi automaton
"moves" from the state q ε Q first into qf and then into
a1
(qf, )f . We extend (as usual) A to the free monoid A* over A
a1 a2
consisting of all finite sequences of elements of A, including
the empty sequence Λ) and get
f
a1a2
fa fa
a1 a2
i.e. the map a ->f, is a monomorphism from A* into the transforma-
a
tion monoid over Q with f. = idQ. In the case of GSA's, we are
also able to study the superposition f, +f (defined pointwisely;
a1 a2
of two "simultaneous" inputs a|,a»EA. Hence it is natural to
consider {f | a ε А) и {f *} and all of its sums and products ( =
composition of maps). The obvious framework for that is, of
course, the structure of a near-ring.
9.234 DEFINITION Let S
^Q,A,F) be a GSA. The subnear-ring
N(S) of M(Q) generated by idn and all f 's (a ε A) is
ч а
called the syntacti с near-ri ng оf S .
Thus N(S) is always a near-ring with identity. If Q is finite
then N(S) if finite, too. We now briefly discuss two special,
yet most important, cases.
9.235 The ho mo morphism case. Let Q and A be additive groups
with zero 0 and F a homomorphism from the direct product
QxA. We then call (Q,A,F) a homomorphic GSA. Because of
qfa = F(q,a) = F(q,0) + (0,a)) '= F(q,0) + F(0,a) = qfQ +
+ 0fa
we get f, = f + f , where f is a homomorphism (i.e. a
а и а о
distributive element in N(Q)), while 7, is the map with
a
constant value Of . If no input can change the zero state
a 3
i.e. if Of, = 0 for all a ε A, then N(S) obviously is a
a
d.g. near-ring, consisting of t -sums of powers of f
(which are endomorphisms).We also get a d.g. near-ring

380
§9 MORECLASSES OF NEAR-RINGS
if F is additive in the first component. If this is not
the case, we have to take a closer look, what N(S) consists
of (go to 9.237). For homomorphiс GSA's one sees by
η
(Τ, f^1 + ...+T
a. о
induction that f, , , = f
a1a2--,an °
where the map in brackets is constant. Each power
f is a homomorphism.
о
f +T
П - 1 П
9.236 The linear case is a special case of the homomorphism
case in which Q and A are abelian groups (or more generally,
R-modules for some ring R) and where F is linear. We arrive
at the case of linear semiautomata, the type of (semi-)
automata which are studied most extensively (see e.g.
(Holcombe) or (Kalman)). This case deserves more words.
Let Q and A be free R-modules with finite bases X,Y
respectively. Let |X| = n, |Y| = m. Then the action of
F can be described by an mx(n + m)-matriχ Ζ = (ζ··) over
R if we replace each element of Q and of A by its
decomposition f = f +7,induces a "decomposition" of Ζ such
α θα
that
F(q ,a ) = Z.(q,a'
=: B.q+C. a
We then get
U
■ΙηΛ ί'\
.Ζ
mm/
V
Zi mii·--Ζ
1 m+1
г ... .ζ
m m+1
1 m+n\
a1a2
Bk.q+Bk~1.Ca
m m+n J
C.ak_1+C.ak
= Bk.q and
If, in particular, С = 0, we get qf
ar..ak
N(S) is a ring, generated by В and the unit matrix I
fa ---a = fa'
a1 ak a1
Anyhow, each f
iff С
'акЛ
, (and hence each f for αε A*) is an affine
a a
map from Q to Q. If Q is free on X with |X| = η then we
can use a method due to Blackett (3) to extend the idea
of matrix representations from linear maps to affine maps.
Let f be an affine map.

9i Near-rings and automata
381
Then f decomposes as f = f + с where f is a homomorphism
r oo
and с is constant. Let F be the matrix for f w.r.t. X.
Invent a symbol e with e+e = ее = е and er
all r ε R. Then
re = e for
f-
(:*:)
establishes an isomorphism between Μ ^(Q) and a subnear-
ring of all (n+1 )x(n+1 ) matrices over Rule} .
9.237 THEOREM Let S
N(S]
(Q,A,F) be a homomorphic GSA. Then
= { Σ ± f |α- ε A*} =: N
a ■ ι
Proof. NsM(S) is clear. Conversely it suffices to show that
N is a near-ring , since obviously N contains all
f (a ε A) and idn = f . In fact, we show that N is a
a g
subnear-ring of M(Q
(a) Take f
g = Σ t f „ ε Ν. It is clear
that f+g ε N. So consider fg:
Σ ± f ε Ν,
i ai
fg - ^ - V4 t 4
Σ + (Σ + f )f
j 1 W
ι ι J J
Hence we only look at the last expression in a
Let 3.: = a -a2 ·
а ε Α*. Then
η
ι ι j 1 -ι 12
We first focus our attention to η
a. = a for a moment.
:ς i f )f
i ai а
r-%,f°
:i t f
a .0
+ 7,
1 and put
ξ t %0) - fo + fa£N-
Therefore we get γ, ε A* with
,f
(Σ i fa )fa fa
i ai a1 a2 an
= (Σ - fv )fa '••fa ·
к Yk a2 an
By induction, this is in N.
Σ ΐ f )f, )f.
We remark that N(S) can also be characterized as the subnear-
ring of M(Q) generatd by {T |a ε Α} υ{f ,id}. But the explicit*
expression in 9.2 37 is much easier to handle.

382
§9 MORE CLASSES OF NEAR-RINGS
9.238 COROLLARY Let S = (Q,A,F) be a GSA.
a) If S is homomorphiс then by the last formula in 9.235
and by 9.237 N(S) is a g. d.g. near-ring with identity.
b) If S is even linear, we can arranqe the f 's so that
' M α
each element of N(S) is the sum of linear and a constant
map. Hence N(S) is an affine near-ring with identity in
this case. So not even in the linear case we net proper
near-rings which are not rings.
We will see in 9.241 that N(S) can be quite general for
arbitrary S. In the homomorphic case we exhibit an interesting
feature of the zero-symmetric part N (S) := (N(S)) .
9.239 PROPOSITION Let S = (Q,A,F) be homomorphic. Then N (S)
consists of all finite sums of elements of the form
с t f - с with f ε {id,f0 ,f^,f^ ,...} and с ε {I ί ?α |й] ε Α*}.
i
Proof. All elements с i f - с as above are in N (S).
Conversely, take g = Σ t f ε Ν (S). Then 0 = Oq =
J 3 α · о у
= 0(Σ ί f ) = Σ ί Of = Σ ί 7 .
αι αι αι
3y standard group tneory, we can arrange g = Σ ί f
ai
= Σ t (f + 7 ) into sums and differences of elements
0 ai ni
of the form с + f - c, where с is the sum of some
7 's.
a.
ι
9.240 COROLLARY Let S be linear. Then (with f^:= id)
Vs> = {νο+ζιΨ···+νοΐζίεΖ'ηεν
Hence N (S) is the subnear-ring of Μ ff(Q) generated by
{id,f }. Since (M f^(Q))Q is a ring, N (S)is a ring, too.
Since there are plenty of near-rings N where N is not a rinq
with two generators, we see that far not every near-ring arises
as a N(S) with linear S. What happens in the general case?
Which near-rings arise as N(S) for various types of S? Compare
the next result with 9.191.

9i Near-rings and automata
383
9.241 TH EOREM For every near-ring N with identity there is some
GSA S with N = N(S).
Proof. By 1.86 we can find a group Q such that N is
isomorphic to a subnear-ring TJ of M(Q). Let A be an index
set for "N, i.e. TT = {f |a ε A}
N sTJ = N(S) with S = (Q,A,F).
set for N, i.e. TT = {fja ε A). Let F(q,a) := qf . Then
Since every near-ring can be embedded in a near-ring with
identity (1.102) we get
9.242 COROLLARY Every near-ring can be embedded in the near-rinq
of some GSA.
9.243 THEOREM For a near-ring N there exists a linear GSA S with
N s N(S) iff
(a) (N , + ) is abelian
(b) N has an identity 1
(c) There is some d e N . such that N is generated by {1 ,d}.
Proof. One implication is covered by 9.240. So let N be a
near-ring with (a)-(c). By 1.91 and 1.96 we know that
U is isomorphic to a subnear-ring TT of M(N,+). Let
"d and Τ be the images of d and 1 in IT. Since d is
distributive, сГ is an endomorphism of (N, + ) and
Τ = id... TJ is generated by id and d\ whence
N0 = {zoid + Ζ]Έ + ...+zn^n|zi εΖ,η eHQ}.
Now let (A, + ) : = (Q,+ ) := (N,+ ) and F(q,a) := qcf + 0a,
Then (Q,A,F) is a linear GSA, since (N,+ ) is abelian.
Furthermore, take
We get f = Of = 0 (Σ t f „, ) = Σ t (Of.
-n- 1
Si псе сГ = f we qet TJ„
о э о
with Of
= f f
a. a, o
• +7,
о
f ε N.(S:....,. . . ._ . . .
с tti a.
a = Of, cin~1 + .. .+07 εΟΝ =
3 „ a. a _
— _ η 1 n_ _
= N. This shows N (S)=N . Conversely, every cεN
(with constant value c) is in N (S]
Hence N(S _
since с = Τ
π ε ν.
It is customary in algebraic automata theory to consider the
semi group-epi morphi sm A*^N(S) given by a ■+ f . The idea of
a

384
§9 MORECLASSESOF NEAR-RINGS
simultaneous inputs (lines before 9.224) enables us to transfer
this e pi morph i sin from semigroups to near-rings. We can, for
instance, interpret a.a„ + 2a^ as being the complex input
"inputsequence a.a„ together with the simultaneous input a?
(in double strength)". We extend A to the free near-ring A
over A. If a
f
= wt a
.a.
is a word in A we define
and F ( q , a ) := qf u. Thus we get an extended simultaneous
sequential GSA S# := (Q,A#,F#). Let I be {a# ε A#If „ is the
aff
zero map}. Then I is a near-ring ideal and we get by the
homomorphism theorem:
9.244 THEOREM Ал
/I
N(S
//
= N(S
If we had used right near-rings, we would have N(S) antiisomorphiс
to Α ι,. Hence N(S) can be viewed as a homomorphic image of A .
It is, however, impossible to give a nice "canonical" form
и
for al1 elements of A .
A possible relief comes from the observation that one might
replace A by A , the free algebra in a variety V of near-rings
containing N(S) (for instance, one might take V as the variety
generated by N(S ) ) .
ATTENTION ! If A already bears some additive structure (as in
the cases 9.235/9.236, for instance), this new addition can
(and in most cases will) be different from the given addition
in A! In particular, our new addition is pne in A
A*.
and not in
In the linear case we saw that N(S) is an affine near-ring.
Since the class of all affine near-rings is known to form a
variety (9.82), it makes sense to look at free affine near-rings,
the more so since we know how this monsters look like.

9i Near-rings and automata
385
9.245 PROPOSITION Let A be a set, A* the free monoid over A and
~K the free affine near-ring over A. Then every element of
Ж is a finite sum of elements ία · with a-c(Ac/{0})*.
Proof. Since x(y+z) = xy+xz, (x+y)z = xz-xzO+yz-yzO+zO
and (-x)y = -xy+yxO+yO are laws in the variety of
affine near-rings, we can bring all expressions into
+-sums of elements which are products of elements
in Al/{0} (observe that we use left near-rings!).
We now turn to the concept of accessibility.
9.246 DEFINITION Let S = (Q,A,F) be a GSA and A# the free near-
ring on A. q * ε Q is accessible from q? ε Q if there is some
#
αεΑ with q~f = q«. S is access i ble if each state q is
accessible from each other state.
N(S) is not only a near-ring, but it also operates on Q.Obviously
Q is an N(S)-group via qf in the usual meaning, q. is accessible
from q„ iff q< EqJ(S). Al tena t i ve ly , Q can be viewed as an
Ας roup via qa:= qf .
9.247 PROPOSITION Let S be a GSA. S is accessible iff Q is an
N:= N(S)-group with ON = Q.
Proof. If S is accessible then obviously ON = Q. Conversely,
suppose that Q = ON = 0NC· If q ε Q then qN = qNQ +
+ qNc = qN0+ONc = qNQ+Q = Q, and S is shown to be
accessible.
It might be most useful to examine the relationship between
generators, primitivity and accessibility more closely. Now
we look at constructions of semiautomata and their corresponding
syntactic near-rings.
9.243 DEFINITION Let S = (Q,A,F) and S' = (Q',A,F') be GSA with
identical input sets. A group homomorphism h: Q-»■ Q' is
called an GSA-homomorphi sm if h(qf ) = h(q)f ' holds for
all qcQ and acA (with f'(q'):=F'(a ,q'), of course).

386
§9 MORE CLASSES OF NEAR-RINGS
We then wri te h:
S'
9.249 PROPOSITION (Cf. Holcombe (11)
Let h: S - S' be a GSA-
epimorphism. Then there exists a near-ring epimorphism
F from N(S) to N(S') with h(qn) = h(q)F(n) for all qeQ
and η ε Ν(S).
Proof.
If η ε N(S
w( a,
η is a word η
w(f.
1
... ,f.
.a,
in f.
Ί'··"4' "1
f ' fol1ov
w
length of w. Define F(f
,f.
Then h(qf
= h(q)f' follows fron, 9.248 by induction on the
w
:= fw·
F is wel1-defi ned
s i nee f.
f
implies h(q)fw = h(qfj = h(qfw,
= h(q)f', for all q e (]. Since h is surjective,
w
fw = f' , follows. Obviously, F is a near-ring
epimorphi sm and h(qn
h(qf,
= h(q)f' = h(q)h(n
W
is also true for all qcQ and л eN(S
If we have more in mind, we have to consider outputs as well.
9.250 DEFINITION An automaton ia a quintuple A = (Q,A,B,F,G),
where (Q,A,F) is a semiautomaton, Β a set (the output
set) and G: QxA+B a function (called the output function
of A). If Q is a group, A is called a group-automaton
(abbreviated by GA_) . Similar to 9.235/9.236 we call A
a homomorphic GA if Q,A,B.are groups and F,G are homo-
morphisms. A is called a linear GA or 1i near automaton
or linear sequential machine if Q,A,B are R-modules for
some ring R and F,G are R-linear maps.
Since for every (group-,homomorphic-,1inear-) automaton A =
= (Q,A,B,F,G), S:= (Q,A,F) is a semiautomaton with the same
attributes, we define N(A) as N(S). Hence the syntactic near-
ri ng of a GA has nothing to do with outputs and output maps!
It is easy to adjust 9.244 and 9.248/9.249 to the GA-case.
In many cases, however, outputs do play an essential role.
For instance, if one wants to connect two (or more) automata
in series. For doing that, consider A = (Q,A,B,F,G) and

9i Near-rings and automata
387
A' = (Q',B,C,F' ,G ' ). The outputs of A shall be the inputs of A'
> С
Series connection As A'
More formally, A s A':= ( QxQ',A,C,F ",G ") with F"(q ,g ' ) ,a): =
= (F(q,a), F'(q' ,G(q,g)) and G "((q,q' ),a):= G'(G(q,a),q' ).
Anybody who knows a bit of automata theory knows that N(A s A']
will not be easily expressable in terms of N(A) and N(A'). For
linear GA's we get with these notations
9.251 THEOREM (Holcombe (12!
If A and A' are linear GA then
N(A s A') is the near-ring N(A) s N(A') additively
generated by all pairs of the form (f0»f,p (kcH ), the
constant-map-pairs (7,, 7„, n ,) (acA) and all (0,kp„)
α blUjal О
(к ε IN,
with pQ:Q - Mc(Q'
G(q,0
N(A) s N(A') can be obtained, as Hoi combe remarked, by means of
a wreath product construction for a.a.n.r.'s. (See also 9.285).
He also goes on to relate N(A) s N(A') to the splitting short
exact sequence 0 -+ Q ' -+ QxQ ' -+ Q ■+ 0 of N(A) s M(A')-groups in the
category of a.a.η.r . ' s-groups.
Now we turn to the input-output behaviour of a GA A = (Q,A,B,F,G]
As in the beqinning of this chapter, let A* and B* denote
the free monoids over A and B, respectively. For q ε Q let
= Λ, s (a,
G(q ,a ) , s (a1 ,a2!
B* be defined by s Ah]
= G(q,a,) G(F(q ,a1) ,a2) = sq(ai)sF(q a )(аг' and Proceed
inductively with s (a ^2 . . . . ,a ) = s ( a1 a2 . . . an_ 1 )G( F(q ,q 1 ,,
1n-1
9.252 DEFINITION s :A*-+B* is called the sequential (input-
output-
function of A at q. If A is a GA, Sg=: s
ι s
called the sequential function of A. Furthermore, call

388
§9 MORECLASSESOF NEAR-RINGS
q ,q ' ε Q equivalent states ( q ~q ' ) if s = s , (i.e. if q
and q' induce the same "input-output-behaviour").
# # #
It might make sense to extend s from A to Β , where A and
В are the free near-rings in a variety which contains the one
generated by N(A) if we define s (a.+a2):= G(q,a.) + G(q,a2) =
= s (a.) + s (a„) and observe the "attention" after 9.244.
s is certainly well-defined.
If A = (Q,A,B,F,G) is homomorphic we get for q,q',q" ε Q:
If q'~q" then s . = sn,, . Let q ε Q. Then sn „, (Λ) = Λ = s " (л)
q q n q + q q + q
sq + q,(a) = G(q+q',a) = G(q,a)+ G(q',a ) - R(0 ,a ) =
= G(q,a) + G(q",a) - G(0,a) = G(q+q",a) = s „(a)
Sq+q'(a1a2) = sq+q ■(a1 )G((F(q,ai ) ,a2) + ( F (q ' ,a 1 ) ,a2 ) ·( F( 0,a χ ),a? ))
= sq + q" (a! )G(F(q,a1 ) ,a2) + F(q",a1),a2) - (F(0 ,a1 ) ,a2)) =
= sq+q»(a1a2)-
and so on, hence s , = s ,, , whence q+q'~q + q" .
q+q q+q
Similarly, if q~q' acA and η = f
a1---ak
N(A) then
sqn(a) = G(qfa ___a ,a) = G( F( q ,a 1 ,. . . ak ) ,a ) = G ( F( q ' ,a χ ... ak ) ,a ) =
I К
= G(q'f, , ,a) = sn,n(a) and induction (use 9.237!) shows
qn~q'n. We therefore get
9.253 PROPOSITION Let A be a homomorphic GA. Then - is a congruence
relation in the N(A)-group Q.
9.254 COROLLARY (Holcombe (11)). If A is a homomorphic GA then
(a) Qo:= {qeQ| q^O} <N(A)Q
(b) G(q,0) = 0 for all qEQn.
We might ask what q^q' means in detail. The last formula in 9.235
suggests the answer for homomorphis GA's.

9i Near-rings and automata
389
9.255 THEOREM (Holcombe (11)). Let A be homomorphic and
g0:Q-B, q-qg0 = G(q,0). Then q - q ' ♦» \f к ε NQ : q(fJgQ) =
■ 4'(fjg0).
Proof. Let q ~ q'. We use induction on к and start with
0 . If a ε A then s (a[
G(q,a) = G(q,0) +
+ G(0,a) = qg + G(0,a). Since s (a) = s ,(a) we get
49,
q'g . Now suppose that 9.255 holds for all
words α = a.a~ .
■ak- 1 bA* °f length k-1 =: t. Then
for all a ε A, s(aa) = s , ( aa ) , hence G(qf ,a) =
4 4 t
G(q'fa,a). By 9.235, G(qfa,a) = G(qfJ+ Σ ^.^~\а) =
t . ι=1 ι
= G(qfj,0) + T, G(f fnt-\0) + G(0,a)
о i=1 ai ° ·
Similarly, G(q'fa,a) = G(q'fJ.O) + Σ G (fa f J""1 ,o ) +
+ G(0,a), hence G(qf^,0) = G(q'f^,0) and we get
к к
^o^o = ^'^o^o' ^e converse ls shown similarly.
9.256 DEFINITION A GA A = (Q,A,B,F,G) is reduced if ~ is the
equality. If A is accessible (i.e. if (Q,A,F) is accessible'
and reduced then A is called minimal.
Obviously, a homomorphic GA is reduced iff G = {0}.
9.257 PROPOSITION Let A = (Q,A,B,F,G) be a GA. Then
(a) Aa:= (Q(N(A))=:Qa> A,B,F/Q хД, G/Q хД) is accessible
a a
(Q is called the access i ble part of Q)
(b) By 9.247, Q = ON(A).
(c) A/. := (Q/., A,B,F_,(L) with F_([q],a): = [F(q,a)]
and G~([q],a) := G(q,a) is reduced.
(d) A /~ is mini ma 1.
a
The nroofs are straightforward and omitted. In 1 ooking for criteria
to decide if a given GA A is minimal or not, we obviously have
to view Q not only as an N(A)-group but also have to care about B.

