Near-Rings and Near-Fields: Proceedings of a Conference Held at the University of Tubingen, F.R.G. 4-10 August, 1985

NEAR-RINGS AND NEAR-FIELDS

NORTH-HOLIAND MATHEMATlCS STUDIES

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD .TOKYO

137

NEAR-RINGS AN D NEAR-FIELDS Proceedings of a Conferenceheld at the University of Tubingen, E R.G. 4IOAugust, 7985

editedby

Gerhard BETSCH MathematicalInstitute University of Tubingen Federal Republic of Germany

1987

NORTH-HOLLAND -AMSTERDAM

NEW YORK

OXFORD .TOKYO

Elsevier Science Publishers B.V., 1987 All rights reserved. Mopart of this publication may be reproduced, stored in a retrieval system, ortransmitted, in any form or by any means, electronic, mechanical,photocopying, recording or otherwise, without the prior permission of the copyright owner.

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Near-rings and near-fields. (North-Holland mathematics studies ; 137; Papers f r o m an international Conference on Nearrings and Near-fields, held a t the Mathematical I n s t i t u t e of the University o f fibingen, Aug. 4-10, 1955. Includes bibliographies. 1. Near-rings--Congresses. 2 . Near-fields--Congresses. I . Betscb, Gerhard, 193411. Conference on Neerrings and Near-fields (1985 : Mathematical I n s t i t u t e Of the University of flbingen) 111. S e r i e s . 512l.4 86- 32931 QA251.5.143 1987 ISBN 0-4k4-70191-5 (U.S.

.

PRINTED IN THE NETHERLANDS

V

PREFACE

T h i s volume c o n t a i n s t h e substance o f two i n v i t e d l e c t u r e s and 30 communications d e l i v e r e d a t an i n t e r n a t i o n a l Conference on Near-rings and N e a r - f i e l d s , which was h e l d i n t h e Mathematical I n s t i t u t e o f t h e U n i v e r s i t y o f Tubingen, 4th

-

1 0 t h August 1985.

A l l papers i n t h i s volume have been refereed. The e d i t o r expresses h i s g r a t i t u d e t o a l l colleagues who helped i n examining t h e manuscripts. Special thanks a r e due t o Gunter P i l z f o r h i s u n f a i l i n g h e l p and a d v i c e , and t o h i s s e c r e t a r y Waltraud E i d l j o r g f o r a l o t o f work on t h e way t o a camera-ready manuscript

.

The p a r t i c i p a n t s o f t h e conference g r e a t l y a p p r e c i a t e d t h e h o s p i t a l i t y o f t h e Tubingen Mathematical F a c u l t y and Mathematical I n s t i t u t e . The e d i t o r (and o r g a n i s e r ) expresses h i s thanks t o Dean P r o f . W.

Kaup and t h e s t a f f o f t h e

I n s t i t u t e f o r t h e i r v a l u a b l e assistance. Several p a r t i c i p a n t s r e c e i v e d f i n a n c i a l support. I n t h i s c o n t e x t t h e e d i t o r g r a t e f u l l y acknowledges generous support by t h e

-

Deutsche Akademische Austauschdienst, Bonn

-

Vereinigung d e r Freunde der U n i v e r s i t a t Tubingen e.V.

and by t h e (Universitatsbund). F i n a l l y , t h e e d i t o r would l i k e t o thank t h e North-Holland P u b l i s h i n g Company f o r i n c l u d i n g t h i s volume i n t o t h e Mathematics Studies s e r i e s and f o r e x c e l l e n t cooperation.