390
§9 MORE CLASSES OF NEAR-RINGS
9.258 THEOREM (cf. Holcombe (11)). Let A be a homomorphic GA
Then A is reduced iff N,.,Q'has no non-zero ideals Ρ with
Pg0 = {Ob
Proof. If N/«\Q has no such ideals then Q = {0} and
A is reduced by 9.253 and 9.254. So suppose that
conversely A is reduced and that f><M(*)Q has G(p,0) =
= P90 = 0 f°r all ρε Ρ, If ρεΡ, we see by similar
arguments as in the lines preceding 9.258 that p~0,
hence ρ = 0, whence Ρ = {0}.
From 9.247 and 9.258 we get
9.259 THEOREM (cf. Holcombe (11)). Let A be a homomorphic GA.
Then A is minimal iff N(A\Q is generated by 0 and does
not contain non-zero ideals which are annihilated bv g .
* 3o
In the linear case, Holcombe (14) continues these ideas by
defining a Jacobson-1ike radical for N(A) (involving the output
map G). A minimal linear GA has zero radical (but not conversely).
Decompositions of N(A) are studied in Holcombe (9), (10) (see
9.232) and (13).
Finally, some comments seem to be in order. If Q bears any
algebraic structure (or even if Q is an algebra with relation
as considered in (Gratzer), for instance), the same carries over
(point-wisely) to the set N = N(S) or N(A) of all f (acA).
a
From the concatenation a.a„ of inputs, composition of mappings
in N enters the area, turning N into a semigroup (if Q is just
a set as in general automata theory) or into a seminear-ring
(if 0 is a semigroup) or into an ordered near-ring (if Q is an
ordered group), and so on. We find us back at the situation
described in 1.118. N always operates on Q, turning Q into
an N-module-type structure. Alternatively, one can study the
free structure on A and let it act on Q (see the lines after
9.246. Hence the theory of "S-acts" (see Weinert (13)) is
avai1able .

9i Near-rings and automata
391
Of course, similar ideas apply, to (linear and non-linear) systems
theory (see e.g. Lidl-Pilz (1)), but the situation there is
more delicate since time-considerations also play their role.
In a very general setting, automata and systems mightbe considered
as (input-output-) functions (cf. 9.252). If, at it often makes
sense, we consider automata or systems F. (i ε I) such that
their input- and output-sets are all equal to Γ, and if г
carries the structure of an additive group (see e.g. (Sain))
then one can observe the following facts:
(a) The parallel connection of F, and F. is given by F.+F..
(b) The series connection of F. and F. is given by F.°F..
Hence one gets the fact
(c) The class of automata of systems which can be constructed
by means of series and/or parallel connections of the
F-'s is precisely the subnear-ring N generated by {F. |i ε 1}
in Μ(Γ).
Up to now, this situation was only considered for linear automata/
linear systems, in which case N turns out to be a ring. The
non-linear situation and the use of near-rings seems to be most
promiss i ng .
These ideas will be pursued in forthcoming papers.
Our world is becoming increasingly complicated and the automata
and systems involved and arising are in many cases far away from
being linear. But in many cases the state sets Q carry a natural
group structure (e.g. Q = Ε ). Hence one might hope that
near-rings can be of use in the non-linear case, thus becoming
an important tool in the understanding of our world.

392
§9 MORE CLASSES OF NEAR-RINGS
j) MISCELLANEOUS TOPICS
In this final section we intend to give brief descriptions of
topics we didn't discuss in our journey through the "nr.-universe1
until rtow. Again it should be noted that being in this section
should not imply any discrimination of this subject (as being
"less important"). We have to reach an end of this monograph -
the reader might be tired.
9.260 SEMIPRIMARY NEAR-RINGS were introduced and studied by
Kaarli in a series of papers. N = NQ is called semi primary
if N contains a finite chain of ideals such that each
factor is either nilpotent or isomorphic to a ring of linear
transformations on a finite-dimensional vector space or
isomorphic to a certain ring of homomorphisms. Semiprimary
nr.'s have the DCCN; if a semiprimary nr. N is a ring then
it is semiprimary in the sense of (Jacobson) (i.e. N/^(N)
has the DCCL). In (7), Kaarli shows that N is semiprimary
iff 31/2(N) is nilpotent, the N-group N/^1/2(N) has the
DCCI and no N-group of type 2 is N-isomorphic to one of its
proper factor N-groups. The structure and the radical theory
of semiprimary nr.'s N and their N-groups was developed
in Kaarli (2), (4), (6) and (7) (and sometimes mentioned in
thi s book ).
9.261 TOPOLOGY IN NEAR-RINSS
The starting point was Beidleman-Cox (1) which contains
definitions and structural properties of topolonical near-
rings .
Topological nr.'s on relatively free groups were considered
by Tharmaratnan (3) (see 6.35(f)).
Betsch (3) considers topological so?ces induced by
-j-primitive iaeaIs (ν = i,2).
Nr.'s of continuous mappings on topological groups
(totally disconnected topological groups, Banach-spaces,
real numbers,...) were considered by Betsch (3),

9j Miscellaneous topics
393
Magi 11 (1)-(3), Hofer (1)-(5), Yamamuro (5), Pal mer-Yama-
иго (1), Blackett (4)-(6), Su (1),(2), Holcombe (3),(4),
H.D. Brown (2), R. Hofer (3),(5), Seppala (1), Su (2) and
Adler ( 1 ).
For instance, Yamamuro obtains the followina result in (5):
Let В be a real Banach-space of dimension >2, and let
N be a nr. of continuous mappinas В -»· Β, containing
Μ ^ЛВ). Then every automorphism of В is inner. This
implies that if Bi>fJi and B?'N2 are two C0UPles as
above and N, = N~ then ?'l and M2 are also topo-
logically isomorphic (honeomorphic) .
See Wefelscheid (1),(2) and (7) for topological near-fields
See Neuberger (1),(2) for applications of nr.'s in
functional analysis. See Magill (9) for an excellent summary.
.262 NEAR-RINGS IN ALGEBRAIC TOPOLOGY
In decomposina polyhedras one meets near-rinqs as
structures which annihilate homoloay groups (see
Curjel (1)).
Curjel (2) contains (anona others) the followina results:
Let A be a finite complex, ΣΑ the reduced suspension
of A and Ν(ΣΑ) =: N the near-rinn (with identity) of
homotopy classes of base-point preserving selfmaps of EA.
Using the induced endomorph.i sms of Η»(ΣΑ), the following
assertions can be shown to be equivalent:
(a) \/ Γη,ηεΝ: mn-nm is of finite additive order.
(b) The group of invertible elements in the monoid (N,·)
(= its group kernel) is finite.
(c) \/ ηεΝ: η nilpotent => η is of finite additive order.
If the Betti- numbers of ΣΑ are known, one can decide
whether or not N has these oroperties by a mechanical
application of Hilton's formula for the homotopy nroups
of a union of soheres. Also,

394 §9 MORE CLASSES OF NEAR-RINGS
9.263 VALUATION THEORY ON NEAR-RIN^S
This is developed in Zenimer (3),(4) and (for near-fields)
in Wefelscheid (6) ,(7) .
9.264 EXTENSIONS AND HOMOLOGY
Maxson (1), Choudhari (1),(2), Seth-Tewari (1), Mason (3),(4;,
Banaschewski-Nelson (1), Oswald (7), Maxson-Oswald (1),
Mel drum (8) and Prehn (1)-(3) consider exact sequences
of N-groups, injectivity, projectivity and the connections
to semi-simplicity (see 5.49, 5.50 and 9.155).
Steinegger (1) describes extensions of near-rings by sets
of functions (similar to the ring case).
For dgnr.'s, homological investigations were carried out by
Frbhlich (5)-(8) ("non-a bel ian homological al gebra ") ·, cf.
Lausch (1),(3) and Lockhart (4).
9.265 NEAR-RINGS AND CATEGORIES
Let С be a category with finite products and a final
object. Let XcC be a group object. Then Mor(X,X) =
- M(X) (cf. 1.4(a)) is a nr. with the obvious operations
(Holcombe (3 ) ,(7),(8)). Holcombe studies these near-rings
in various categories. Homology and cohomolony groups
can be viewed as certain N-nroups for some nr. N.
Similar considerations (in additive catenories) can
be found in Huq (1) and Aijaz-Huq (1).
A categorical investigation to radical theory is in
Holcombe (7) and Hoicombe-Wa1ker (1).
In (15) , (16), (17), CI ay gives a detailed account on nr.'s
("fibered product near-rings") arising in the study of
categories with group or cogroup objects.
Frbhlich (4)-(8) studied dgnr.'s by means of categorical
considerations. Mahmood (1)-(4) continued these studies and
showed (among other results) the surprising facts that
products (cf. 6.9(d) and the fact that the direct product of
dgnr.'s is not d.g. in general!), coproducts, limits and
colimits exist in the category of dgnr.'s (N,D) (with

9j Miscellaneous topics
395
(N ,D)-homomorphisms as in 6.17 as norphisms ). Mahmood-
Meldrum (1) showed that several categories are linked by
functors arising from dgnr.'s. Mahmood-Meldrum (2) applied
several of those ideas to study subdirect products of
dgnr. 's .
9.266 NEAR-RINGS ON A GIVEN SEMIGROUP
In this situation one studies a problem "dual" to the one
studied in §9 d). Given a multiplicative semigroup (N,.),
which additions + can be defined on N in order to turn
(N,+,.) into a near-ring (with certain properties). For
instance, Ligh (20) classified all finite groups (G,.)
such that G, and all subgroups of it, are multiplicative
groups of near-fields. It turns out that G is exactly one
of the four types: (a) 2 , such that every divisor d of η
is of the form d=pnl-1 (p a prime), (b) the quaternion group
of order 8, (c) a metacyclic group of order 24, (d) a bi-
tetrahedral group of order 24. See all papers in the
bibliography which are labelled by M'.
9.267 CONDITIONS FOR N TO BE FINITE
Ligh (1) has shown that if N contains η right zero divisors
о
(at least one of them ε Nj) then |N| Sn , hence N is
finite. See also Linh-Malone (1).
For rings, the DCC and ACC on subrinqs force the rinn
to be finite. Bell-Lioh (1) extended this result to
dgnr.'s and obtained similar other finiteness conditions
(mainly for dgnr.'s). See also Bell (3),(11), Bell-Ligh
(1), Feigelstock (1) and John (1) and cf. 9.268.
9.268 RESIDUAL FINITENESS
Call an algebra A res idual1 у fi η i te if for all a,bcA, afb,
there is a finite algebra A . in the variety generated by
A and a homomorphism h : A -+ A . with h(a)4=h(b). Free near-
rings in U are residually finite (and word problems in

396
§9 MORE CLASSES OF NEAR-RINGS
them are solvable). If 2f is a variety of groups in which
all free groups are residually finite and if D is a re-
sidually finite semigroup then the "free d.g. near-ring"
F„ ,j, of 6.21 is residually finite, too. See John (1).
9.269 NON-ASSOCIATIVE NEAR-RINGS
In Ramakotaiah-Santhakumari (2),(3) and Santhakumari (1),
zero-symmetric 1oop near-rings N are studied (which means
that (N,+) is a loop). Loop nr.'s arise from the study of
mappings of a loop into itself (cf. 1.118). Among other
results, the authors obtained a density theorem for v-primi-
tive loop nr.'s similar to 4.30. Cf. also 8.41 and 8.42.
Timm (5)-(7) studied multipiicativelу non-associative near-
rings. Cf. 8.48. See also Stefanescu (1)-(10).
9.270 COMMUTATORS, DISTRIBUTORS AND SOLVABILITY
Distributors are defined in 9.79. For a detailed study of
these concepts see Esch (1) and confer H.D. Brown
Esch (1) also contains results due to Frohlich (1),(2)
on distributors and "weak distributivity" in dgnr.'s
(cf. 6.16). See also Mason (1),(2) and Maxson (1).
Nr.'s generated by the commutators of a (non-abelian)
group are studied in Gupta (1). See also Curjel (1).
Dasic (1)-(9), Dasic-Peric (1), Kuz'min (1). Meldrum (13),
Oswald (1),(5), Roberts(l) and Scott (7).
9.271 DISTRIBUTIVE NEAR-RINGS
This is the place where the theories of near-rinas and
semirings meet. We mentioned these nr.'s already in 1.15,
1.107 and 1.108. All of §6 is applicable. Taussky (1) also
showed that in a distributive nr. N either each element
is a zero divisor or N is a rino. A simple distributive
nr. is also a rinq (Ferrero (1), Linh (13)).
For more details see Heatherly (4),(6), Heatherly-Ligh (1),
Heatherly-01ivier (3), L i gh (8),(15), Malone (7) and (a
unifying presentation) Weinert (7)-( 10) .

9j Miscellaneous topics
397
N is said to be n-di stributi ve ( η ε Ш) if (Ν ,+) is
abelian and Ц χ , у,,..., у , ζ,,..., ζ ε Ν:
η η
x( Σ У-:2-.-) = X xy ^ ζ - N is pseudo-di stri buti ve if N is
n-di stri buti ve for all ηεΙΝ .
If one considers the nxn-matrices Μ (Ν) with entries
from some nr. N together with the usual addition and
multiplication then (Heatherly (4), Ligh (17)) Μ (Ν)
is a nr. iff N is n-distributive. Also, one can study
polynomials, formal power series, group near-rinns and
"Gaussian near-rings N(i)". These sets are (under the
usual operations) always near-rinas iff "1 is pseudo-
distributive (see Heatherly-Linh (1) for this and many
other results concerninn pseudo-distributive near-rinas).
Confer also Beidleman (1) and Supta (l), as well as 9.160.
Sieno-Stefano (1) showed that all ^ coincide in a
distributive nr.
For this and 9.271, see also all other papers marked by D and
D ' in the bibli ography.
9.272 CHARACTERIZING SERIES
Let ΝΓ be a unitary N-group. An N-seri es of .,Γ is a series
of length n: Г = Τ qsl Τ ^ . . . =>T n = {o} with ri + 1 ^N ri for
each i<n. If I a N then this series is said to be a
n-1
characterizing series for I if I = f\ (Г· . : Г·) and
i = 1 1 ' 1
ΙΓ ,· ^ Γ · о for Oiifn-2. I has a characterizing series
only if I ^ s (ο : Γ ) for some к ε IN. All characterizing
series for I й N have the same length n, and η is just the
nilpotency class of I/(o:r). For this and many other
results see Lyons (7), Lyons-Mel drum (1),(2) and Meldrum (7).
9.273 CENTRAL N-SUBGROUPS are studied in Scott (22).
Δ s:N г is cental if Δ is contained in the center of (r,+ )
and \/ η ε Ν \/ γ ε Γ tf ί ε Δ: η(γ + δ) = ηγ + ηδ. If Δ is central
then Δ з Г. If г = Γ1 β Г2 and Г й„ г has intersection {о}
with г, and Г0 then Δ is cental. If Γι and Го have cental
ι ί. Ν ' Ν £
N-subgroups Δ. ,Δ2 with Δ1 = N Δ2 by h then Γ1 a Γ2/Δ with

398
§9 MORE CLASSES OF NEAR-RINGS
Δ := ί(δ-,1ι(δ^))|δ^εΔ1} is called a central product of г.
and Гл. If Г =N r« a Γ~ then any N-homomorphiс image of
N
Г is a central product.
9.274 C-Z-TRANSITIVE AND C-Z-DECOMPOSABLE NEAR-RINGS
N is "C-Z-transitive" if \/ η.εΜ* \j η'εΝ. 3 ηηεΝη :
nonc = nc·
In this case, M Nr is stronnly tnonoaenic. N is
uo c
"C-Z-decoinposable" if Nc <l N (these nr.'s are closely
related to a . a . η.r.'s !).
Heatherly (2) developes an ideal theory for these near-
rinqs. Cf. also Pi 1 ζ (1) ,(6) .
9.275 Η-MONOGENIC NEAR-RINGS were already touched in 9.122.
о
If HsN = N , N is H-monogen ic if N с Η and Η is "integral"
(i.e. h 1 h, = 0 => h1=0vh2=0)· If N is H-monogen i с with
Η = {0} then N has zero multiplication. On the other hand,
if N is N-monogenic then N is integral, Η-monogenic near-
rings can be constructed by a generalization of Ferrero's
method presented in 1.4(b). For this and other topics see
Heatherly-01ivier (3) and Olivier (1),(2).
9.276 N-SYSTEMS
Λ nr. ΝεΉ0 with rinht cancellation law and a "halvable
idempotent e 4= 0" (i.e. 3 hcN: h+h = e) is called
N-system.
Every N-system is abelian (see the proof of 9.13(b)) and
integral (so 9b)2) is at hand). A finite N-system is a
near-field, but there do exist infinite N-systems v/hich
are neither rinqs nor near-fields (see Lioh-Malone (1).
Ligh-McQuarrie-Slotterbeck (1) and 'IcOuarrie (1),(3)).
If Ν < Μ (Γ) and N is an N-system containinn id., then
every function of N is odd (cf. 9.152(b)).

9j Miscellaneous topics 399
9.277 AUTOMORPHISM GROUPS OF NEAR-RINGS
Scott (18) studied the group Aut(N) of all (near-ring-)
automorphisms of a near-ring ΙεΊ),. If η ε Ν . is invertible
then α : χ ->· nxn is in Aut(N). Inn(N) := {α |η ε Ν , л
л η invertible} is the (normal) subgroup of all i nner
automorphisms of N. As for groups we get D(N)/Z(N) = Inn ( N ) ,
where D(N) = {neNd|n invertible} and Z(N) = {η ε D(N) |an = id}.
In another analogy to groups, a nr. N is called complete
(cf. 9.100(b)) if Z(N) = {1}, Νε1)ο and if all automorphisms
of N are inner. If г is a complete group and N =A(r) such
that N г is monogenic then N is a complete near-ring with
Aut (N) s Aut(r) . For instance, I(S ), nf6, is of this type.
By 7.16, Μ (г) is complete and Aut(MQ(r)) = Г. See also
7.59 and 9.226.
Magill (7) studies a nr. (N,+,.), chooses some aeN and
calles N := (N,+,-a) with η -a m := nam the near-ring
laminated by a. For N=M_„„+(IR), Aut (N,) is determined.
con ι a
In a more general frame, automorphisms are studied by
Nobauer (12) and Plotkin (3).
9.278 DICKSON-NEAR-RINGS
The definitions of couplinn maps, derived nr.'s and
Dickson nr.'s can be found in 9.90. For a detailed study
of these concepts see Maxson (8) and Timm (6),(7).
Of course, a Dickson near-rina (=: DNR) is abelian.
One may write a DNR as (D,+,«,o), where (D,+,·) is
a ring and (D,+,o) the derived nr. (o = о , ) .
Maxson shows e.g. in (8) that (D,+,o) has an identity
iff (D, + ,·) has one and \/ dcD*: i>d j= o. A finite
DNR with identity is a nf..
The ideal structure of a DNR is also considered by
Pieper (1) in comparing the left ideals of (D,+,·) and
(D,+,o). The connection between homomorphisms of (0,+,·)
and (D,+,o) are studied in Maxson (13).
Kerby (5) settles the question in which cases the nr. of
quotients of (D,+,o) is a Dickson one w.r.t. the rinn
of quotients of (D,+,·)·

400
§9 MORE CLASSES OF NEAR-RINGS
Aside from these considerations, Magill (2),(7) also studies
"changed multiplications". See also 9.277.
9.279 NEAR-RINGS AND NUMBER THEORY
Connections between near-rings of formal power series and
number theory were pointed out by Frbhlich (9). Other
connections are established in Mazzola (1) and Ligh (20).
9.280 NEAR-VECTOR SPACES
It seems not to be quite clear how to define a near-vector
space. Beidleman (1) defined it as a 2-semisimple N-group
(N a nf.), and developed a kind if "nearly-linear" algebra.
Other approaches to this concept are made by Andre (3),(5),
(6), Bachmann (2). Hule-Muller (1) study algebraic equations
over nr .'s.
9.281 SYLOW-TYPE THEOREMS; p-SINGULAR NEAR-RINGS
Ferrero (1),(2) shows that |Π! = m, pk/m but pk+1/m
and N = N. implies the existence of a two-sided invariant
subgroup of N of order ρ . If \H\ = p-q (p,qeIP , p<q)
and N is not abelian then N has no subnear-rinn of order p.
If И is finite and ρ ε IP , N is called ρ - s i η π u 1 a r i f ρ
properly divides |N|, but И has no subnear-rinn whose
order is divisible by p. So p-sinoular nr.'s are "minimal
for not fulfilling the Sylow-theorems". A p-sinoular nr.
N is zTff and ..N is stronoly mononenic.
See Ferrero (4 ) , ( 5 ) , (7 ) , ( 18 ) , (19 ) and Scott (8).
9.282 LOCAL NEAR-RINGS
Нг710г\У>1 is called local if L: = L(N): = {χεΝ|χ has no
left inverse} <., N. (this happens iff L is a subnroup).
Maxson (1) , ( 3) shows:
A local nr. is indecomposable. Hence a 1-semisimple one
with DCC is simple. A nr. N is local iff N has a unique
maximal N-subgroup (namely L). L is qr. and if U is not
2-radical then N is local iff *J2(N) = L. So L <1 N.

9j Miscellaneous topics
401
If N is local then N/L is a nf., hence a simple nr. is
local iff it is a nf.. ft local nr. has only 0 and 1 as
idempotents. The additive nroup of a finite local nr. is
a p-group.
Maxson (6) goes on to determine all local nr.'s of order
ρ and ρ . In (9) and (12) he presents local non-nnns on
non-cyclic abelian (p-)nroups of order >5 and гчоге
results in this direction.
Other examples of local nr.'s are aiven by Μ -^(V)
(Maxson (1),(3)), F[x], where F is a field with |Fl>3
(Clay-Doi (2) - 7.98), and Ε(Γ), Α(Γ) and Ι(Γ), where
Γ is a generalized quaternion group (Malone (7)).
See Karzel-Meissner (1), Pieper (1),(2) and
Armentrout-Hardy-Maxson (1) for applications of local
nr.'s to geometry (coordinatisation ) .
Local nr.'s with DCC on principal N-subgroups are studied
in Ramakotaiah-Reddy (1).
9.283 ASSOCIATED RINGS
Let N be abelian and A(N) the subnear-ring of M(N,+))
generated by all h : N -» N. Then A(N) is a ring
m -» ran
(the "ring associated to N") and was investigated in
Williams (1). N and A(N) are closely related.
9.284 SHEAFS OF NEAR-RINGS
A sheaf У of η ear-rings is a disjoint union N. (i ε I) of
near-rings (called the stalks of У ) if I and У are
topological spaces together with some assumptions on these
topologies.
Sheafs of near-rings are studied in Betsch (5) (a Dauns-
Hofman-type result) and (for certain IFP-near-rings) in
Szeto (8)-(10) and Szeto-Wong (1).

402
§9 MORE CLASSES OF NEAR-RINGS
9.285 FULLY GENERATING SUBSETS of a nr. N are subsets S such
that each left ideal of N can be generated by elements of
S. This concept was introduced and studied by Van der Walt
(2),(3). As one might expect, several properties can be
transferred from S to N in this case. The elements of S
must be "evenly and densely distributed throughout N" if
S fully generates N. If S SN fully generates N then the
following properties transfer from S to N: zero-symmetry,
being a near-field, simplicity, DCCL and ACCL. For more
results, see Van der Walt (3).
9.286 NEAR-RINGS AND THEIR "CHILDREN"
In many places, relations between a nr. or an N-group and
all (or some) of its "children" (= substructures and
factors) are studied. We met this topic just before in
9.285 and will meet it again in the next number 9.287.
Ferrero-Cotti (9) studied critical and cocri ti cal near-
rings and N-groups (that are those ones which are not
simple, but do not belong to the variety generated by its
proper factors (those ones with proper substructures which
do not belong to the variety generated by its proper
substructures, respectively)). Near-rings ε 7? in which every
proper subnr. is a near-field are studied in Pellegrini-
Manara (1). If such a nr. is not integral, but without
nilpotent elements =(= 0 then it is isomorphic to a product
of two fields of prime order. If it is integral then N is
simple and 2-primitive. In any case, such a nr. can be
generated by 2 elements and all proper N-subgroups and
left ideals are maximal. See also 7.102(f) and the lines
after 1.40.
9.287 ULTRAPRODUCTS
Let I be a set and N. a near-ring for each i ε I. If % is
a filter on I (i.e. У \ 0, 0 ψ "^ , F ^, F^ ε У => F. л F „ ε ?,
F.c F„ a F. ε "У => F„ ε?) then Π Ν, is the subnear-ring of
Π N. consisting of all ( . . . ,n . ,. . .) with {i ε I|n. = 0} ε ',

9j Miscellaneous topics
403
called the У-fi 1 terproduct of the N.'s. For -instance, if
{J £ I I\ J is finite} then Π N.
N.. A maximal
ι ε I
filter is called ultrafliter, the corresponding products
are the ultraproducts. J is an ultrafilter if for all J SI
either Jc? or I\j ε?. If к ε I then \ ■= {J£l|M J}is
an ultrafilter with Π N. s N, . Other ultrafilters exist,
γ 1 К
but nobody has ever seen them. Many results from model
theory are applicable. For example, every near-ring can be
embedded in an ultraproduct of its f.g. subnear-rings.
See e.g. (Gratzer). Also, one can show that if Ν. (ιεΐ) is
2-primitive on Γ. and if If is an ultrafilter on I then
Π Ν· is 2-primitive on Π г.. Ultraproducts are related to
7 λ Г ^
direct (co-)limits (see Mahmood (3)) and might prove useful
in future research.
9.2i
STILL MORE TOPICS (and still incomplete!)
Semigroup near-rings are introduced in Banaschewski-Nelson
(1). Near-rings with involution are studied in Ferrero-
Cotti (4-),(8), Scott (13) and Suppa (1). Wreath products
of near-rings were already mentioned in §9 h) and §9 i).
See Cla/ (14),(17), С lay-Maxson-Meldrum (1), Holcombe
(9)-(14) and Velasco (1). A connection between near-rings
and difference equations is established in Lopez (1).
Finally, we mention algebraic equations in near-rings and
their relation to extensions, the amalgamation property
and injectivity (Hule-Muller (1), Ki ss-Mark i -Prb'h 1 e-Thol en
(1П.

APPENDIX
Near-rings of low order
222 remarkable examples and counterexamples
List of open problems
Bibliography
Supplementary works
List of symbols and abbreviations
Index

405
NEAR-RINGS OF LOW ORDER
Now we give description of all near-rings of orders 7 and of
several classes of near-rings of order 8 and 12. The whole
discussion is due to Clay (2), (4), (7), (8), (9), Anoerer (1),
Angerer-Pilz (1) and Yearby (1). Because of 9.92, this ammounts
to the description of all mappings 4>:r+End(r), γ -+■ Φ with the
property mentioned in 9.92, where г is a group of "small" order.
The multiplication ·ψ in Γ is then given by γ ·. δ = φ,(γ). This
will be done in the following way:
(a) If Γ = {γ.,.,.,γ }, we list the endomorphisms α.,...α. of Γ.
(b) Every isomorphism class of near-rings of order η is
determined by the η-tuple (k.,...,k ) of elements of Ν ,
where φγ^ = ak_. So r1 ·φ Yj = Φγ.(Ύ,·) = ak-(V·
(c) The numbers following this η-tuple denote the numbers of
those automorphisms of (a) which yield isomorphic near-rings
on Γ.
For the near-rings listed below, much more information is given.
Letters following (a) - (c) have these meaning for the near-ring
N considered:
A...N is abstract affine
С . . N is comrautati ve
D...N is distributi ve
F...N is a near-field
G...N is distributivelу generated
I... N is i ntegral
N. . . N is nil potent
0...N is planar
P. ..N is prime
Q. . .N is quas i regular
R. ..N is regular
W...N is without non-zero nilpotent elements
VJe observe that N is zero-symmetric iff its η-tuple starts with
entry 0. N is constant iff its η-tuple is ( 1 ,1 ,.. . , 1 ) .The n-tuple

406
APPENDIX
(0,0,...,0) is the zero-nr. on r. The letters are followed by
I = γ if γ is an identity in N. After that, two sets of η 0-1-
numbers appear if η is not a prime. They represent the ^ -
and the X- radicals. In the first η numbers, a "1" at the
i-th position means that γ. e ^ (Γ) , while "0" means γ. (j; X(N).
The second η numbers mean the same for X> (N). The case η εΡ
and the radicals ^wo апс' Л f°ll°w from the results in 5.67(v^
Example: The near-ring N
4) (0,14,2,1); 1,2,3,4,5,6; ACDG; I = c; 1100, 1100
on Klein's four group {0,a,b,c} means that φ is given by
и , Φ = α,. The multiplication table
ψο " "ο' ψ3 ~ u14' % '
i s then the foil owing:
0
«14(o:
a14(t>:
a14U,
«2(o:
a2ia -
a2(b ]
a2ic ,
•Φ
0
а
b
с
0
0
0
0
0
а
0
0
а
а
b
0
а
с
b
с
0
а
Ь
с
α.,α?,...,α6 are isomorphisms yielding isomorphic near-rings.
This near-ring N is affine, commutative, distributive(ly
generated) and has с as identity. But N is neither a near-field,
nor integral, nil potent, etc. The radicals are given by
?0(N) = 22(N) = {0,a}. Hence ?1/2(N) = ^(N) = {0,a} as well.
We close with several near-rings of order 8 and 12 and a list
of the total numbers of near-rings on most groups of order i 12.
Д) Ж. = {0): This case is trivial
8) Έ2 = {0,1):
+
0
1
0
0
2
1
1
0
0
1
αοαι
ο η
О 1
1) (0,0); 1;ADGINQR
2) (0,1); 1;ACDFGIPQR; 1=1
3) (1,1); 1;AIPQRW

Near-rings of low order
Z3 = {0,1 ,2}:
1)
2)
3)
4)
5)
[0,0
:o,o
:o,i
:i.i
:o,i
,0)
,1)
,D
,1)
,2)
+
0
1
2
\__Q 1 2
0 1 2
1 2 0
2 0 1
0
1
2
α ο α ι α ?_
0 0 0
0 1 2
0 2 1
1 ,2; ACDGINQR
1,2; PQ
1,2; IPQRW
1,2; A IPQRW
1 ,2; ACDFGIOPQR; 1 = 1
D) TL^ - {Q. 1,2,3}:
+
0
1
2
3
0
0
1
2
3
1
1
2
3
0
2
2
3
0
1
3
3
0
1
2
0
1
2
3
ao
0
0
0
0
aia,2
0 0
1 2
3 2
2 Π
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
Π)
12)
(0,0,0,0)
(0,1,0,0)
(0,3,0,0)
(0,0,1,0)
(0,1,1,0)
(0,1,0,1)
(0,2,0,1)
(0,1,1,1)
(1.1,1,1)
(0,1,3,1)
(0,1,3,2)
(0,3,0,3)
1;
1
1
1
1
1
, 1
1
1
1·
1
ι·
ACDGNQ
7 ·
— 1
2; NQ
Ρ
2; Ρ
2;
IPRW
AIPRW
2; ACDG;I =
ACDGNQ
1111 1111
1000 1111
1111 1111
1000 1000
1000 1000
1010 1010
1010 1010
1000 1000
1010 1010
1010 1010
=1 1010 1010
1111 1111
Ε) Klein's four nroup {0,а,Ь,с}:
+
0
а
b
с
0
0
а
Ь
с
а
а
0
с
Ь
b
ь
с
0
а
с
с
Ь
а
0
0
а
Ь
с
а0
0
0
0
0
αϊ
0
а
Ь
с
а2
0
а
с
Ь
а3
0
Ь
а
с
αϊ»
0
Ь
с
а
as
0
с
а
Ь
as
0
с
Ь
а
а7
0
а
0
а
as
0
а
а
0
а3
0
с
0
с
aioaiiai2ai3aiijai5
0 0 0 0 0 0
0 0 Ь Ь 0 с
с Ь Ь 0 а с
с Ь 0 Ь а 0

408
APPENDIX
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
(0,1,1,1)-
(0,14,1,1)
(0,0,1,1)
(0,14,2,1)
(0,0,2,1);
(0,4,5,1)
(0,7,11,1
1; IPRW;
; 1,3,4;
1,3,4; ADG
; 1,2,3,4,5,6; ACDG; I=c
1,2,3,4,5,6;
1,2,5; ACDFGIOPR; I=c
; 1 ,2,5; ACDGRW; I=c
(0,14,11,1); 1,2,3,4,5,6; I=c
(0,7,0,1)
(0,0,0,1)
(0,7,13,7
(0,7,0,7)
(0,7,13,9
(0,7,0,9)
1,2,3,4,5,6;
1,2,5;
; 1,2,3,4,5,6;
ACDG
; 1,2,3; ADG
1,2,3;
(0,13,0,13); 1,2,3; ACDGNQ
(0,0,0,14
(0,0,0,0)·
(7,7,1,1)
(7,7,7,1)
(7,8,1,2)
(7,7,1,7)
(7,7,7,7)
(1,1,1,1)
; 1,2,3,4,5,6; NQ
1; ACDGNQ
1,2,3,4,5,6; ARW
1,2,3,4,5,6;
1,2,3,4,5,6; APRW; I=b
1,2,3,4,5,6; W
1,2,3,4,5,6; A
, 1; AIPRW
1000 1000
1100 1100
1100 1100
1100 1100
1100 1100
1000 1000
1000 1000
1100 1100
1010 1010
1000 1111
1010 1010
1010 1010
1010 1010
1010 1010
1111 1111
1111 1111
1111 1111
1000 1000
1010 1010
1000 1000
1010 1010
1010 1010
1000 1000
F) Zc = {0,1,2,3,4}: Addition is modulo 5.
0
1
2
3
4
схоахагИзСч
0 0 0 0 0
0 12 3 4
0 2 4 13
0 3 14 2
0 4 3 2 1
1) (0,0,0,0,0)
2) (0,1,0,0,0)
3) (0,1,1,0,0)
4) (0,0,1,1,0)
5) (0,1,1,1,0)
6) (0,0,4,1,0)
7) (0,1,4,1,4)
8) (0,1,1,1,1)
9) (1,1,1,1,1)
10) (0,1,2,3,4)
1; ACDGNQ
1,2,3,4; Ρ
1,2,3,4; Ρ
1,2; Ρ
1,2,3,4; Ρ
1,2,3,4; OP
1,2,3,4; IOPRW
, 1; IPRW
, 1; AIPRW
, 1,2,3,4; ACDFGIOPRW;

Near-rings of low order
G) Z6 = {0,1,2,3,4,5}: Addition is modulo 6.
0
1
2
3
4
5
αοαια2«3θΐι(α5
0 0 0 0 0 0
0 12 3 4 5
0 2 4 0 2 4
0 3 0 3 0 3
0 4 2 0 4 2
0 5 4 3 2 1
) (о
) (°
) (o
) (o
) (0
) (0
) (o
) (o
) (0
) (o
) (o
) (o
) (0
) (o
) (o
) (°
) (o
) (°
) (°
) (o
) (3
) (3
) (3
) (3
) (0
) (0
) (0
) (3
) (3
) (3
) (0
) (4
) (4
) (0
) (3
) (°
) (3
) (3
1,0,0,0,0)
0,1,0,0,0)
ι, ι; ο, ο, π)
1,0,1,0,0)
0,1,1,0,0)
1 ,1,1,0,0)
1,0,0,1,0)
1,1,0,1,0)
0,5,0,1,0)
1,0,1,1,0)
1,1,1,1,0)
1,0,0,4,0)
4,0,0,4,0)
5,0,0,0,1)
1,1,0,0,1)
1,1,1,0,1)
3,0,3,0,1)
5,0,3,0,1)
,3,4,3,0,1)
5,5,0,1,1)
3,1,3,1,1)
1,3,3,1,1)
,3,3,3,1,1)
.5,5,3,1,1)
,2,4,0,2,1)
,5,4,0,2,1)
,1,2,3,4,5)
,3,1,3,3,1)
,3,3,3,3,1)
,5,3.3,3,1)
4,4,0.4,1;
4,4,1,4,1)
4,4,4,4,1)
5,1,0,5,1)
5,1,3,5,1)
4,2,0,4,2)
3,3,3,1,3)
3,5,3.1,3)
·,1,5;
;i,5
;i,5
;i,5
;1,5;
;1,5;
;1,5
;i,5
;i,5
; 1,5;
•J, 5;
;i,5
;i,5
;1,5
•,1,5
; 1,5;
;i,5
; 1,5
;1,5
;i,5
;1,5;
;1,5
;i,5
·, ι, 5;
•,1,5
; 1,5
; ι, 5;
; 1,5
;i,5
; 1,5
;1;5
;l;5;
;l;5;
; 1; 5
;l;5;
;l;5;
;1;5
;1;5;
Ρ
Ρ
Ρ
Ρ
Ρ
W
RW
ACDGRW; 1=
W
D
ARW
ACDG
W
100000
100000
100000
100000
100000
100000
100100
100000
100000
100000
100000
100100
100100
100000
100000
100000
101010
101010
100000
100000
101010
101010
101010
101010
100100
100100
1 100000
100000
101010
101010
100100
100100
100100
100100
100000
100100
101010
101010
111111
111111
111111
111111
100000
100000
100100
111111
111111
100000
100000
100100
100100
111111
111111
100000
101010
101010
100000
111111
101010
101010
101010
101010
100100
100100
100000
100000
101010
101010
100100
100100
100100
100100
100000
100100
101010
101010

-τ— τ— От— ΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟ
■τ— τ— От— -с— τ— Ο Ο Ο Ο τ— Ο Ο Ο Ο τ— τ— τ— ΟΟΟ
■τ— τ— Ο*— Ο Ο τ— Ο Ο Ο Ο ·<— Ο Ο Ο Ο Ο Ο τ— г— τ—
■τ— τ— От— τ— τ— ΟΟΟΟ"*— Ο Ο Ο Ο ·<— τ— τ— ΟΟΟ
■τ— τ— Ο·*— ΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟΟ
χ
Ω
2
■οοοοοοοοοοοοοοοοοοοοο
■ОООО-гт-т— ΟΟΟΟ-*— ΟΟΟΟτ— ч— τ— ΟΟΟ
■ΟΟΟΟΟΟ'-ΟΟΟΟ'-ΟΟ'-ΟΟΟτ-t-^-
■ Ο Ο Ο Ο τ— τ— Ο Ο Ο Ο τ— Ο Ο Ο Ο τ— τ— τ— ΟΟΟ
■οοοοοοοοοοοοοοοοοοοοο
ο-
Q αϊ Ω_ з: Q
cj Ω- κ-« 3 ο: s ο
<c cud- ^<coi <с ос <r:s<c :з <t
СГ с: С с HHHHHHrHrtHHH^rotnr.^JW-cr
О СГ *—< «—< СГ СГ СГ HHrHHfo<3--i«jfl-Ci~nr^J t»i
сг нснс«нтс HiHnfocnrTjmrmcH^
ССннССС ^^гНг^гОгГ^ГгГ^С^ГОчГчГ^
crCCCCCCCrifOCOC'JC'iCr.nC^^
о
■ О *— О О О О О
■ О *— О О О О О
■ О ·<— О О О) О О
>>
X
и
-Q
Л5
о
+
>>-Q U (00 X
X О ПЗ -Q >>0
О X >>0 «J _Q
_0 >>0 >i U fO
«3 О) X X jD О
О fOO U X >>
О ЛЗ -Q О X >>
ООООО-- От— т— т— т— ,—
ООООО^- От— т— т— т— ,_
ОООООООООООО
ОООООООООООО
оооооооооооо
D--D_D_D_D_ 0_0_0_
λ
сп
л
CD
<Ч
t^
es
ю
о
^
ΰ
ΰ
«
rt
es
О)
О)
О)
о
О)
О)
О)
О)
О)
О)
О)
с)
о
<П
и
о
_Q
-Q
(Π
(Π
о
тз
с)
π
ю
_0
«ί
и
лз
С )
η
О)
_Q
с)
η
«ί
лз
η
лз
и
η
и
о
и
О О)
о о
о о
>> χ
X >>
X >>
>> X
>> X
X >>
О О)
X >>
LO
LD
«а- ^а-
го
CSJ
CSJ
«—<
CSJ
ГО
CSJ
C\J
CSJ
«—<
1П
LO LD
ч- «- ч-
го го го
CSJ
LD
«*
го
LD
ч-
го
LD
ч-
LT> LD
«a-
-tiГО ГО ГО
LT?
«а-
го
LO
ч-
го
C\JC\JC\JC\JC\JC\JC\JC\JC\J
о о
мгсг
о
о
ΟΟΟ
о о
о
—1 О
О О
CSJ
г-~
о
CSJ
о
О О О О
о
о
г-~
er г~~
О «-Н
OJ
~
ОЭ «-Н СО
Г— 1— 1—■
Г"~
·—'
«—<
г-~
г-~
О О Г-~ Γν. Γν.
rHCOm4-Lf)lONCOCnOr4CO