G. Betsch

vi

LIST OF REGISTERED PARTICIPANTS

Johannes Andr6, Saarbrucken, F.R. Germany Howard E. B e l l , S t . Catherines, Ontario, Canada Gerhard Betsch, Tubingen, F.R. Germany Donald W. B l a c k e t t , West Newton, MA, USA Anne Buys, P o r t Elizabeth, South A f r i c a James R. Clay, Tucson, AZ, USA Simonetta D i Sieno, Milano, I t a l y U l r i c h Felgner, Tubingen, F.R. Germany Celestina Ferrero C o t t i , Parma, I t a l y Giovanni Ferrero, Parma, I t a l y Yuen Fong, Tainan, Taiwan Peter Fuchs, Attnang-Puchheim, A u s t r i a Gert K. Gerber, P o r t Elizabeth, South A f r i c a A r t u r G r i g o r i , Northridge, C a l i f . , USA N.J. Groenewald, P o r t Elizabeth, South A f r i c a The0 Grundhofer, Tubingen, F.R. Germany Werner Heise, Munchen, F.R. Germany Gerhard Hofer, Linz, A u s t r i a Helmut Karzel , Munchen, F.R. Germany Hermann Kautschitsch, Klagenfurt, A u s t r i a W i l l i a m E. Kerby, Hamburg, F.R. Germany Suraiya J. Mahmood, Riyadh, Saudi Arabia Carlton J. Maxson, College S t a t i o n , Texas, USA John D.P. Meldrum, Edinburgh, U.K. J.H. Meyer, Stellenbosch, South A f r i c a Rainer M l i t z , Wien, A u s t r i a Hans Ney , Saarbrucken, F. R. Germany Dorota Niewieczerza?, Warsaw, Poland Alan Oswald, Middlesbrough, U.K. S i l v i a P e l l e g r i n i Manara, Brescia, I t a l y Gunter P i l z , Linz, A u s t r i a D. Ramakotaiah, Nagarjunanagar, I n d i a Maic Sasso-Sant, Saarlouis, F.R. Germany Raffaele Scapellato, Parma, I t a l y M i r e l a Stefanescu, l a s i , Romania Alberta Suppa, Parma, I t a l y V . Tharmaratnam, Thirunelvely, S r i Lanka S. Veldsman, Port Elizabeth, South A f r i c a Andries P.J. van der Walt, Stellenbosch, South A f r i c a H e i n r i c h Wefelscheid, Duisburg, F.R. Germany Hanns Joachim Weinert, C l a u s t h a l - Z e l l e r f e l d , F.R. Germany Richard Wiegandt, Budapest, Hungary

vii

INTRODUCTION By Gerhard Betsch The papers in this volume provide a fairly complete picture of current trends and problems in the theory of near-rings and near-fields. In the sequel we shall describe a few condensation points. But first we shall mention some .. Basic defilunons A is a triple (N,+,.), where N is a set and + and . are binary operations on N such that (i) (N,+) is a group (not necessarily commutative); (ii) multiplication is associative; (iii) (a+b)c=ac+bc for all a,b,ce N. A Mbear-ring is defined in an analogous manner: (iii) has to be replaced by the other dismbutive law (iii') c(a+b)=ca+cb for all a,b,ce N. In the sequel we shall - with few exceptions - simply speak of "near-rings", because frequently it is inessential for the purpose of this introduction, which type of near-rings (right or left) is investigated in the paper under discussion. Examples of near-rings are abundant: They arise in a natural way whenever one deals with systems of "non-linear" mappings. E.g., let (r',+)be a not necessarily commutative group. Then all mappings of r into itself (written as left operators of r)form a (right) near-ring M(T) with repect to pointwise addition and multiplication by composition. In most cases, M(T) has many subnear-rings. Conversely, it can be proved, that any (right) near-ring N may be embedded into a suitable near-ring M(T). A near-ring N is called m o - s v 'm if nO=On=O holds for all nE N (0 is the neutral element of the additive group (N,+,)). A near-field is a near-ring N with the property that (N\(O),.)is a group.