Near-rings of low order
411
13)
14)
15)
161
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
37)
33)
39)
(7,7,7,1,1,7)
; 1 ,2 , 3 ,4 ,
(7,7,1 ,2,1,2);1,2,3,4,
(7,7,1,7,1 ,7);1,2,3,4.
(7 ,7,7,7,1 ,2) ;1,2,3,4
(7,7,7,7, 1 ,7);1,2,3,4
(7,7,2,1,7,7);1,2,3,4
( 7 , 7 , 7 ,1 , 7 , 7 ) ; 1 , 2 , 3 , 4
(0,2,2,1,0,1);!,2; Ρ
(0,0,0,0,0,1)
(0,0,1,1,1,1)
(0,0,0,1,1,1)
(0,7,1,1,0,0)
(0,0,1,1,0,η)
(0,4,5,1,0,0)
(0,7,8,1,0,0)
(0,0,0,1,0,0)
(0,7,7,7,0,0)
(7,7,1,1,1,1)
(7,7,7,7,1,1)
(7,7,1,1,7.7)
(7,7,7,7,7,7)
(0,1,1,1,1,1)
(0,Π,η,η,1,1)
(0,1,1,1,0,0)
(0,7,8,9,0,0)
(0,0,0,0,0,г,)
(1,1,1,1,1.1)
1,2;ρ
1,3,5; Ρ
,1,2,4 ; Ρ
1,3,5
1,3,5
1,2,4
1 ,2,4
1,2,4
1,3,5; CDC
1,3,5; PRb
1,3,5; PW
1,3,5
1,3,5
1; IPRW
1; ρ
1; G
1
1; CDGNQ
1; IPRW
5
5
5
5
5
κ
5
,6;
Λ;
.6;
,6;
,6;
,0
,6;
Ρ
PRW
Ρ
PW
Ρ
Ρ
100011
100011
100011
100011
100011
100011
100011
100000
100000
100000
100000
100011
100000
100011
100011
100000
100011
100011
100011
100011
100011
100000
100000
100011
100011
111111
100011
100011
100011
100011
100011
100011
100011
100011
100000
111111
111111
111111
100011
111111
100011
100011
111111
100011
100011
100011
100011
100011
100000
111111
100011
100011
111111
100011
I) 1-j = {0,1 ,2,3,4,5,6}: Addition is modulo 7.
0
1
2
3
4
5
6
аосиагазсчо^'ч
0 0 0 0 0 0 0
0 12 3 4 5 6
0 2 4 6 13 5
0 3 6 2 5 14
0 4 1 5 2 6 3
0 5 3 16 4 2
0 6 5 4 3 2 1
1
2
3
4
5
6
7
8
9
10
11
12
13
0,0,0,0,0,0
0,1 ,0,0,0,0
0,1,1,0,0,0
0,1,0,1,0,0
0,1,1,1,0,0
0,1,1,0,1,0
0,2,4,0,1,0
0,0,0,1,1,0
0,1,0,1,1,0
0,0,1,1,1,0
0,1,1,1,1,0
0,0,0,6,1,0
0,1,1,1.0,1
.0)
,0)
.0)
.0)
,0)
,0)
.0)
,0)
,0)
.0)
,0'
,ο
,0)
1 ; ACDGNQ
1,2,3,4,5,6; Ρ
1,2,3,4,5,6; Ρ
1,2,3,4,5,6; Ρ
1,2,3,4,5,6; Ρ
1 , 3; Ρ
Γ,2',3,ί ,5,6; 0Ρ
1,2
1,2
1,2
,6; Ρ
,6; Ρ
,6; Ρ
,6; ΟΡ
,6 ; Ρ

412
APPENDIX
14) (0,0,1,1.1
1 5 ) ( П , 1 ,1,1 ,1
16)(0,0,6,6,1
17) (Ο,η ,6,1,6
18)(0,1,1,1,1
19) (1,1 ,1,1,1
20)(0,6,6,6,1
21)(0,6 ,6 ,1 ,6
22)(0,2,4,4,1
23)(0,1 ,2,3,4
24)ί0,1 ,2,4,4
1.°)
1,0)
ι.ο)
КО)
ΚΙ)
κΐ)
ΚΙ)
ΚΙ)
2 Л )
5,6)
,2,1)
1,2,3; Ρ
1,2,3,4,5,6;
1,2,3,4,5,6;
1,2,3,4,5,6;
1; IPRW
1; AIPRW
1,2,3,4,5,6;
1,3; I0PRW
1,2,3,4,5,6;
1,2,3,4,5,6;
, 1 ,2,3; I0PRW
Ρ
0Ρ
ΟΡ
IOPRW
IOPRW
ADGIOPRW
1 = 1
JL·?:
{0,1,2,3,4,5,6,7}: Addition is nodulo
-о
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
И)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
(0,0,0,0,0,0,0,1
(0,0,0,0,0,0,0,4
(0,0,0,0,0,0,0,0
(0,0,0,0,0,0,4,4
(0,0,0,0,0,1,0,3
(0,0,0,0,0,1,1,0
(0,0,0,0,0,1,1,1
(0,0,0,0,0,4,4,0
(0,0,0,0,0,4,4,4
(0,0,0,0,1,0,0,1
(0,0,0,0,1,0,1,1
(0,0,0,0,1,1,1,η
(0,0,0,0,1,1,1,1
(0,0,0,1,0,0,0,5
(0,0,0,1,0,1,0,1
(0,0,0,1,0,1,1,0
(0,0,0,1,0,1,1,1
(0,0,0,1,П,7,О,η
(η,η,0,1,1,1,0,1
(0,0,0,1,1,1,1,0
(0,0,0,1,1,1,1,1
(0,0,0,2,0,4,4,2
(0,0,0,4,0,4,0,4
(0,0,0,4,0,4,4,0
(0,0,0,4,0,4,4,4
(0,0,1,0,0,0,1,1
(0,0,1,0,0,1,3,3
(0,0,1,0,0,3,3,1)
0
1
2
3
4
5
б
7
α ο αιθί.2α 3αι,
0 0 0
0 1 2
0 2 4
0 3 6
0 4 0
0 5 2
0 6 4
0 7 6
;КЗ,5,7
-.1,3,5.7
-.1.3,5.7
-.1,3.5,7
;К3.5,7
;КЗ,5,7
;КЗ,5,7
-.1,3,5,7
;1 ,3.5,7
;КЗ,5,7
; 1.3. 5, 7
-,1,3,5,7
-,КЗ,5,7
;К3,5.7
Л , 3,5 ,7
; К 3 ,5 .7
Л,3,5.7
Л,3,5,7
;КЗ,5,7
Л,3,5,7
Л ,3,5,7
Л , 3 ,5 ,7
Л,3,5,7
Л,3,5,7
; К 3 .5 ,7
-.1,3,5,7
-.1.3,5,7
Л
,3,5,7
0 0
3 4
6 0
1 4
4 0
7 4
2 0
5 &
NQ
α5
0
5
2
7
ά
ι
6
3
ADGNQ
NQ
NQ
NQ
Ρ
Ρ
Ρ
Ρ
Ρ
Ρ
Ρ
NQ
NQ
NQ
NQ
α6α7
0 0
6 7
4 6
2 5
0 4
6 3
4 2
2 1
10000000
11111111
11111111
11111111
10000000
10000000
10000000
11111111
11111111
10000000
10000000
10000000
10000000
10001000
10000000
10000000
10000000
10000000
10000000
10000000
10000000
11111111
11111111
11111111
11111111
10000000
10000000
10000000
11111111
11111111
1111 1111
11111111
11111111
11111111
11111111
11111111
11111111
10000000
10000000
10000000
10000000
11111111
11111111
11111111
11111111
11111111
10000000
10000000
10000000
11111111
11111111
11111111
11111111
11111111
11111111
11111111

Near-rings of low order
413
29)(0,0,l,0,l,O,l,l);l,3,
30) (0,0,1,1,0,1,0,1);1 ,3,
31)(O,O,l,l,0,l,l,l);l,3,
32)(0,0,1,1,0,7,7,0);1,3,
33)(0,0,1,1,1,1,0,1);1,3,
34) (0,0,1,1 ,1,1,1,1);1,3,
35)(0,0,1,7,0,1,7,П);1,з,
36)(O,n,4,O,O,0,4,4);1,3,
37)(0,0,4,4,O,4,O,4);l,3,
38) (0,0,4,4,0,4,4,4);1,3,
39) (0,0,4,6,0,4.0,6);1,3,
40)(0,1,0,1,4,5,4,5);1,3,
41)(0,l,0,3,0,5,o,7);l,3,
42) (0,1,0,3,4,5,0,7);1,3,
43) (0,1,0,5,4,5,4,1);1,3,
44) (0,1,1,1,0,7,7,7);1,3,
4 5 ) ( 0 ,1,1, 3 ,0 , 3 , 3 ,1) ; 1 , 3 ,
46) (0,1, 2,3,4,5,6,7);1,3,
47) (0,1 ,4 ,3,4,5,4,7);1,3,
48) (0,2,4,2, 0,2,4,6);1,3,
49) (0,2,4,6,0,6,4,6);1,3,
50) (0,0, 0,0,0,0,1,0);l,5;
51)(0,0,0,0,0,0,4,0);1,5;
52)(0,П,0,0,0,1,0,П;1 3 ·
5 3 ) (0 . ι, ι, ι . n . *., τ , '. ; ■ ' ' ~. ■
54) (0,0, 0,0,1,0, i,0);U5:,
55Η0,Ο,0,η,1,1,ο,1)·1 3 ·
56Hl,0,0,l,n,n,n,l)jl,'5;"
57)<0.0,Λ,1,0,0,1,1);1,5;
58)(O,0,0,l,O,l,n,0);l,7;
59)(0,0,0,1,1,О,П,1);1,5;
60)(0,0,0,1,1,0,1,1);1,5;
61)(0,0,O,l,l,l,0,0);l,7;
62)(0,О,П,2,0,О,4,2);1,5;
63)(0,0,О,4,О,0,0,4);1,5;
64) (0,0, 0 ,4, η,η,4,4);1,5;
65)(Ο,0,Ο,4,Ο,4,0,Ο);1,7;
66)(0,0,1,0,П,П,3,О);1,5;
67)(0,0,1,0,0,0,7,0);1,5;
68)(0 ,0,1,0,П , 1,1 ,1) ; 1 ,3 ;
б9)(0,0,1,о,1Л,1,1);1,3;
70)(П,0Л,1,0,0,ОЛ);1,5;
71)(0,П,1,1,П,п,1,1);1,5;
72)(0,О,1,1,п,1,1,0 ;1,7;
73)(0,η,1,ΐ,1,ο,η)ΐ);1ι5.
74)(0,О,1 ,1,1.0,1Л);1,5;
75) (0, 0,1,1,1,1,1,0);1, 7;
7 6 ) ( 0 , 0 ,4 , 0 ,0 , 4 , 4 ,4 ) ; 1 , 3 ;
77)(О.О,4,4,О,О,0,4);1,5;
78)(0,О,4,4,О,0,4,4);1,5;
79)(0,0,4,4,0,4,4,0);1,7;
80)(0,0,4,6,0,0,0,6);1,5;
81)(0,1,0,1,0,1,1,1);!,5;
82)(0,1,0,1,О,5,О,5);1,3;
5,7;
с 7 -
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7;
5,7:
NQ
NQ
Ρ
Ρ
Ρ
Ρ
Ρ
NQ
NQ
NQ
NQ
AD
NQ
NQ
Ρ
Ρ
Ρ
NQ
NQ
NQ
NQ
Ρ
Ρ
Ρ
NQ
NQ
NQ
NQ
10000000
10000000
10000000
10000000
10000000
10000000
10000000
11111111
11111111
11111111
11111111
10101010
10101010
10101010
10101010
10000000
10000000
10101010
10101010
11111111
11111111
10000000
11111111
10000000
11111111
10000000
10000000
10001000
10000000
10000000
10000000
10000000
10000000
11111111
11111111
11111111
11111111
10000000
10000000
10000000
10000000
10000000
10001000
10000000
10000000
10000000
10000000
1
1
1
10000000
10101010
10000000
11111111
11111111
11111111
10000000
10000000
11111111
11111111
11111111
11111111
11111111
10101010
10101010
10101010
10101010
11111111
11111111
10101010
10101010
11111111
11111111
11111111
11111111
11111111
11111111
10000000
10000000
11111111
11111111
11111111
10000000
10000000
10000000
11111111
11111111
11111111
11111111
10000000
10000000
10000000
10000000
11111111
10001000
11111111
10000000
10000000
10000000
11111111
11111111
11111111
11111111
11111111
11111111
10101010

414
APPENDIX
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
9Я
99
100
101
102
103
101
105
106
107
108
\r\a
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
/0,1,0,1
0,1,0,1
0,1,0,1
0,1,0,1
0,1,0,1
0,1,0,3
0,1,0,3
0,1,0,3
,0,7,0,7)
,1,1,1,1)
,4,1,4,1)
,4,5,0,5)
,4,7,0,7)
,0,1,0,3)
,0,3,0,1)
,4,1,0,3)
0,2,0,3,4,3,0,1)
0,l.o,5
0,1,0,5
0,1,0,7
0,1,0,7
0,1,1,3
0,1,1,7
0,1,2,1
0,1,2,3
0,1,2,5
1,2,7
1,3,3
1.4,1
1,4,1
0,1,4,3
0,1,4,3
0,1,4,5
0,1,4,7
0.1,6,1
0,1,6,5
0,1.7,7
0,2,4,4
0,2,4,6
0,2,4,2
0,2,4,6
0,4,0,4
0,4,4,6
1,1,1,1
0,0,0,0
0,0,0,0
0,0,1,0
0,0,1,0
0,0,4,0
0,1,0,1
0,1,0,1
0,1,0,1
0,1,1,1
0,1,1,1
0,1,2,1
0,1,4,1
0,1,6.1
0,2,4,2
0,4,0,4
0,4,4,4
0,6,4,6
0.
,4,
,0,1
,4,1
,1,1
,0,1,7,7
, 4 , 5 , 2 , ^
,4,1,6,3
,4,5,2,1
,4,1,6,7
,0,1,1,3
,4,5,4,5
,4,7,4,7
,4,1,4,3
,4,3,4,1
,4,5,4,1
,4,1,4,7
,4,5,6,5
,4,5,6,1
,0,1,1,7
,0,2,0,4
,0,2,4,6
,0,6,4,6
,0,6,4,2
,0,4,4,4
,0,4,0,6
,1,1,1,1
,0,0,0,0
,1,0,0,0
,0,0,1,0
,1,0,1,0
.0,0.4.0
,0,1,0,1
,1,1,0,1
,4,1,0,1
,0,1,1,1
,1,1,1,1
,4,1,2,1
,4,1,4,1
,4,1,6,1
,0,2,4,2
,0,4,0,4
,0,4,4,4
,0,6,4,6
1,3
1,5
1.5
1,3
1,3
1,5
1,7
1,5
1,7
1,7
1,7
1,5
1,5
1,5
1,5
1,3
1,5
1,7
1,5
1,5
1,3
1,3
1,5
1.7
1,7
1,5
1 ,3
1,7
1,5
1,5
1,5
1,3
1,7
1,5
1,5
NQ
ACDGNQ
NQ
NQ
NQ
NQ
Ρ
NQ
1; AIPRW
1; ACDGNQ
1; Ρ
1;
1;
1:
1;
l; p
l;
l;
l; Iprw
l;
l;
l;
l; nq
1; ACDGNQ
l; nq
1; NQ
0101010
0000000
0101010
0101010
0101010
0101010
0101010
0101010
0101010
0101010
0101010
0101010
0101010
0000000
0000000
0101010
0101010
0101010
0101010
0000000
0000000
0101010
0101010
0101010
0101010
0101010
0101010
0101010
0000000
0101010
1111111
0000000
0001000
0000000
1111111
0101010
0000000
0101010
0001000
0000000
0101010
0101010
0101010
1111111
1111111
1111111
1111111
10101010
10000000
10101010
10101010
10101010
10101010
10101010
10101010
10101010
10101010
10101010
10101010
10101010
1111111
1111111
10101010
10101010
10101010
10101010
11111111
11111111
10101010
10101010
10101010
10101010
10101010
10101010
10101010
10101010
1111111
10000000
10001000
10000000
11111111
10101010
10000000
10101010
10001000
10000000
10101010
10101010
10101010
1111111
1111111
1111111
11111111

Near-rings of low order
415
K) The dihedral qroup D„ = {0,a,2a ,3a ,b,a+b,2a+b,3a+b}:
=—. 5 : —^
For more compact typing, we list the endomorphisms row-wise
from now on.
?a+b 3a + b
a
0
си
«2
Ct3
cu
Cts
Ctc
a7
a«
a9
α-ι ο
α 11
α -ι 2
α ι 3
Cti n
C«1 5
Cti 6
C«17
Cti а
a 1 э
Ct20
Ct21
Ct22
Ct23
Ct24
Ct25
Ct26
Ct27
Ct29
Ct29
Ct30
Ct31
Ct32
Ct33
a3u
Ct3S
0
0
0
0
0
0
0
0
η
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
а
а
а
а
За
За
За
За
0
0
О
0
0
2а
b
а + b
2а+Ь
За + Ь
2а
b
а + b
2а + Ь
За + Ь
2а
2а
2а
2а
а + Ь
а + Ь
За + Ь
За + Ь
Ь
b
2а + Ь
С 3" о
0
2а
2а
2а
2а
2а
2а
2а
2а
0
0
0
0
о
0
0
η
0
0
0
О
0
0
0
о
η
0
0
0
0
0
0
0
η
1
η
9
За
За
За
За
а
а
а
а
0
0
0
0
0
2а
b
а + b
2а + Ь
За + Ь
2а
b
а + b
2а + Ь
За + Ь
2а
2а
2а
2а
а + Ь
а + Ь
За + Ь
За + Ь
b
b
2а + Ь
2а + Ь
0
b
а + b
2а + Ь
За + Ь
b
а + b
2а + Ь
За + Ь
2а
b
а + b
2а + Ь
За + Ь
2а
b
а + b
2а + Ь
За + Ь
0
О
О
О
0
b
а + b
2а + Ь
За + Ь
2а
За + Ь
2а
а + Ь
2а
2а + Ь
2а
b
О
а + b
2a + h
За + b
b
3a + b
b
a+b
2a + b
2a
b
a + b
2a + b
3a + b
0
0
0
0
0
2a
b
a + b
2a + b
3a + b
2a + b
3a + b
b
a + b
3a + b
2a
a + b
2a
2a + b
2a
b
2a
0
2a + b
3a + b
b
a + b
2a + b
3a + b
b
a + b
2a
b
a + b
2a + b
3a + b
2a
b
a + b
2a + b
3a + b
0
η
0
η
η
b
a + b
2a + b
3a + b
2a
3a + b
2a
a + b
2a
2a + b
2a
b
0
3a + b
b
a + b
2a + b
a + b
2a + b
3a + b
b
2 a
b
a + b
2a + b
3a + b
η
η
О
О
О
2а
b
a + b
2a + b
За + b
2a + b
За + b
b
a + b
3a + b
2a
a + b
2a
2a + b
2a
b
2a
2a
3a
a+b 2a+b 3a+b
0
a
2a
3a
b
a + b
2a + b
3a + b
О а 2а За
a 2a За О
2a За О а
За О а 2а
b За+b 2a+b a+b
a+b b За+b 2a+b
2a+b a+b b 3a+b
За+b 2a+b a+b b
3a
a+b 2a+b 3a+b
a+b 2a+b 3a+b
2a+b 3a+b
За + b b
О За
a 0
2a a
2a
b
a + b
2a
3a
О
a
a + b
2a + b
a
2a
3a
0

416
APPENDIX
Not all near-rings on Dg will be listed (altogether there are 1447
isomorphism classes !). But we give a complete list of representatives of
isomorphism classes of near-rings on D„ which are either non-zerosymmetric or
have an identity or are integral.
)(io,
)(10,
)no>
)(ю,
)(io,
)(io,
)(io,
)(io,
)(io,
)(io,
)(io,
)(io,
(10,
)(ю,
)(io,
)(ю,
)(io,
)(io,
)(io,
)(io,
)(io,
) (io,
)(io,
)(io,
)(io,
)(io,
)(io,
)(15,
)(15,
)(15,
)(15,
)( 15,
)(15,
)(io,
)(10,
)(in,
)(i°»
)(10,
)(i°,
)(10,
)(io,
)(io,
)(io,
)(in,
)(io,
)(10,
)(10,
)(io,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,
1,10.
1,10.
1,10,
, ι
, 1
,10
,m
,10
,10
,10
,10
,10
,10
, 1
1
5
5
5
1,10
1,10
1,10,10
1,10,10
1,10,10
1,10,10
1,10,10
1,10,10
1,10,10
1,10,10
1,24, 1
1,24
1,24
1,24
1,15
1.15
1,15
1,15
1 .15
1,15
10,24
10, 1,10
10, 1,10
10,10,10
10,10,1η
10,10,in
10,10,10
10,10,10
10, 10,1П
10,10,24
1П,10,?4
10,10,24
10,10,24
10,10,24
24,10,24
,10, 1,
,10, 1,
,10, 1,
,10, 1,
,10, 1,
,10, 1 ,
,10,10,
,10,10,
,10,10,
,10,10,
,10, 1,
.10, 1,
,10, 1,
,10, 5,
,10,10,
,10, 1,
,10, 1,
,10, 1,
,10, 1,
,10,10,
,10,10,
,10,10,
,10,10,
,10,10,
,10, 1,
,10, 5,
,10,10,
,15, 1,
,15.15.
,15, 1,
,15, 5,
,15,15,
,15,15,
,10,24,
,10, 1,
,1°, 1,
,10, 1,
,10, 1 ,
,10, 1,
,ιο, ι,;
,10,10,
,10,10,
,10, 1,,
,10,10,
,10,10,
,10,24,
,10,24,
,10, 1,
, 1
,10
, 1
, 1
,10
,10
, 1
, 1
,10
10
1
10
10
10
10
1
1
10
10
1
1
10
10
10
25
24
in
15
35
15
15
15
35,
10.
1,
10,
1,
10,
io,
24,
Ю,
10,
24,
ln,
10,
10,
10,
24,
,10); Ρ 1
,10); ρ ι
, i); ρ 1
,10); Ρ 1
, ΐ); ρ 1
,ιο); ρ ι
, ΐ); ρ 1
,ιο) ; ρ ι
, ΐ); ρ 1
,10); ρ ι
,10) 1
,10) 1
, 5) 1
, 1) 1
,10) 1
» 1) 1
10)
1)
10)
1)
10)
1)
10)
24)
5)
1)
10)
15)
35}
1)
15)
15)
2Λ>
10); Ρ
1°); Ρ
10)
54
10)
5)
10)
24)
1)
ιη)
24 )
Ю)
24)
δ)
1110000
1110000
1110000
1110000
1110000
1110000
1110000
1110000
1110000
1110000
1110000
1110000
0100000
0100000
1110000
1110000
1110000
1110000
1110000
1110000
1110000
1110000
1110000
1110000
1110000
0100000
1110000
0100101
0100101
0100000
0100000
0100101
0100101
1110000
1110000
1110000
1110000
1110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
10100000
10100000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
10100000
11110000
10100101
10100101
10100000
10100000
10100101
10100101
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000

Near-rings of low order
417
49)
50)
51)
52)
53)
54)
55)
56)
57)
58)
59)
60)
ol)
5 2)
63)(
r λ \
65)
66)
67)
68) (
69) (
70) (
71)
72)(
73) (
74)(
75) (
76)(
77)(
78) (
79)(
8П)(
81)(
82) (
83)(
84)(
85){
86)(
87)(
88) (
B9)(
90) (
91)(
92)(
93)
94)
95)(
96)
97)(
98)
99)
100) (
101)
102)
103)
(10
( 15
(15
[15
[15
15
15
15
15
15
15
15
ι D
1 '"
15,
15,
15
15
15
15
15
15
15
15
15
15
15
15,
15,
15,
15,
15,
15,
15,
15,
15,
15,
10,
10,
10
10,
10
10,
10
10
10
10,
10
10,
10
15
15
15
15
15
,24,10
,1,1
,1,1
, ι, ι
1, 1
1, 1
1, 1
1, 1
1, 1
1, 1
ι, ι
1,15
1 ,35
15, 1
15, 1,
ι = Λ Ξ,
15,15
15,15
1,15
1,15
1,15
1,15
1,15
1,15
1,15
1,15
1,15
1,35,
15,15,
15,15,
15,15,
15,15,
15,15,
15,35,
15,35,
3 5,15,
3 5,35,
1, 1,
1, 1
ι, ι
ι, ι
1,1°
1,10,
1,10,
1,10
1,24
1,24,
24,10
24,10
24,10
1, 1
1, 1
1, 1
1 , 1
1,15
,24,10
, 1,15
, 1,15
,15,15
15,15
15,15
15,15
15,15
15,15
15,15
15,15
1,15
5,15
15,15
15,15
15,15,
15,15
15,15
5,15
15,15
15,15
15,15
15,15
15,15
15,15
15,15
15,15
5,15
15,15
35,15
35,15
35,15
35,15
15,15,
35,15
35,15
35,15
1,10
1 ,10
1,10
1,10
1,10
1,10
1,10
1,10
1,10
1,10
24,10
24,10
24,10
1,15
1,15
1,15
1,15
1,15
,10,10,
, ι, ι,
, 1,15,
,1,1,
1,1,
1,15,
1,15,
15, 1,
15, 1,
15,15,
15,15,
1, 1,
5.75.
1 ■ » .
Ϊ Л5,
1 1
ί', 15 *
1,15,
35,35,
1,1,
1, 1,
1,15,
1,15,
15, 1,
15, 1,
15,15,
15,15,
1,35,
15,15,
15,15,
15,15,
35,15,
35,15,
1,15,
1,15,
15,15,
1,15,
1,1,
1,10,
10, 1,
10,10,
1, 1,
l,ln,
10, 1,
10,10,
1,24,
10,10,
1,24,
10,10,
24,10,
1, 1,
1,15,
15, 1,
15,15,
1,1,
24)
15);
15).
1)?
15);
l);
15);
i);
15);
i);
15>;
15);
1 "
' - i
15) ;
ι с ^
15)
15)
35)
1)
15)
1)
15)
1)
15)
1)
151
5)
35)
15)
35)
15)
35)
5)
1)
35)
5)
i);
1);
10);
10);
1)
1)
10)
Ю)
1)
10)
1)
10)
24);
1)
i);
15);
15);
1)
Ρ
Ρ
PW
Ρ
PW
Ρ
Ρ
Ρ
Ρ
Ρ
ρ
PRW
PW
PW
PW
PRW
PW
Ρ
Ρ
11110000
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100000
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100000
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
11110000
11110000
11110000
11110000
11110000
10100000
11110000
11110000
10100000
11110000
11110000
11110000
11110000
10100101
10100101
10100101
10100101
10100101
11110000
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100000
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100000
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
11110000
11110000
11110000
11110000
11110000
10100000
11110000
11110000
10100000
11110000
11110000
11110000
11110000
10100101
10100101
10100101
10100101
10100101

418
APPENDIX
104)
105
106
107
108
109
110
Ш
112
11Г
1!ί
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
(15,
(15,
(15,
/15.
1,15,
15,
15,
15,
15,
35,
24,
(Ю,Ю, 1,
(15
(10
1,
(1°.
(1°.
ПО,
(1°.
(1".
(ΐη·
(1^·
(15.
(15.
(15,
(15.
(15,
(15,
(15;
10
Ю,
И,
10,
in,in
10,10,1
ιο,ιο,ι
10,1П , 1
1,1
10,1
1
,15, 1,15, 1)
,15,15, 1,15)
,15,15,15,15)
,15,15,35,15)
,15,35,35,35)
,15, 1,35
,10,24,10
1,H, 1. 1
0,10, 1,10
0,10,10, ]
0,10,10
1,10, 1,
0,10, 1
15
15
15
15
15
15
15
Π 5 ,15
(15,15
(15,15
(15,35
(15,35
(15,35
( Ο ,
( Π,
( Ο,
( Ο,
( Ο,
ί ο,
( ο,
( ο ,
( ι.
1,1
1,1
1,1
15,1
15,1
15,1
15,1
15,1
3 5,1
15,3
15,3
15,3
1,14,
14,
14,
Η,
14,
14,
14.
1,
1.
0,10,
ο,ιο,
ο,ιο,
5,15,
5,15,
5,15,15
5,15,15
5,15,
5,15,
1 .24
10,
24,
1,
1.
η
24)
ΐ);
ΐ);
1ΐ);
1ΐ);
1)
1)
Μ
10)
10,24)
ι, ΐ);
15, 1);
1,15);
15,15)
1η,
1,
10,
1
ι, ι,
1,15,
1)
1)
15)
lb,
15.
15.
15.
,15,
15,
,15.
Γ
PW
PW
Ρ
Ρ
5,15,15, 1
5,15,15,15,15
5,15,35,15,35)
5,15, 1,15, 1)
15,15,15,15)
35,15,35)
1,15
26,17
21
16
21
16
7,15,16,17,31);
7,15,21,17,30);
1, 1, 1, 1, 1):
1, 1, 1, 1
1)
,18);
,17,23);
,17,18);
,17,23);
.35,18);
I=a
I=a
I=a
I=a
I=a
I=a
G; 1 =
IPRW
IPRW
10100000
10100101
10100101
10100101
1010C101
10100000
1111C000
11110090
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
1010C101
10100101
10100101
1010010"
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100000
10100101
10100000
10100101
10100101
1O1000O0
10000000
10100000
10100000
10100101
10100101
10100101
10100101
10100000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
11110000
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
10100101
101001C1
10100101
10100101
10100101
10100000
10100101
10100000
1P1001U1
10100101
10100000
10000000
10100000
Number 86,140 ,141
near-rings ом Dp.
and (15,11,1,1,15,1,1,1) are the Boolean
L) The quaternion nroup Q = {0,a,2a,3a,b,a + b ,2a + b, 3a + b}:
2a
3a
a+b 2a+b 3a+b
0
a
2a
3a
b
a + b
2a + b
3a + b
0
a
2a
3a
b
a + b
2a + b
a
2a
3a
0
3a + b
b
a + b
За+b 2a+b
2a
3a
0
a
2a + b
3a + b
b
a + b
3a
0
a
2a
a + b
2a + b
3a + b
b
b
a + b
2a + b
3a + b
2a
3a
0
a
a + b
2a + b
3a + b
b
a
2a
3a
0
2a + b
3a + b
b
a + b
0
a
2a
3 a
3a + b
b
a + b
2a + b
3a
0
a
2a

Near-rings of low order 419
ag
«1
«2
«3
»„
«5
«6
«7
«8
«9
«10
«1 1
«12
«13
«14
«1 5
«16
«17
«18
«1 9
«20
«21
«22
«23
«24
«25
«26
«27
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
о
0
0
о
0
0
0
0
0
0
0
0
0
о
a
0
a
a
a
a
3a
3a
3a
3a
b
b
b
b
2a + b
2a + b
2a + b
2a + b
a + b
a + b
a + b
a + b
3a + b
3a + b
3a + b
3a + b
η
2a
2a
2a
0
2a
2 a
2a
2a
2a
2a
2a
2a
2a
2a
2a
2a
2a
2a
2a
2a
2a
2 a
2a
2a
2 a
2a
2a
2a
0
0
0
3a
0
3a
Зэ
За
За
a
a
a
a
2a + b
2a + b
2a + b
2a + b
b
b
b
b
3a + b
3a + b
3a + b
3a+b
a + b
a + b
a + b
a + b
η
2a
2a
b
0
b
2a + b
a + b
3a + b
b
2a + b
a + b
3a + b
a
3a
a + b
3a + b
a
3a
a + b
3a + b
a
3a
b
2a + b
a
3a
b
2a + b
2a
0
2a
a + b
0
a + b
3a + h
2a + b
b
3a + b
a + b
b
2a + b
3a + b
a + b
a
3a
a + b
3a + b
3a
a
b
2a + b
3a
a
2a + b
b
a
3a
2a
2a
0
2a + b
0
2a + b
b
3a + b
a + b
2a + b
b
3a + b
a + b
3a
a
3a + b
a + b
3a
a
3a + b
a + b
3a
a
2a + b
b
3a
a
2a + b
b
2a
0
?a
3a + b
0
3a + b
a + b
b
2a + b
a + b
3a + b
2a + b
b
a + b
3a + b
3a
a
3a + b
a + b
a
3a
2a + b
b
a
3a
b
2a + b
3a
a
2a
2a
0
If we impose the same restrictions as for Dp, there are
just two near-rings (both are Boolean):
1) (0,1,1,1,1,1,1,1); IPRW 10000000 10000000
2) (1,1,1,1,1,1,1,1); IPRU 1010000010100000

420
APPENDIX
Some near-rings on TL.xTL,, = {0 ,1 ,2 ,. . . ,7 }
+
0
1
2
3
4
5
6
7
0
0
1
2
3
4
5
6
7
1
1
2
3
0
5
6
7
4
2
2
3
0
1
6
7
4
5
3
3
0
1
2
7
4
5
6
4
4
5
6
7
0
1
2
3
5
5
6
7
4
1
2
3
0
6
6
7
4
5
2
3
0
1
7
7
4
5
6
3
0
1
2
with
(0,0)
(1 ,0)
(2,0)
(3,0)
(0,1)
(1,1)
(2,1)
(3,1)
= 0
= 1
= 2
= 3
= 4
= 5
= 6
= 7
Let ac
id and αϊ be given by oi(x,y) = (x,0]
1) (ο,ο,ο,ο,ο,ο,ο,ο:
2) (0,0,0,0,1 ,1 ,1 ,1:
A·' 1o = 1г
AOR; 30 = Ί
= {0,2,4,6}
, = {0,2}
Some near-rings on Ζ,χΖ,χΖ, = ^ '1 '2 '''' ·7^
+
0
1
2
3
4
5
6
7
0
0
1
2
3
4
5
6
7
1
1
0
3
2
5
4
7
6
2
2
3
0
1
6
7
4
5
3
3
2
1
0
7
6
5
4
4
4
5
6
7
0
1
2
3
5
5
4
7
6
1
0
3
2
6
6
7
4
5
2
3
0
1
7
7
6
5
4
3
2
1
0
wi th
:o,o,o:
:o,o,i:
:o,i ,o:
:o,i ,i:
:ι ,ο,ο:
:i ,0,1:
:i,i,o:
:i,i,i:
Some endomorphisms
0
1
2
3
4
5
6
7
a0
0
0
0
0
0
0
0
0
ai
0
1
2
3
0
1
2
3
a2
0
0
0
0
2
2
2
2
a3
0
0
0
0
4
4
4
4
ai>
0
1
2
3
4
5
6
7
0
0
0
0
6
6
6
6
0
1
0
1
4
5
4
5
0
0
2
2
4
4
6
6
0
1
2
3
1
0
3
2
1) (0,2,0,3,0,2,0,3); ^Q = {0}; ^]/2 = {0,4};
7i - Ь - N
Remark: for this near-ring N we have
?,( ^(N)) = {0} + ^(N).
2) (0,2,3,3,4,1,5,5); ^ = ^1/2 = {0};"^ = ^2 = {0,1,2,3}
3) (4,6,7,1,4,4,7,7); 0; All radicals = {0,1}
4) (2,2,8,8,2,2,8,8); A; All radicals = {0,1,4,5}

Near-rings of low order 421
0) Some near-rings on Ал = {0,1,...,11}
+
0
1
2
3
4
5
6
7
8
9
10
11
0
0
1
2
3
4
5
6
7
8
9
10
1 1
1
1
0
3
2
5
4
7
6
9
8
11
10
2
2
3
0
1
6
7
4
5
10
1 1
8
9
3
3
2
1
0
7
6
5
4
11
10
9
8
4
4
7
5
6
8
1 1
9
10
0
3
1
2
5
5
6
4
7
9
10
8
11
1
2
0
3
6
6
5
7
4
10
9
1 1
8
2
1
3
0
7
7
4
6
5
1 1
8
10
9
3
0
2
1
8
8
10
1 1
9
0
2
3
1
4
6
7
5
9
9
11
10
8
1
3
2
0
5
7
6
4
10
10
8
9
1 1
2
0
1
3
6
4
5
7
1 1
1 1
9
8
10
3
1
0
2
7
5
4
6
Again, we list endomorphisms row-wise.
a0
Cti
a2
a3
α ι»
a5
aG
a7
a8
a9
Cti 0
an
Ctl2
Cti 3
an»
Cti 5
«1 G
Cti 7
Cti β
Cti 9
Ct2 0
a2 ι
a22
a2 3
Ct2 i»
a2 5
a2 g
a2 7
a28
a2 9
a3o
Ct3 1
Ct3 2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
2
0
0
0
0
0
0
0
0
0
2
2
2
2
3
3
3
3
2
1
1
1
3
3
3
3
1
1
1
1
2
2
2
2
3
0
0
0
0
0
0
0
0
0
3
3
3
3
2
2
2
2
3
3
3
3
1
1
1
1
2
2
2
2
1
1
1
1
4
0
4
5
6
7
8
9
10
1 1
4
5
6
7
8
9
10
1 1
8
9
10
1 1
4
5
6
7
4
5
6
7
8
9
10
1 1
5
0
4
5
6
7
8
9
10
1 1
5
4
7
6
9
8
11
10
10
1 1
8
9
6
7
4
5
7
6
5
4
11
10
9
8
6
0
4
5
6
7
8
9
10
1 1
6
7
4
5
1 1
10
9
8
9
8
1 1
10
7
6
5
4
5
4
7
6
10
1 1
8
9
7
0
4
5
6
7
8
9
10
1 1
7
6
5
4
10
1 1
8
9
1 1
10
9
8
5
4
7
6
6
7
4
5
9
8
1 1
10
8
0
8
10
1 1
9
4
7
5
6
8
10
1 1
9
4
7
5
6
4
7
5
6
8
10
1 1
9
8
10
1 1
9
4
7
5
6
9
0
8
10
1 1
9
4
7
5
6
9
11
10
8
5
6
4
7
6
5
7
4
10
8
9
11
1 1
9
8
10
7
4
6
5
10
0
8
10
1 1
9
4
7
5
6
10
8
9
1 1
7
4
6
5
5
6
4
7
1 1
9
8
10
9
1 1
10
8
6
5
7
4
1 1
0
8
10
1 1
9
4
7
5
6
11
9
8
10
6
5
7
4
7
4
6
5
9
1 1
10
8
10
8
9
1 1
5
6
4
7

422
APPENDIX
Subgroups
A)
B)
О
D)
E)
{0}
{0,1}
{0,2}
{0,2,3}
{0,4,8}
F)
G)
H)
I)
J)
{0,5,10}
{0,6,11}
{0,7,9}
{0,1 ,2,3}
{0,1 ,... ,11}
A, I, and J are normal; I is the commutator subgroup;
A i s the center.
1) (0,0,0,0,0,0,0,0,1,1,1 ,з:
Nilpot
Non-ze
Idempo
Left i
Right
Regula
Centra
Q u a s i r
Left d
Right
D i s t r i
Subnea
N-subg
Left i
Monoge
Ideals
Nil le
Modula
Prime
Maxi ma
Maxi ma
0-modu
1-modu
2-modu
0 - r a d i
1/2-ra
1 -radi
2-radi
Nil id
Quasi-
T h i s η
2 - s e m i
α0,αι ,
are th
automo
ent e
ro d i
tent
d e η t i
i den t
r el e
1 ele
e g u 1 a
i ν i s о
d i ν i s
b u t i ν
r - r i η
roups
deal s
ni с N
ft id
r 1 ef
ideal
1 N-s
1 lef
lar 1
lar 1
lar 1
cal :
dical
cal :
cal :
eal s :
regul
ear-r
si mpl
a3 ,a9
e onl
rphi s
1ements:
visors :
elements:
ties:
i t i e s :
ments :
ments :
r elements:
rs of zero:
ors of zero
e elements:
gs :
0,1 ,2,3,4,5,6,7
0 ,8,1 1
0,11
none
none
0,8,1 1
0,1 ,2,3,4
1 1
none
none
0,1 ,2,3,4
A,B,C,D,E,G,I,J
A,B,C,D,I ,J
A,I,J
A,E,G
A,I,J
A,i
E,G,I
-subgroups:
eal s:
t idea 1s :
S :
ubgroups:
t i deals :
eft i deals
eft ideal s
eft ideals
A,I
ar left i deal s : none
ing is zero-symmetric, subdirectly irreducible,
e, but not 2-primitive and not d.g..
= id are near-ring-endomorphisms; they
у N-endomorphisms as well. Only a9 is an
m.

Near-rings of low order
423
2) (0,0,0,0,1 ,1 ,1 ,1 ,5,5,5,5
0,1 ,2,3,4
0,4,8
0,4
none
none
0,4
al 1
al 1
al 1
al 1
all
elements
elements
elements
elements
elements
В ,C,D,Ε , I ,J
B,C,D,E,I ,J
I,J
Ε
I,J
I
except 4
Nilpotent elements:
No η - ζ e г о divisors:
Idempotent elements:
Left identities:
Right identities:
Regular elements:
Centra 1 e1ements:
Quasiregular elements:
Left divisors of zero:
Right divisors-of zero:
Distributive elements:
Subnear-rings:
N-subgrouos:
Left ideals:
Monogenic N-subgrouos:
Ideals:
Nil left ideals:
Modular left ideals:
Prime i deals:
Maximal N-s ubgroups:
Maxima 1 left ideals:
0-modular left ideals:
1-modular left ideals:
2-modular left ideals:
0 - r a d i с a 1 :
1/2-radical :
1 - radical :
2-radical:
Nil ideals:
Quasi-regular left ideals: none
This near-ring is zero-symmetriс
tive and subdirectly irreducible.
α0,αϊ,019,0121,0125 near-ring endomorphisms; αο,αι,α5,α9,
cti 3 ,c«i 7 ,c«21 ,'-t2 5 ,ч 2 q N - e η d о mo r ρ h ί s m s . Automorphisms are
o.9,ct2i (order 3, inner automorphism determined by "N
and a,25 (order 3, inner automorphism determined by
distributive, commuta-
5).
1 ,9,9,9,1 ,1 ,9,9,1 ,9,9,9!
Nilpotent elements:
Non-zero divisors:
Idempotent elements:
Left identities:
Right identities:
Regular elements:
Centra 1 elements :
Quasiregular elements:
Left divisors of zero:
Right divisors.of zero:
Distributive elements:
0
all elements except 5
all elements except 5
none
1 ,2,3,5,6,7,9,10,1 1
all elements
none
0
0,1 ,2,3
0,4,8
0

424
APPENDIX
Subnear-ri ngs:
N-subgroups:
Left ideals:
Monogenic N-subqroups:
Ideals:
Nil left ideals:
Modular left ideals:
Prime i dea1s :
Maximal N-subgroups:
Maximal left ideals:
0-modular left ideals:
1-modular left i deals:
2-modular left i deals:
0-radical :
1/2-radical:
1 -radical:
2-radical:
Nil ideals:
A,B,C,D,E,I,J
Α,Ε,ϋ
A,I,J
E,I
A,I,J
A
A,I
I
Ε
I
I
none
none
A
J
J
J
A
Quasi-regular left ideals: none
This near-ring is weakly commutative (=(P3)), without
nilpotent elements, subdirectly irreducible; NN is simpli
and cyclic.
Near-ring-endomorphisms are α0,αι,α5,αιο = id;
am are N-endomorphi sms .
αϊ and
Some more near-rings on A
4
5
6
7
8
9
10
1 1
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,0
0,0,9
0,9,2
1 ,1 J
1,1,1
1 J ,1
9,9,9
,0,0,
,0,0,
,0,0,
,0,0,
,0,0,
,0,0,
,0,0,
,0,0,
,0,0,
,0,0,
,0,1 ,
,0,1 ,
,0,1 ,
,0,1 ,
,0,1 ,
,0,1 ,
,0,1 ,
,0,9,
,0,9,
,0,9,
,13,9
1 ,25,
,1 ,1 ,
,1 ,1,
,9,1 ,
,9,9,
0,0,0
0,0,0
0,0,0
0,0
0,9
0,9
0,9
9,9
9,9
9,21 ,
1 J ,1
1 ,1 ,4
1 ,3,3
1 ,4,4
1 ,9,4
2,9,9
9,21 ,
9,11 ,
9,12,
10,20
,13,9
0,0,0
1 J ,1
9,21,
9,9,9
9,9,9
,0,9,
,1 ,4,
,9,9,
,0,0,
,0,19
0,0,9
0,18,
,0,13
9,19,
25,0,
,1 ,9,
,5,6,
5 5
,1,4,
,5,6,
,1 ,9,
25,1 ,
11 ,9,
12,32
,18,9
,13,1
,0,0,
,1,1 ,
25,1 ,
,1 ,9,
,9,9,
4·
21,25)
9,12)
10,10)
0,17)
,0,19)
,0,10)
0,20,0)
,13,13)
9,0,19)
21 ,25,9
21 ,25)
6,6)
5,8)
9,12)
29,5)
2,9)
21 ,25,9
9,11,11
,31 ,31 ,
,18,10,
3,9,9,1
9,21 ,25
1 ,1 )
21 ,25,9
9,9)
9,9)
32:
20:
3)

Near-rings of low order
P) A remarkable near-ring of order 32:
Let N be the direct product of the near-ring 1) of N)
and of 10) in Ε). Ν is of minimal order w.r.t. having a
Jacobson-type near-ring radicals different. See also
5.67(v).
Q) The number of near-rings on some groups of small order:
Μ is the metacyclic group of order 12.
Order
1
2
3
4
5
6
7
8
9
10
1 1
12
Group Γ
Z1
Z2
Z3
Z4
Z2x22
Z5
Z6
S3
Z7
Z8
Z4xZ2
(z2f
D8
Q
Z9
Z3xZ3
Z10
D10
Z11
Z12
A4
Μ
Number of
nr ' s on г
1
3
7
17
99
29
98
160
112
350
6982
a lot
9308
4692
1 190
8907
1200
3454
1312
5522
8728
6571
Number of isomorphism
classes of nr's on г
1
3
5
12
23
10
60
39
24
135
115
many
1447
281
222
264
329
206
139
1749
483
824

426
222 REMARKABLE EXAMPLES AND COUNTEREXAMPLES
The following list contains some examples of near-rings with
remarkable properties. They are ordered according to their
"dominant" property, following the course of this book.
Some remarks: It is very difficult to define the borderline
between examples and theorems. It is equally difficult to decide,
which property is "dominant". So I tried to do my best and I
appologize for all possible cases of disagreement. Covered are
only papers which deal with near-rings (and not the ones on
near-fields, ets.).Excluded are examples which are already
mentioned in this book."E" means: English version and "M13"
means that the page number is 13 in the manuscript of this paper
(when it has not appeared yet).
No claim is made concerning completeness.
1) Berman-Si1verman 1, 29/30: Various examples of invariant (but not normal) subgroups etc.
2) Pilz 10, 100: Another multiplication on R[x] turning it into a near-ring
3) Choudhari 1, 10: Near-rings on TL^ χ ΖΖ_ with N-subgroups which are not left ideals
1) Choudhari 1, 13: h £ Ηοπι(Ν,Ν'), where h(N) is not a left ideal of N'
5) Heatherly 7, 350: N = (ZZ g ,+ ,«) is simple with x3>x, regular and N,= {0}.
6) Heatherly 7, 351: N=(ZZ? , + ,») is simple with x4»x.
7) Malone 1, 31: |N|=18, Νς<3 Ν, the automorphism of N can be "mated" by the ones from N
and N .
с
8) Malone 4, 37, 39- N near-rings, where the homomorphisms on N and N can and cannot be
"mated" to give a homomorphism on N.
9) Prehn 3, M3: N=ZZ2 has an identity, but „S, is not unitary
10) Ferrero-Cotti-RinaIdi 1, 78 : N finite, simple, all proper right ideals ire maximal,
but there sre more than 2 of them

222 remarkable examples and counterexamples
427
17)
N dg. without Ore condition, but with left cancellation law
Π: Ν nr. on S3 with left- Ore condition, without left cancellation law
N nr. on TL . : properties as above; not embeddable into a near-field
10 Γ [ x ] J ■ properties as above
N ■- IR Q I [ χ . ] : Left Orp condition, left cancellation, no identity,
multiplicative center = 0
Baskaran 1, 351: (N,+) finite, simple, (0:0)=N, (0:x)=N f<?r some x/0, but some y/0 with
(0:y)-0 isnot one-sided identity.
Ugh 2, 1368 and 4, 11: N near-ring on 7Z4wiih Ν / ί 0 i , ?x/0 iy/0: xy^O, no left ideals
11)
12)
13)
14)
15)
Graves ]
Graves 1
Graves 1
Graves 1
Graves ]
. 7:
, 11
, 11
, 12
, 30
19)
20)
21)
22)
23)
24)
24
26)
27)
28)
29)
30)
31)
33)
34)
form a subgroup
Ugh 6, G67: N finite with 1, (-1)χ=χ«φ χ-0, Ν abelian, but not integral
Ligh-Malone 1, 375: N near-ring υ η 11 г , not every element has a unique square root
Marin 1, 136: N = N near-ring on /I^ with 1 and vx/0 iy: yx=l, but N =)01
Maxson 1, 26: N a near-ring on 71 - with some χ which is nor divisor of zero, but not left
cancel IabIe.
McQuarrie 2 (all): N infinite, ηon-abelian, with 1 and r (-1) - r ■%► r = 0.
Pilz 3, 165: N -_ R|x] with a,c L N but no h r N with ,■ r l N: br = a(r+c)
Plasser 1, 111.16: A non-ring on ΖΖ-,χ TL „ with left identity e, where - e ь Ν ,
; i dleman 3, 383
Beidleman 1, 49: г with DCCI, but not DCCN; N=N
n о
Beidleman 1, 50: E(rjr's with DCCI dnd /or ACC1 and with DCCI, but not ACCI.
Beidleman 9, 203: N dg. with 1, tin DCCI, по АСС1. The sum of all nilpotent ideals is not
nilpotent. J2(N) is qr. and (N,+) is nilpotent.
Heather Iy Э , 205: Fq[ χ , (F a fiel d) has not the DCCI , is integral , but. not regular
Ligh-Ramakotaiah-Reddy 1, 123: N near-ring on the nilpotent group D , but all (0:χ)
are =[0; or =N.
Beidleman 1, 103: N t ?' л К, dg, N a direct sum of simple left ideals, but not a direct
sum of N-simple left ideals
J, but Nr does not split
Choudhari-Goyal 1, Mb: N nr. on S without proper left ideals (so L =L) for all left
Poli η 2, 71: For M,M"
ideals), but not Μ =Μ for all N-subgroups
N let Mo M' be the N-subgroup generated by MM', о is not
associative.
Oswald 1, 61: N d.g. which is "strictly semiprime" (i.e. {0'» is the only N-subgroup Μ
M={0J *
with Μ =(01), but not "strictly prime" (v M,M" <N Ν: ΜΜ'= Ο
ν Μ'-ίΟ})
Beidleman 9, 204: N dg . with 1, Φ{ Ν ) - . . .=}2(N) is nilpotent, the sum of all nilpotent
left ideals is an ideal
Betsch 3,38: N <_ M(ZZ4)
nilpotent.
is of type 0, not type 1. 31/2(N)»0\ bu t "3 χ (Ν) =?2 (N) is not

428
APPENDIX
39} Laxton-Machin 1, 22 В: N dg. with just one N-subgroup of type 0, but not of type 2 .
40} Rhabari 2, 21: N dn., "weakly prime", not "weakly riant simple", without faithful minimal
41}
42}
43}
44)
45)
46)
48)
49)
51)
52)
Blackett 1, 32: N=N finite, eNe not closed w.r.t.
о
multi pii ca tions .
not all endomorphisms are right
and has N-subgroups which are not left ideals.
Fain 1, 57: 3 orthogonal idempotents in M( 2~} which induce a direct decomposition
Lyons 4, 5B2: N d.g.n.r. on DR with idempotent e, where the summands A and Η in the
Peirce decomposition induced by e have AM=iO)
Maxson 1, 4B: Example of an idempotent e in Μ ( 2 , ) , such that 1-е is not idempotent
Holcombe 2, 22: N"N0» finite, 0-primitive, but without right identity
Beidleman 7 , 101: N dg, finite with 1. N is 0-primitive and prime, but neither simple
nor strictly prime nor 2-primitive.
Betsch 3, 37: Ν < Μ (2^) , l-primitive, simple with right identity, but 32(N)=N.
Ramakotaiali 3. 25: N <_ Μ {2-} , N 1-primitive without identity, not dense in M, (2.} .
Beidleman 7, 103: N dg., infinite with 1 . N is 2-primitive and all (0: χ) ( x/0 ) are prime
and modular left ideals.
Ramakotaiah 2, 135: N <_ Μ (2.) =: Μ, Ν 2-primitive with 1; Μ is completely reducible.
Polin 1, 270 (E 263): N 2-primitive, simple with 1 and a minimal N-subgroup 'and left
ideal), but without 2-sided invariant subgroups, containing a
dense non-simple subnear-ring N1 which contains a minimal N-subgroup,
but no non-trivial 2-sided invariant subgroups.
53) Betsch 3, 38: N < M(24) , N 2-primitive: left ideals: 0 , (0 ; 1) , (0:2),N .
54} Holcombe 2, 20: N=N finite, 2-semisimple but without right identity,
55} M. Johnson 7, 337: N not regular, but 2-semisimple with 0CCN.
56} Hartney 4, 20: "3U(N) k J2 (N} = S(N) (cf. p. 165) =:1, where N/1 has no nilpotent left
ideals.
57} Hartney 4, 21: N do., JQ(N} * S(N} * J2^)m
5B} Betsch 3, 39: N=NQ finite, 0 = ^0(N} / 3J/2{N}, which is nilpotent
59} Laxton 4, 16: N finite, dg., ^{N} not nilpotent, 31/2(N} not an ideal
60} Beidleman 1, 103: Ν ε?),, finite, dg., 32{N) is neither nilpotent nor small nor strictly
small, with a max. left ideal which is not regular or with ^{N} not qr.
61} Beidleman 2, 228: N dg., finite with 1, 32(N) neither nilpotent nor qr.
62} Beidleman 2, 228: N dg., finite with 1, ?2(N} is qr. and = radical subgroup {= 0 max.
N-s ubg roups}
63} Beidleman 8, 94 + 97: N d.g., finite with 1; 32(N) is not nilpotent, but "semi -ni1 potent"
and a small ideal, but not a small left ideal.

222 remarkable examples and counterexamples
429
65)
66)
67)
70)
Beidleman 8, 100: N dg. with 1, countable, U2(N) is not nilpotent, 3j/2(N)i ^2(N); N nas
only finitely many N-subgroups.
Beidleman 9, 204: N dg. with 1, finite, ί (N)=0, 3j/2(N) is not a" ideal, 32(N) is not
nilpotent, the sum of the nilpoitent left ideals is not an ideal.
Beidleman 9, 215: N dg. with 1, S2(N) is qr. and nilpotent.
Laxton 3, 41: Finite d.g. near-rings with and without nilpotent 32(N).
Laxton-Machin 1, 229: N dg., where tt|,2(N) is a direct summand of 32(N)
Maxson 1, 66, and 3, 200: N a near-ring on ZZ?x ZZ2 , 32(N) / {0), N/32(N) is a near-
field { = 2?) , N is not local and J2(N) is not strictly small .
Scott 12,3; N distrib. generated by a single ηί Ν, with n2=1, N finite, 0-primitive,
but ^ 2 ( N } Is not nilpotent.
beidleman 3,383: N finite d.g. with 1 and 02(N) strictly small {i.e. N is the only
72) Choudhari-Tpwari 3, M7: N dg., finite with 1, JQ(N)cJ2(N) = G-radical
73} Hartney 2, 219: N finite dg. with 1, not simple, but s-primitive
74} Laxton-Machin 1, 227: Examples of finite d.g. near-rings with "critical radical" - {0}
(there is no non-zero ideal containing a non-zero nilpotent left
i deal)
75) Meldrum 4, 336: Examples of group d.g. near-rings
76) Meldrum 7, 294: N d.g. with Nr simple, non-trivial, faithful, but N has a nilpotent ideal
t (0).
77) H.Neumann 2, 51-69: Ideals in endomorphism near-rings Endr on reduced free groups.
78}
79)
80)
81}
82)
84)
85}
86}
B7)
88)
89)
90)
91)
92)
93)
Ligh-Luh 1, 21: Examples of D-near-rings which are not d.g.
2
Meldrum 7, 292: N dg., N = {0}, N has only trivial d.g. representations
Tharmaratnam 4, 137/38: A d.g. near-ring N with (Nd,.) a group, but „N is not simple
(and conversely)
Mahmood 3, 80: D.g. near-rings without identity over inverse semigroups
Heatherly 1, 48: N < E(S3), but N is not d.g.
Meldrum 3, 474: N d.g. without faithful representation
Blackett 4, 602: 4 simple subnear-rings on H( щ
Blackett 4, 606: Ν < Μ (t), N simple
Blackett 5, (all): N countable and dense in Mcont(IR )
Blackett 6, (all): dense s ubnea r-r i ngs of Μ nt(]R )
Clark 1 , 390: M{ZZ4) in detail
Clark 1, 391: M(Z6) in detail
Clark 1, 394: M(S3) in detail
Gorton 1, 75: Near-rings of constant and "nearly constant" mappings and polynomial
near-rings {λ-completeness, simplicity)
M. Johnson 3, 389: Ν -- Μ (Γ), where left ideals and normal N-subgroups coincide
M. Johnson 3, 390: Ν = Μ (г), where left ideals and normal N-subgroups do not coincide.

430
APPENDIX
94) Gurthrie 1, (all): E(Dg)
95) King 1, (all ): E{qg)
96) Lyons-Malone 1, 75: E(Sj)
97) Fong 1 and 2, (all): E(S )
9B) Malone 10, (all): Examples of groups г where E{r) is a ring
Maxson 14, 296: Examples of groups r with E(i) a ring, but E. г not cyclic.
99)
1D0) McQuarrie 5, last page: Examples of г of order 27, 32, 64 and 81 where A{r) is a ring
McQuarrie 1, 8,9: Subnear-πngs of R[x)
So 1, (all): Examples of substructures, ideals, homomorphisms, radicals etc. of polynomial
near-π ngs.
103} Pilz-So 1, 'i'i : R-subgroups of Rlx] which are not left ideals
104) Pilz-So 1, 65 : ι < R[χ] which is even a maximal subgroup, but not a ring ideal.
1°5) Brenner 1, (all): Ideals of ZZ2[x]
101
102
106)
107)
108)
109)
110)
111)
Ugh 4, 10: N near-ring on ZZ2 χ ZZ2 , N-simple, but not a near-field
Anshel-Clay 2, 172: Planar (integral and not integral) near-ings on ZZ
Clay 10, 324: N finite, planar, non-abelian
Lawyer 3, 89: Integral and non-integral near-rings on 1{ъ")
Szeto 3, 271,273: Examples of non-planar, but strongly uniform near-rings
P11 ζ 22, M5: A planar near-ring N on ( IR ,+ ) with ?,/2(N) = (0) , but ?2(N) = N
112) Pi 1 ζ 22, M5: A planar near-ring N on (IR,+ ) with } (N) = 3,(N) / (0) and N.
is planar and integral.
113) Ferrero-Cotti-Pel legrini 1, M3: N planar, 1 < N, N/ not planar
114) Ferrefo-Cotti-Pellegrini I, M7: N finite, planar, not integral, but with proper ideals
t A (8.86!)
115) Ferrero 13, 429: New block designs from planar near-rings
116) Bell 1, 367: N = N , xn'x^ = x, (N,+) abelian, but not every homomorphic image has a
non-zero central idempotent.
117) Bell 1, 367: N = N , x"'*' = x, no identity, (N,+) not abelian.
118) Choudhari 1, 126/127: 2 non-trivial duo-near-rings .
119) Choudhari-Goyal 1, M6 . Duo near-rings on ZZ ^ which are not "strictly duo" (= not every
N-subgroup Ts an ideal).
'20) Heatherly 7, 203: N near-ring on ZZ without nilpotent elements, but all χ = χ. Ν is
N-simple and intergal, but not a near-field.
121) ugh-Utumi 1, 113: Near-rings with aN = aNa for all a, but not Na = aNa and conversely.
122) Plasser 1, I1I.3: Examples of IFP-near-rings
123) Plasser 1, III.8: A strong IFP-near-ring which is not weakly commutative
124) Plasser 1, III.39: Ν φ Ν near-ring on ZZ 2 χ ZZ2 , where each element is a power of itself.
125) Ramakotaiah 2, 132: Ν < Μ (ZZ.) , a non-ring which is "π-requl ar"

222 remarkable examples and counterexamples
431
126)
127)
130)
131)
1 32)
133)
134)
1 35)
136)
1 37)
138)
139)
HO)
141 )
142)
143)
144)
145)
146)
Rati i ff 1, 5: Examples of 3-near-πngs (x = x, 3x = 0)
Rati i ff 1, 13: Examples of 5-near-rings
Rati i f f 1, 26: N a Boolean near-ring, η .- Η, ρ(π) : - (p| pn = ρ), Ρ(1 + η) ^ P(1+m), but
P(n) f P(m).
Adams 2, 182: Non-integral domains cm nilpotent and non-nι 1potent groups (finite and
infinite) and on non-abelian groups of order < 1000.
Heatherly 5, 151: Integral near-ring on ZZg (hence of composite characteristic)
Heatherly-01ι νier 1, 218 and Olivier 2, 15: Integral near-rings on ZZ~ , 7Z 1,- and ZZr.,3
Hea ther 1 y-01 i ν i e r 1, 220 and Olivier ?, IB: Some integral near-rings of order ·- 32
Heather1y-01iνier 2, 88' Two non-isomorphiс integral near-rings on ZZ.- with the same
"Sylow-decompos i t ion",
Hea the rl y-01 i ν i e r 2, 90: N finite, integral, but (N, + ) is not mlpotent
Ligh 4, 32: N integral near-ring on ΖΖΓ , not idempotent and not a near-field
Olivier 2, 22: N d.g. with 1, (N,t) not nilpotent, N integral with left cancellation
Whittington 1, {all}: All integral near-rings of order -- 32
Choudhari-Tewari 3, M6: N a . a . n . r . , not every G-regular element is in the G-radical
Lawyer 1, 375: A near-ring on ZZ (S")
Lawyer 1, 376: "Trivial" near-ring multiplications on groups
Malone 8, (all): All d-g. near-rings on the infinite dihedral group
Kiisel 1, 27: Archimed. ordered near-ring on ( IR, +) without identity
Iz 1, 46: A vector-ordered near-ring
Iz 3, 162
Iz 3, 167
Iz 7, 341
Η 7) Ρ iι ζ 8, 2 5 3
149)
15D)
151)
152)
153)
154)
155)
156)
157)
N ordered with a < h, с > 0 but ca > cb
Examples of fully ordered near-pings N with proper Nc.
N fully ordered, N , Nj go ^ _ b(]t ^ and ^ are „n-;ero-near-rings .
N., N? ordered near-rings. An example anda counterexample that these orders
can be continued to order Ν ^ q ν ? -
K.B.P. Rao 1, 242: A strictly (partially) ordered Nr
K.B.P. Rao 1, 243: A strictly fully ordered Nr ·
Rhadhakrishna-Bhandari 1,2: A fully ordered near-ring.
Rhadhakrishna-Bhandari 1,4: A lattice-ordered non-ring with ал b * 0, c;0 -?аслЬ =
- сал b = 0 .
Graves 1, 37: N a Oickson near-ring associated with D|x] (D a division ring), which is
Euclidean.
Maxson 8, 156: A Dickson near-ring which is not a near-field .
Maxson 8, 161: Examples of Dickson near-rings from "ί-rings" .
Maxson 13, 410: Examples of homomorphisms of Oickson near-rings .
Choudha ri-Goyal 1, M3: A strongly regular near-ring on Ί...
Heatherly 8, 353: The number of regular near-rings of order < 7

432
APPENDIX
158) Μ. Johnson 7, 333: N near-ring on ZZg with Ν φ Nq , N regular, ^1/2(N) = 3>2(N) = (0:0) =
- unique non-trivial left ideal. The intersection of the maximal
N-s ubgroups is - {0,3).
159) Mason 5, (-): Μ (г) and M(r) are regular, but not "strictly regular".
160) Szeto 4 , 68: N integral, regular, Ν , = {0}, N not a subdirect product of near-fields
S61) Maxson 1, 12 and 3, 203: N = F [xl (F a fi eld) is local with J~(N} t N and distributor
2 3
ideal = {a 2x + a τχ + ■ ■ ■ *
16?) Esch 1, 16: N nr. on S,, where the additive commutator of two left ideals is not a left
ι dea 1
163) Esch 1, 31+61: N nr. on S^, where the distributor subgroup of N is not normal; iterated
distributors never yield 0.
164) Esch 1, 32+37+85: N nr. on D., where the distributor subgroups of left ideals are left
ideals, the third distributor subgroup is 0; another kind of distributors
does not terminate at 0.
165) Ferrero-Cotti 5, 266: N finite, N1 :- [Ν,Ν] Δ Ν, but N/N1 does not fulfill xy = zt φ yx=tz
Daric 3, Ml2: N distributive on D.; its defect and commutator
Ferrero 1, 10: N distributive nr. on S~ with exactly one "Sylow" ideal
Heatherly 6, 65: Distributive near-rings on ποη-abelian groups
Heatherly 6, 66: Smallest non-trivial distributive near-ring (on S^).
Heatherly-Ligh 1, 450/451: Examples of pseudodistributive near-rings which are not
di stri buti ve
Jones 1, 5: Distributive near-rings on Q
166)
167)
168)
169)
170)
Π1)
172)
173}
174}
175)
176)
177)
178)
179)
180)
181}
182)
Jones 1, 6: Distributive near-rings on A ^
2
Jones 1, 7: N a non-distributive near-ring on Dg with χ = 0
Jones 1, 10: Distributive near-rings on D-2
2
Jones 1, 13: N distributive, N a cyclic group
Jones 1, 16-3 В: All distributive near-rings of order < 15
L i gh 17, 383: Near-rings which are "n-distributive", but not distributive; the nxn-
matrtces over N form a near-ring.
Stefanescu 1, 439: A proper non-associative, but distributive "near-ring"
Ligh-McQuarrie-Slotterbeck 1, 89: An N-system, not a near-field
McQuarrie 1, 13: A proper N-system
McQuarrie 1, 26: AnN-system N in Ε [χ] with a homomorphic image which is not an
N-sys tern
McQuarrie 1, 30: Two examples of N-systems N f N with modular ideals 1 such that N/1
is an N-system
McQuarrie 1, 36: A near-ring on ZZ3 χ ZZ 3 with a halvable idempotent e, but the
multiplication by (-e) is not fixed-point-free.

222 remarkable examples and counterexamples
433
185) Be i dleman 8, 100: N d.g. with 1, uncountable, but with only finitely many N-subgroups
186) Deskins 1, 827: N infinite with exactly one minimal left ideal.
187) Su 2, 148: Examples of topological near-rings
188) Tharmaratnam 2, 301: Examples of topoloqical (R,S)-groups
189) Tharmaratnam 3, 121,122: Examples of topological d.g. near-rings
19D) Tharmaratnam 3, 124: A topological d.g. near-ring, which is no d.g. near-ring
191) Tharmaratnam 3, 134: Embeddings of topological (R,S)-groups
192) Tharmaratnam 4, 137: N a discrete topological division d.g. near-ring
193) Ma gill 8 (all): Examples of near-rings on topological groups.
194) Banaschewski-Nelson 1, 21: N dg., without non-trivial injective unitary N-groups
195) Sanaschewski-Nelson 1, 22: N dg., with injective N-groups, but not every unitary
N-group has an injective extension
196) Mason 3, 46: Injective objects in the category of ZZ -groups
19?) Maxson 1, 53: Nr projective, but not "strictly projective"
198) Oswald 6, 268: N d.g. with 1 with a left ideal L and some г such that
VhtHom (L.rpfcr luL: f(l)-vl, but N f is n°t injective
199) Maxson-Smith 1, 32/38: Some maximal N-subgroups of Ν = ΜίΛι(Γ) (αεΑυΐ(Γ)); Ν has a
ni1 potent 2-radi ca 1 .
Maxson-Smith 5, 223: A bicentralizer- near-rinn %([), which is 2-seimisimple,
but not simple.
Maxson-Meldrum 1, M3: A near-field N M°("), where S is not a subnroup of Aut(l').
4axson-Mel drum 1, Μ 15: M.([) a near-field, but -1 is not abelian.
200)
201)
202)
203)
204)
205)
Maxson-Oswald 2, M2: Ms(-) a near-field, S^End(^), each
nene ra tes
VAv) ■
207}
208)
209 }
Maxson-Smith 1С, 147: A simple subnear-rinn Μ of Mi") such that there is no A£Aut(")
with N - Цд(ι ).
Maxson-Smith 10, 148: A subnear-fιe1d N of Μ (г), where W:=Mfl . . Лг) is a near-field
о ftut|,ii )
properly contain inn N .
Maxson-Smith 10, 151: A field F мд(!'Ь such that г iS not a vector space over F.
Mel drum-Zeller I, 187: Мд(г) non-simple with s t (av) - st i'x ) ■·· st(at) = st(v) and >2
conjuqacy classes of stabilizers, but with some >ε ι such that
Π
st (aY) -f hd].
210) Holcombe 8, 24-27: Εndomorphism near-rings in various categories
211) Pilz-So 1, 150: Examples of generalized d.g. near-rings which are not d.g.
?\2) Lockhart 4, 151: A non-exact sequence of endomorphism near-rings
213)
214)
215)
Heatherly-Olivier 3, M2: N a commutative near-ring with some Η <z N such that N C_ Η
and Η is " i ntegra1 "
Jones-Ligh 1, M3: Examples of near-rings on multiplicative semigroups
Olivier 2, 28: Two Η-monogenic near-rings on S ~ with different H, which is not an
additive subgroup

434
APPENDIX
216) Kaarli 9 (all): A countable abelian near-ring N, where ^ (N) has a four-element
1-primi ti ve homomorphic image (hence J0 (Ν ) ^ "}χ{ /'„(Ν)) ) .
217) Kaarli g, Μ 11: N*4o( Zg) , ^(N) = N, but N has a 1-primitive ideal.
218) Frohlich 9, (-): Examples of near-rings arising from formal groups.
219) Maxson 1, 68: A local abstract affine near-ring.
220) Scott 15, 62: A compatible near-rinn on a simple r,-qroup and a Zariski-type topolonv
on it.
221) Scott 22, H2: Examples of central N-subnroups of an N-cirouP.
222) Yearby 1, 111-127: Many more near-rinns on A^.

435
LIST OF OPEN PROBLEMS
1) Generally, determine the structure of our special classes
of near-rings (radicals, complete reducibi1ity , semisimpli-
city, primitivity,...). For instance, what can one say about
the radicals of planar near-rings ?
2) Are all restrictions to zero-symmetric near-rings in this
book really necessary ?
3) Study measure and integration in near-rings (this is
motivated by the Μ (T)-type near-rings).
4) Do 2.63, 2.85, 5.54 and 5.62 hold for arbitrary ideals ?
5) Is Μ (Γ) a near-ring if it contains a subnear-ring of Μ(Γ)
which is dense in Μ (Γ) (cf, 4.53(e)) ?
6) Let N be 2-primitive with ACCL or ACCN. Does N contain a
(right) identity ? Is the topology J discrete (cf, 4.29) ?
7) Are the radicals of direct products the direct product of
the radicals (cf. 5.20) ?
8) Suppose that Ι(Γ) has ACCN, Is Γ necessarily finite (cf,
7,60(m)) ?
9) Recall the 4 problems on page 178,
10) Determine the ideal structure in the polynomial near-rings.
Which ones have the DCCL, DCCN, et cetera ?
11) Do there exist proper near-domains (i.e. those ones which
are notnear-fields - cf. p. 247) ?
12) Find some examples of infinite near-fields which are not
Dickson near-fields.
13) Is 9.21(a) correct without the assumption that N has an
identity (Bel 1) ?
14) Study lattice-ordered near-rings, F-near-rings and vector-
near-rings (cf. p. 343).
15) How can one characterize those near-rings which can be
fully ordered ?
16) Which (partial) orders in a near-ring can be extended to a
full order (cf. (Fuchs)) ? Cf. 9 .152(g ) , (h) .
17) Is 9.133 true without the assumption that N is abelian ?

436
APPENDIX
18) Is a 0-semis imp 1e nr. (with or without finiteness conditions)
tame (cf. 9.171(d)) ?
19) If N is tame, does 70 (N) = ^2^) always hold ? Is N 0-semi-
simple iff it is 2-semisimple ?
20) Is every tame nr. embeddable in (isomorphic to) some S(r)
or С(Г) (see 7.121, 9.166(b)) ?
21) (S.D. Scott) Call N semi tame if there are tame N-groups, the
intersection of whose annihilators is zero (cf. 5.14(a)).
What can be said about semitame near-rings ?
22) (Meldrum-Zel1er) Let М.(Г) be regular such that all
stabilizers are conjugate. Is М.(Г) then simple ? Conversely, if
М.(Г) is simple, is МД(Г) necessarily regular and are all
stabilizers conjugate ? Cf. 9.218.
23) (Betsch) Which subnear-rings N of M.(r) are O-primitive
(on Г) ? What about Ν=Μ.(Γ) ? What happens if N is dense
in МД(Г) ?
24) Under which conditions does МД(Г) s Μβ(Δ) hold (cf. 7.38) ?
25) Is an ultraproduct Π Ε(Γ.) isomorphic to Е(П Г.) ? The same
for А(Г.), 1(1^), Ρ ( Γ Ί- ) and С ( Γ Ί-) . Can a zero-symmetric
near-ring be embedded in an ultraproduct of d . g.near-rings,
and is this ultraproduct d.g. again (cf. 9.285, 6.11, and
problem 9).

437
BIBLIOGRAPHY
PLEASE NOTE
The classification scheme for the рорегь is as follows:
Mear-rings: Л... Additive groups of near-rings, near-rings on given groups
A1.· Affine near-rings
β. . . Boolean near-ring^ and qciiei\ili:ations f p- near-rings, IFP-near-ring?,...)
C... Constructions (Sums and products, subdirect products,...'
С'·· Computer-aided inνesligations
0... Distribut lvely generated near-rings
D1 .. Distributors, distributive elements, commutators, solvability.
D".· Dickson near-rings
Π... Distributive near-imgs
E.,. Elementarv, examples, axiomatics, chain conditions, lattice of ideals, ...
E'.. Embeddings
E". . Endomorphism near-ring (E[!'l, Α(Γ), ί(Γ))
F... Near-fields
F'.. Free near-ring and N-groups
G... Geometric interpretations (coordinatisation, incidence groups, ...)
H... Homological and categorical aspects, extensions, injectivity and projectivity
I... Idempotents, biregular near-rings
I'.. Integral near-rings, near-integral domains and generali;ations
L.. . Local near-rings
M... Modularity
M'.. Multiplicative semigroups of near-rings
N... Nilpotence and non-nilpotence
0... Ordered near-rings
p... Primitive near-rings, N-groups of type у
Ρ'.. Prime (semiprime, completely prime, ...) ideals
P".. planarity
P°.. Polynomial near-rings, near-rings of formal power series
0... Quasi-regularity
Q1.. Near-rings of quotients
R. .. Radical theory
R'.. Regular near-rings
5... Simplicity and semisimpiicitv
5'.. 5ylow-type topics
S".. Relations to sharply transitive groups
T... Transformation near-rings (M(r), MQ(r), MG(r))
Γ'.. Topological considerations
V... Valuations
W... Near-rings without nilpotent elements
X... Other topics
Structures related to near-rings:
Cr.. Composition rings ίTO-Algebras)
Na . . Near-algebras
Nd.. Near-domains (in the sense of "non-associative near-fields"!
Rs.. Other related structures (seminear-rings, ...)
Ua.. Universal algebraic context
Combined classifications give more information on the paper; for instance:
P",F. . . Planar near-fields or
D',R... Radical theory for distributively generated near-rings
The bibliography also contains abstracts of talks or napers nresented at the near-ring
conferences un to 19B2. If vou want to obtain these abstracts or naoers, please write
to Dr. O.Betsch or the autor of this book for the Cberwo 1 fach-abs tracts, to the author
for the Edinburgh-abstracts and to Prof, Ferrero for the Saη-Benedetto-Proceedings,
Near-rinq and near-field conferences un to 1^32:
Oberwol fach .... 19*8 Edinburqh 19 78
Oberwolfach .... 1^72 Oberwolfach 1980
Oberwolfach .... 1976 San Benedetto del Tronto ... 1981
Starred (*) paners denote books or survey articles on near-rings.

438
APPENDIX
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21-36.
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1· Projektive Ebenen liber Fastkorpern, Math. Z. 62 (1955), 137-160. MR 17-73. F,G,Rs
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5. Bemerkungen uber Fastvektorraume, FU Berlin, Lenz-Festband (1976), 28-36. F,G,X
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ANGERER. Josef, Chemie Linz AG, A-4020 Linz, Austria
1. Radikale kleiner Fastringe, Diss. Univ. Linz, 1978, R,A,P',N,Q,D,0,F,
SEE ALSO ANGERER-PILZ P",A',R\C,I'
ANGERER, Josef and PILZ, Gunter
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BASKARAN, S., Ramanujam Institute, Univ. of Madras. Madras-5, India
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8EAUM0NT. Ross Α.. Dept. Math. Univ. of Washington, Seattle. Wash, geigs. USA
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BETSCH, Gerhard and CLAY, James R.
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BETSCH, Gerhard and KAARLI, Kalle
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E.F.O.S.R
R.P.S l
E,P,R,S,M,
P,T
X
Ρ,Τ,Ε1
P.T.I.D'
Τ
Τ,Ι,Ρ
T,E",D'
Ρ
R,S
BETSCH, Gerhard and WIEGANDT, Richard
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to appear.
R,S
BHANDARI, Mahesh Chandra, Dept. Math., Indian Institute of Technology, Kanpur, 208016 India
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BHANDARI, Mahesh Chandra and RADHAKR1SHNA, A.
1. On partially ordered near-rings. Math. Student 43 (1975), 113. 0
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BLACKETT, Donald W., Math. Oept-, Coll. of Lib. Arts. Boston Univ., Charles River
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BRDWN, Harold Oavid, Serre House. Сотр. Science Dept., Stanford Univ., Stanford, Calif.
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CARTAN. H.
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CHAN, G.H., Oept. Math. Nanyang Univ., Sinqapore ?2, Singapore
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CHAN, G.H. and CHEW, Kim L.
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CHANDY, Attupurathuvadakkethil J., 1269 Drift Road, Westport, Mass. 0279D, USA
1. Rings generated by inner automorphisms of non-abelian groups, E"
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CHAD, Dale Zao-Tzu, Inst, of Math., Nat. Tsing-Hua Univ. Hsinchu, Taiwan, R.O.China
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SEE CHAN-CHEW
CHDUDHARI, S.C., Dept. Math. Univ. Alger, Alger, Algeria
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SEE ALSO CHDUDHARI-GOYAL, CHDUDHARI-JAT, CHDUDHARl-TEWARl
CHDUDHARI, S.C. and GOYAL, A.K.
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CHDUDHARI, S.C. and JAT, J.L.
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MR 8Dj-16024.
CHDUDHARI, S.C. and TEWARI, K.
1. On strictly semisimple near-rings, Ahh. Math. Sem. Univ. Hamburg 4D (19 74) S,P
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2. (NB)-prOperty in near-rings, Riv. Mat. Univ. Parma 4 (1979), 29-36. X,N,E,R
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SEE ALSO CDURVILLE-HEATHERLY
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442
APPENDIX
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SEE ALSO DASIC-PERIC
DASIC, Vucic and PERlC, Veselin
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SEE ALSO DI SIEND - DE STEFAND
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SEE ALSO DI SIEND - DE STEFAND.
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3.

444
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P.
P.
H.
P,
P,
Q'
R.
H,
X.
χ
X.
X,
X.
R
,T
.T,R,Q'
.F,T
,T,R
,T,R,Rs
',E"
■ H
,E",E' ,P,R
.1
.1
,A',D,P°
,A',P°
■ A
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KARZEL, Helmut and KIST, Gunter
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KAUTSCHITSCH, Hermann, Math. Inst. Univ. Klagenfurt, A-9D10 Klagenfurt, Austria
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SEE ALSO KAUTSCHITSCH-MOLLER
KAUTSCHITSCH, Hermann and M0LLER, Winfried
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KERBY, William E., Math. Sem. Univ. Hamburg, Bundesstr. 55, 2000 Hamburg, Germany
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6. Near domains and sharply 2-transitive permutatibn groups, Oberwolfach, 1972. Nd,S"
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KIM, W.J., Dept. Math., Kyungpeok Natl. Univ., Taegu, Korea
SEE KIM-PARK
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LIGH, Steve, McQUARRIE, Bruce and SL0TTERBECK, Oberta
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LOPEZ, Kathleen D., Oept. Math. Univ. of Southwestern Louisiana, Lafayette,
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454
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456
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SEE ALSO ANGERER-PILZ, HOFER-PILZ, LI0L-P1LZ, MELDRUM-P1LZ-S0, P1LZ-SC0TT,
P1LZ-S0
PILZ, Giinter and SCOTT, Stuart 0.
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PILZ, Giinter and SO, Yong-Sian
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458
APPENDIX
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SEE ALSO BHANOARl - RA0HAKR1SHNA
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SEE ALSO L1GH - RAMAKOTAIAH - REDDY, RAMAKOTAIAH - RAO, RAMAKOTAIAH - REDDY,
RAMAKOTAIAH - SANTHAKUMAR1
RAMAKDTA1AH, Davuluri and RAO, G. Koteswara
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RAMAKOTAIAH. Oavuluri and REDDY, Venkateswara, Y.
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RAMAKOTAIAH, Davuluri and SANTHAKLWAR1, С
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RAO, G. Koteswara, Oept. Math. Andhra Univ., Postgraduate Center, Guntur-52205 (A.P.), India
SEE RAMAKOTAIAH - RAO
RAO, V. Sambasiva, Dept. Math. Nagarjuna Univ., Nagarjunanagar 522510 (AP), India
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SEE ALSO RAO-SATYANARAYANA
RAO, V. Sambasiva and SATYANARAYANA, Bh.
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REDOY, Venkateswara Y., Math. Oept. A.U.P.G.Centre, Guntur-522005 (A.P.), India
SEE L1GH-RAMAK0TA1AH-RED0Y, RAMAK0TA1AH-RE00Y
R1E0L, Christiane
1. Radikale fur Fastmoduln, Fastrin9e und Kompositionsrinqe, Ooctoral Dissertation, R,E,M,0,Cr,Rs
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SEE ALSD FERRERO-COTTI - R1NAL0I
ROBERTS, Ian, Oept. Math., Univ. of Edinburgh, Mayfield Rd., Edinburgh EH9 3JZ, Scotland
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SEF LIC-H-McQUARRlE-SLOTTERBECK
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N
R.N,I,E
1,N
E.S.N
C,D,D ,E,
S,S',T,W
E",D,1,R.
E,E",M'
Ε,Ν,Χ
l.N
P.D.R
X,T
T.E
Ρ,Χ,Τ,Τ'
Ν,Ε
Χ,Ε,Ν
Χ,Ε
E.F
Χ,Ρ°,Ε"
Χ,Ε
Χ,Ε,С
М.С.Е
,Ε'
,Μ,
,E",1,M,N,P,R,
,P,S'

460
APPENDIX
Rs, Χ,D,
Rs.A',1
X.Rs
X.Rs
X,S,Rs
X.S.Rs
Rs,D
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SO, Yong-Sian, Dept. Math. Tunghai Univ., Talchung, Taiwan 400, Rep. of China
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VAN DER WALT, Andries P. J., Dept. Math. Univ. Stellenbosch, 7600 Stellenbosch, Rep. of
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1. Prime ideals and nil. radicals in near-rings, Arch. Math. (Basel) 15 (1964), P",R,N
408-414. MR 30-3g00.
2. Fully generating subsets of near-ring, San Benedetto del Tronto, 1981, 123-130. X,E,1,S
3. Dense subsets of near-rings, submitted- X,E,1,S
VAN HOORN, Willy G., Math. Dept. Agricultural Univ. de Dreijen, Wageningen, Holland
1. Some generalizations of the Jacobson radical for seminear-rings, Oberwolfach, Rs,R,P
1968.
2. Some generalizations of the Jacobson radical for semi-near-rings and semirings, Rs,P,R»M,N,S
Math. Z. 118 (1970), 6g-82.
3. The direct sum for seminear-rings, Techn. Note 79-03, Dept. Math. Agricultural C,Rs
Univ. Wageningen, 1979.
SEE ALSO VanHOORN - VanROOTSELAAR
VAN HOORN, Willy G. and VAN ROOTSELAAR, B.
1. Fundamental notions in the theory of seminear-rings, Composition Math. 18 Rs
(1966), 65-78.
E,
E,
A,
D"
E,
D"
D"
Na
D",Rs
D",Rs
Rs
,Rs
F,P'
.Rs
,Rs
,F'
'.A,
,Ua
,Rs
.P0,
,0

462
APPENDIX
VAN ROOTSELAAR, В., Van Nijerodeweg 914, Amsterdam 1, 11, The Netherlands
1. Die Struktur der rekursiven Wortarithmetik des Herrn V. Vukovic, Indag. Math. Rs
24 (1962), №-200.
2. Algebraische Kennzeichnung freier Wortarithmetiken, Compositio Math. 15 Rs
(1963), 156-168.
3. Zum ALE-FasthaJbringbegriff, Nieuw Archief voor Wiskunde 15 (1967), 247-249. Rs
SEE ALSO VanHOORN - VanROOTSELAAR
WAHLING, Heinz, Math. Sem. um'v. Hamburg, Bundesstr. 55, 2000 Hamburg 13, Germany
1. Einige Satze uber Fastko'rper, Oberwolfach, 1968. F,D"
2. Invariante und vertauschbare Teilfastkorper, Abh. Math. Sem. Univ. Hamburg 33 F
(1969), 197-202. MR 42-1869.
3. Automorphismen Dicksonscher Fastkorper, Oberwolfach, 1972. F,D"
4. Zur Theorie der Fastkorper. Habil itationsschrift, Hamburg, 1972. F,D".G
5. Automorphismen oicksonscher Fastkbrperpaare mit kleiner Oicksongruppe, F.n"
Abh. Math. Sem. Univ. Hamburg 44 (1975), 122-13R. MR 53-8033.
6. Bericht uber Fastkorper, Jahresben cht Ot. Math. Ver. 76 (1975), 41-103. F,D".G
MR 58-2599.
7. Automorphismen Dicksonscher Fastkbrperpaare mit kleiner Dicksongruppe, F,D"
Abh. Math. Sem. Univ. Hamburq 44 (1975), 12?-138. MR 53-8033.
8. Normale Fastkorper mit kommutativer bzw. zweiseitiger lnzidenzgnjppe, F,G
Math. I. 147 (1976), 65-78. MR 53-3879.
9. Ein Zassenhauskriterium fiir unendliche Fastkorper, Arch. Math. (Basel) ?8 F,D"
(1977). MR 55-Ш1?.
10. Normale TeiIquasikorper eines Fastringes. Oer Satz von Cartan-Brauer-Hua, Math. F,Rs,X
Z. 158 (1978), 55-60. MR 57-383.
WALKER, Roland, Dept. of Pure Math., Queens Univ. of Belfast, BT-7 INN, Northern Ireland
SEE HOLCOMBE-WALKER
WEFELSCHEIO, Heinrich, Fachber. Math., GHS Duisburg, Postfach 919, 4100 Ouisburg 1, Germany
1. Vervollstandigung topoloqisch-al gebrischer strukturen, Doctoral Dissertation,
Univ. Hamburg (Germany), 1966.
2. Vervollstandigunq topologischer Fastkorper, Math. Z. 99 (1967), 279-298.
MR 36-5112.
3. About a connection between order and valuation in near-fields, Oberwolfach, 196Й.
4. Zur Konstruktion scharf 3-fach transitiver Permutationsgruppen mit Hilfe von
Fastkorpern, Oberwolfach, 1972.
5. Untersuchungen uber Fastkorper und Fastbereiche, Habil itationsschrift,
Hamburg, 1972.
6. Zur Konstruktion bewerteter Fastkorper, Abh. Math. Sem. Univ. Hamburg 38
(1972), 106-117. MR 46-5295.
7. Bewertung und Topologie in Fastkorpern, Abh. Math. Sem. Univ. Hamburg 39
(1973), 130-146. MR 48-8577.
8. Ober eine Orthogonalitatsbeziehung in Hyperbel strukturen, Abh. Math. Sem. Univ.
Hamburg, to appear.
9. Zur Planaritat von K-T-Fastkorpern, Arch. Math. (Basel) 36 (1981), 302-304.
10. Sulla immensione di quasi-corpi non planari in quasi-corpi planari, San
Benedetto del Tronto, 1981, 219-224.
WEINERT, Hanns Joachim, Math. Inst. Techn. Univ. Clausthal, Frzstr. 1, 3392 Clausthal-
Zellerfeld, Germany
1. Halbringe und Halbkorper I, Acta Math. Acad. Sci. Hungar. 13 (1962), 365-378.
MR 26-3634.
2. Halbringe und Halbkorper II, Acta Math. Acad. Sci. Hungar. 14 (1963), 209-227.
MR 26-6219.
3. uber Halbringe und Halbkorper III, Acta Math. Acad. Sci. Hungar. 15 (1964),
177-194. MR 28-4012.
4. Ein Struktursatz fur idempotente Halbkoper, Acta Math. Acad. Sci. Hungar. 15
(1964), 288-295. MR 29-4775.
5. Zur Theorie Levitzkischer Radikale in Halbringen, Math. Z. 128 (1972), 325-341.
MR 47-3467.
6. Halbringe mit aufsteigender Kettenbedingung fur Annulatorideale, J. Reine Angew.
Math. 274/275 (1975), 417-423. MR 52-13948.
7. Ringe mit nichtkommutativer Addition I, Jahresber, Dt. Math. Ver. 77 (1975),
10-27. MR 57-12618a.
8. Ringe mit nichtkommutativer Addition II, Acta Math. Acad. Sci. Hungar. 26
(1975), 295-310. MR 57-12618b.
9. Related representation theorems for rings, semirings, near-rings and
seminear-rings by partial transfomations and partial endomorphisms, Proc.
Edin. Math. Soc. 20 (1976/77), 307-315. MR 56-8637.
10. Dn distributive near-rings, Oberwolfach. 1976.
11. A concept of characteristic for semigroups and semirings. Acta Math. Acad. Sci.
Math. (Szeged) 41 (1979), 445-4M.
12. Multiplicative cancel 1ativity of semirings and semigroups, Acta Sci. Math.
Hungar. 35 (1980), 335-338.
13. S-sets and semigroups of quotients, Semigroup Forum 19 (1980), 1-78.
14. Seminearrings, seminearfields and their semigroup-theoretical background,
Semigroup Forum 24 (1982), 231-254.
15. Zur Theorie der Halbfastkbrper, Stud. Sci. Math. Hungar.. to appear.
WH1TTINGT0N, Robert J.
1. Computer aided determination of near-domains, N.S. Thesis, Univ. of Southwestern
Louisiana, Lafayette, 1973.
WIEGANOT, Richard, Math. Inst. Eotvbs Lorano Univ., Budapest. Hungary
1. Near-rings and radical theory, San Benedetto del Tronto. 1981, 49-58.
SEE ALSO STEINFELO-WIEGANOT
Г
F,
F,
Nc
F,
V.
F
G
F
P'
,F
J'
,0,
i,s
,Nd
.F,l
,r
,S"
,P"
',F
,0",Rs
,0"
V
,D"
D"
,V,D"
,Nd
,S"
,D"
П,
rj,
D,
D,
0,
0.
0.
0.
0,
,Rs
Rs,Q'
,Rs,F
,Rs,F
,Rs,R
,Rs,E
,C,E',B,A
,Ε,Η
,E,T,Rs
О.Е'.Й
D\Rs,E
0
0
0
I'
,Rs,E
>,0'
,E,F,Rs
,E,F,Rs
,A

Bibliography
463
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SEE ALSO PALMER-YAMAMURO
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SEE ALSO MELDRUM-ZELLER
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464
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465
LIST OF SYMBOLS AND ABBREVIATIONS
V.3.3 ι
«=,s 1
0, 2A, A\ В 1
(....a,.,...) 1
И, W0, Р.1.Ц, R.C 1
f/A ι
ι:B - A 1
A* 1
V An 2
Zn 2
DCC, ACC 3
|N| 7
77 8
6 8
nr. 8
М(Г).М0(Г),МС(Г) 8
М°(Г),М5(Г) 8
MCont(r>'Md1ff(R> 8
Maff<r> 9
nf. 11
dgnr. 11,171
Е(Г) 12,171
ΝΓ. tf 13
Ν + Ν1, Hom(N,N') 15
Hom^r,!"),!^!" 15
Ker h, Im h 15
N <* N' 15
^,<lr,<l,<,<lN 16
N/I, Γ/Δ 16
Ω 19
(Δ1:Δ2), (ο:Δ) 20
HNi , ЩП-, N. 24
Ns 26
wn. ^T ,wr
30
TOA 41
Пк 44
DCCI.DCCL.DCCN... 50
ST, Sn 61
(n) 62
f(l) 68
^V(N) 86,143
L. 88
Lz 89
0(N) 101
С„(Г) 106
G0,G°,r,° 106
FN(T) 106
θο, Θι 107
Ό,^ 109
f(Μ) 110
/ , X in
МН(Г) 112
(D) 116
GA 119
CQ 127
?v(N) 136,137
^(N) 143
W("). 7_ ! ( N ) 160
s(0) 161
f(H), ?_2(N) 161
Lr(N), m(0) 161
7.3(N) 164
0(H) 16 4
S(N) 165
Rs(N) 165
CX(N) 165
?3 167
(N,D) 171
do. 171
End Γη 171
Φ 172
Α(Γ), Ι(Γ) 172,206
FD)V 180
0 183
1 (N) 184
Μ°(Γ) 188,189
е. 189
Li, L 190
dimH(r) 192
e6, L6, L 197
Zm 200
£(Γ),£ 200
rk(m) 202
C(Y) 206
End(r),Aut(r) 207
Ιηη(Γ) 207
m(i)% Ε(ι)(Γ) 209
A(X, V) 215
П 216
R[x], PC) 216
Г[х], Р(Г) 216
N[xJ, V[x] 217
Ip! 218
R0E*b RCW 221
(J) , <J> 222
f[x2j 226
V, Τ 227
Γ\Γχ1 231
о »· -

466
APPENDIX
(A), <&> 231
AV[x]
P(A)
Aj[x]
VA>
CfA)
LnP(A)
LP(A)
RCWJ
char Η
C(N)
dimMN
·,, N'
mi,
233
233
234
234
235
238
238
247
251
253
254
255,405
258
N)
259
Ir.(Ir)'
260
L || Μ 261
"pq 261
Coll (P) 262
dim Ρ 262
V*/K» 263
'a ,b
265,266
266
267
268
269
В
269
*t 276
С^И) 277
BIBD 279
IFP 288
(P0)> ···-(%) 239
η i d . 310
и (г) 313
α
a.a.η.г. 313
D(n;n',n") 316
φ 321
Υ
φ*, · 322,405
С(ι,к) 32 5
D 3 29
OO
329
2n
η11 η' 334
p.o., f.o. 334
P< 334
%' So 335
« 33 7
|a| 337
P-0[[xJ] 3 38
S(r) 350
soc(r) 358
ΜΑ(Γ) 36 5
st(y) 365
γ|δ ι γ~δ 366
Ύ
H(Na)
367
369
375
376
Ά^Γ) 376
GSA 378
S 378
d
N(S) 379
NQ(S) 382
A#,N(S# )
GA 386
A,A' 387
A*'VSA
384
387
VAa^a
389
Σ A 393
Aut(N),Inn(N)
Z(N),D(N) 399
Na 399
α
L(N) 400
Π N. 402
У 1
399

467
INDEX
abeli an near-ri ng
abstract affine near-ring
accessible
accessible part
addition of the first type
second type
affine complete
affi ne map
affi ne plane
almost small
α-near-ri ng
anni hi 1ator
anti chai η
anti1exicographiс order
anti tone element
archimedian order
ascending chain condition
associated isomorphism
as soci ated ring
automaton
Baer radical 68,139
balanced block design
base 31
base of equali ty
basic block
β-near-ri ng
Bets ch's 1emma
bicentraliζer near-ring
biregular
block 276
Boolean near-ri ng
Boolean semi ring
11
313
385
389
172
172
239
9
261
292
297
20
327
338
343
337
3
194
401
386
,233
279
,262
34
276
290
366
361
94
,279
300
290
cancel 1able el ement
cancel 1ati on 1 aw
Cebyshev polynomial
cente r
central idempotent
centrali zer
central N-subgroup
central product
chai η
chain rule
characteristic
characterizing series
class λ
cleavable near-ri ng
Cx-modular 1 eft i deal
C^jCj - near-ring
lOf
1
11,4
24
25
9
,36
39
39
24
24
25
39
16
34
16
29
соcr i ti
col 1i ne
commuta
compati
compati
complet
complet
complet
complet
complet
complet
compos i
compos i
compos i
conjuga
cons tan
cons tan
constan
convex
cou pii η
с г i t i с а
C-Z-dec
C-Z-tra
ca 1
ati
ti ν
ble
ble
e
e b
e g
ely
el у
e η
ti о
ti о
ti о
te
t η
t ρ
t ρ
on
e near-ring
triples
279
lock design
roups
non-abelian
reducible
ear-ring
η i deal
η ring
η sequence
chai η
ear-ring
art
olynomi al
235
,324
336
g map
1
omposable
ns i ti ve
243
245
decomposable
degenerated
derivation
Desarguesian
descending chain cond
D-grou ρ
η group
η near-fi eld
η near-ring
η numbers
a 1 group
ckso
ckso
ckso
ckso
h edr
mens
rect
rect
rect
rect
s tri
s tri
255,
262,
i ti on
12,
ι on
complement
product
sum
summand
butive element
butively genera
near-ri η
butive near-rin
butive sum
butor
s
on dgnr.
-ring
24
di s tri
di s tri
dis tri
divide
di vi s i
D-near
D-ri ng
duo (near-)ring
ted
g
9
11
316
402
262
11
240
119
,399
279
324
355
54
399
222
41
53
246
10
10
221
333
,399
402
398
398
53
261
399
263
3
132
255
255
399
256
415
262
46
24
,45
46
10
,171
11
49
396
366
187
295
312
290

468
APPENDIX
efficiency of a BIBD
embeddabl e
enclosing ideal
endomorphism near-ring
equivalent elements
equivalent multiplier
equivalent states
even element
external direct sum
facto
facto
facto
faith
field
filte
f i η i1
f i η i t
f i rs t
F i s h e
fixed
F-nea
forma
free
free
free
Frobe
Frobe
full
fully
fully
r near-rιng
r N-group
r of a sequence
ful N-group
e interpolation
property
e topology
category
r's i nequali ty
-poi nt-free
r - r i η g
1ly real
abelian near-ring
near-ri ng
N-group
ni us group
nius kernel
i deal
generati ng
ordered
dgnr.
polynomi al
near-ring
generali zed
generalized
generalized
generated
genera to r
£J- radical
G-regular
group-automaton
group-semi automaton
GSA-homomorphism
H-base
H-dimensi on
hereditary class
H-group
H-monogeni с
holonomy group
homomorphic GA
homomorphic GSA
homomorphi sm
homomorphism theorem
279
15
223,232
171
366
268
388
342
24
16
16
52
21
2
402
133
111
277
280
,363
343
343
30
29
29
267
267
222
401
332
187
247
320
2,23
75
164
165
386
378
385
192
192
143
192
398
376
386
379
15
17
idea
i dem
i den
i nci
i nci
i η с i
i nco
i nco
i nde
i nde
i ndu
i nne
i npu
i nse
1
pote
tity
dene
dene
dene
mpar
mpl e
comp
pend
cti ν
r au
t se
rti о
nt element
integral
i nternal
i nterpol
i η ν a r i a n
i η ν a r i a n
i nverse
i η ν e r t i b
i somorph
i somorph
i somorph
e gr
e sp
est
able
te d
osab
ence
e Mo
tomo
t
n-of
η r.
d i r
at ίο
t se
t s u
we 11
1 e e
ic d
i с s
i sm
oup
ace
ructure
es i gn
le
ore-sys te,
rphi sm
-factors-
proper
ect sum
η
quence
bnear-ri ng
-order
1ements
ecompos i ti о
equences
theorems
53
31
ty
ns
Jordan-Holder theorem
Jordan-Hb'l der theory
ke rne1-free
kernel of a near-field
Klein's four group
Krull-Schmidt-Theorem
k-tame
Kurosh-Amitsur radical
15,16
10
10
263
261
261
334
279
218
110
2
399
378
288
305
45
133
52
18
3
10
59
52
17,45
52
53
35
253
407
353
352
143
laminated near-ring 399
lattice-ordered near-ring 343
λ-complete 165
left ideal 15
1 eft near-ring 7
length of a sequence 52
length of a series 397
lexicographic order 338
1i near automaton 386
1i near polynomial 219
linear semiautomaton 380
linear sequential machine 386
lines 261
L-near-ring 289
locally affine complete 239
1ocally polynomially
complete 239
local near-ring 400

Index
469
local polynomial function 238
loop near-ring 396
lower nil radical 166
ma
ma
me
mi
m
m
m
m
mo
mo
mo
mo
Mo
m-
ximal i deal (...)
ximum condi ti on
tacyclic
nimal automaton
nimal ideal (...)
nimal non-ni1 potent
nimal prime ideal
nimum condition
du 1 ar
dular 1 aw
nogeni с
notone element
ore-system
sys tern
84
20
3
244
389
20
95
64,65
3
166
48
75
343
2
65
(N,D)-group 182
n-distributive 397
(N.D)-(N',D')-homomorphism 180
near-algebra 41
near-domain 247
near-field 11
near-integral domain 298
nearly cons tant 337
near-module 13
near-ring 7
near-ring of quotients 26,28
N-group 13
N-homomorphism 15
nil 69
nil group 323
nil potent element 7,69
nilpotent subset 69
ni 1 radi cal 160,166
v-modular 86
Noetherian quotient 20
non-field 11
non-generator 119
non-ring 11
non-trivial integral nr. 306
normal endomorphism 214
normal near-field 264
normal sequence 52
normed polynomial 219
v-prime 72
v-primitive ideal 103
v-promitive near-ring 103
v-radical 136
v-semi simple 145
N-series 397
N-s i m ρ 1e 19
N-system 398
Ω-composition group
odd element
orbit
ordered near-ring
order-preserving map
Ore condition
orthogonal idempotents
output function
output set
Pair
para
pa rt
P-ch
Pei r
perm
perf
plan
nl an
plan
p-ne
poi η
poly
poly
poly
pos i
pos i
π- re
prim
prim
prim
prim
pr i η
pr i η
Pj-r
prod
pro j
proj
prop
prop
ps eu
pseu
p-si
of
mete
ial
ai η
ce-d
utat
ect
ar η
ar η
e
ar- r
ts
nomi
nomi
nomi
ti ve
ti ve
gula
e i d
e ne
e ra
i t i ν
ci pa
ci pa
ing
ucti
ecti
ec ti
er к
er r
do-d
do i
ngul
Dickson numbers
rs of a BIBD
order
ecompos i ti on
ion polynomial
group
e a r - f i e 1 d
e a r - r i η g
ing
al
al
all
со
de
r
eal
ar-
dic
e
1 s
1 о
function
у complete
ne
finite
216
219
ring
al
equence
rbi t
103
ve near-field
ve plane
ve space
-tupel
efi nement
istributive
ntegral domain
ar near-ring
Q - r a d i с а 1
quasi regular
quaternion group
quotient near-ring
quotients
radical (map)
radical subgroup
rank map
rank of a map
42
342
9
333
335
26
91
386
386
256
279
333
245
11
245
169
265
269
262
298
261
215
,217
,239
334
343
431
62
63
,161
,105
53
275
298
265
261
262
258
53
397
288
400
165
89
418
28
26
140:
143
165
110
202

470
APPENDIX
reduced automaton
refinement of a sequence
regular near-ring
regular permutation
residual 1 у fini te
right distributive law
right ideal
right identity modulo L
right invariant subnear-
right near-ring
r-i ndependent
ring-free
HL- radi cal
HL-semi prime
π ng
satura
semi au
semi-1
semi ne
s em i pr
semi pr
s e m i ρ r
semi ri
semi ta
sequen
series
sharpl
sheaf
s i m i 1 a
simple
smal 1
socle
s - ρ r i m
sp-sys
s ρ e с i a
s tabi 1
s tal ks
s tate
s tate-
s tri ct
strict
strict
strict
strict
strong
s trong
strong
strong
subdi r
subdi r
subspa
sum of
s urn ru
syntac
ted polynomial
tomaton
i near map
a r - r i η g
i m a r у η r .
i m e ideal
ime near-ring
ng
me
ti al function
connection
у transi ti ve
r multiplications
80,
11 i ν e
tern
1 Boolean near-ri η
i ze r
set
tra
iy
iy
iy
iy
iy
IF
iy
iy
iy
ect
ect
ce
id
le
ti с
nsition function
maximal
mimimal
ordered
prime
primitive
Ρ
monogeni с
regular
uni form
ly irreducible
product
eal s
near-ring
389
52
345
258
395
7
15
84
192
7
110
353
140
140
225
378
194
41
392
66
67
290
436
387
387
258
401
322
19
337
,358
,165
67
301
,365
401
378
378
86,168
368
341
68
186
276
751
348
284
25j
24'
261
44
245
,386
187
105
g
253
379
tactical configuration
tame
TO-ideal
topologi cal dgnr.
transitive
translation
tri-operational algebra
trivial ideal
187
tri vial
tri vial
trivial
trivial
type ν
type 3
type I .
integral near-ring
multiplication
о rde r
subdirect product
IV
ultrafi1ter
ultraproduct
uni tary
units
upper nil radi ca1
vector near-ring
weak commutatiνity
well-ordered set
Zassenhaus-criterion
Z-dis tri buti vi ty
zero divisor
zero set
zero-symmetri с
zero-symmetr i с
ze ro-symmetri с
79
50
22
87
58
65
41
16
306
8
334
25
77
80
?90
403
403
14
184
161, 166
ri ng
near-
part
polynomial
343
289
3
256
310
10
200
10
10
221