All near-fields with more than 2 elements are zero-symmetric. The additive group of a near-field is commutative [ I I , Prop. 8.1 and Theorem 8.111. (Numbers in square brackets refer to the short bibliography at the end of this introduction, while names in capital letters refer to the paper of the quoted author(s) in thisvolume.) The &,&of a near-ring N may be defined as the kernels of (near-ring) homomorphisms of N. If N is a right near-ring, then an N-e~plapis the near-ring analogue of a left module over a ring. (N,+) is an N-group by left multiplication. The of N are precisely the kernels of N-group

...

Vlll

Introduction

homomorphisms of (N,+). The concept of a primitive ring, the various radicals, etc. can be generalized to near-rings, but in such a way, that e.g. different Jacobson type radicals J, (v=0,1,2,3) of near-rings correspond to the Jacobson radical J of rings. If N is a ring, then J,(N)=J,(N)=J,(N)=J,(N)=J(N). For further definitions and details we refer to the monograph [ 111. Now we turn to the subjects treated in this book.

The investigation of near-fields is the oldest branch of the theory. In 1905 L.E. Dickson gave the first known example of a near-field which is not a field, and proved a series of basic results [51, [6].One of the main questions in near-field theory is, whether a given abstract near-field is a Dickson near-field, i.e. can be constructed by a method going back essentially to Dickson. Zassenhaus [13] determined all finite near-fields. There are exactly seven isomorphism types which are Dickson. Until now all known infinite near-fields are Dickson near-fields. GRUNDHOFER proves, that every non-discrete locally compact (topological) near-field of characteristic zero is a Dickson near-field. KARZEL in his paper provides a far-reaching generalization of Dickson's construction to "smcturegroups". These are multiplicative groups with an additional structure such that left multiplication by group elements induces an automorphism of the structure. "Dickson structure groups" are then obtained from structure groups by means of coupling maps. The referee of Karzel's paper suggested, that it might be advisable now to look for a general theory of coupling maps, which covers all known applications. In studying infinite near-fields, suitable finiteness conditions proved very useful. FELGNER presents a series of results on near-fields which are pseudo-finite (= infinite, and elementarily equivalent to an ultraprcduct of finite near-fie!ds). Felgner's results give an example of the impressive power of model theoretic methods. In this context we also mention the paper by FUCHS, in which ultraproducts of affine n e a r - n i ill also be shown in a forthcoming monograph by are investigated. The vitality of near-field theory w H. Wiihling, to appear in 1987. wth Geomegy As early as 1907 near-fields have been applied to geometry [12]. In the 1960s Karzel and his students investigated near-fields in connections with incidence groups [lo]. KARZEL's paper in the present volume is still related to this line of research. G. Ferrero, the editor, and J.R. Clay established the interplay between planar near-rings and Frobenius groups with combinatorial structures, like Balanced Incomplete Block Designs (BIBD's), PBIBD's, etc. [7], [3]. In this volume, AND& presents a survey of non-commutative geometry, which he initiated in 1975 [2]. The paper by SASSO-SANT deals with relations between near-rings, non-commutative geometry, and combinatorial structures, thus combining non-commutative geometry with the problems and methods of [71 and [31.

fnrroduction

ix

Also, SCAPELLATOs contribution fits into this context. Finally, in MAXSONs paper near-rings are associated to generalized translation spaces, and the influence of the geometry on the algebraic structures (near-rings under consideration) is investigated.

Radicals A lot of work has been done to define and investigate concrete near-ring radicals corresponding to the various known ring radicals. Most results, which were obtained until 1983 in this context, may be found in the monograph by G. Pilz [ll]. The paper by GROENEWALD "On the completely prime radical in near-rings" follows this line a bit further, applying methods from general radical theory. Since zero-symmetric near-rings are R-groups in the sense of Higgins [S], the paper by GERBER on "Radicals of R-groups defiied by means of elements" bears immediate relevance for near-rings. Having studied various concrete radicals it was then fairly natural to turn to abstract Kurosh-Amitsur radicals (KA-radicals) of near-rings. In [4], the editor and R. Wiegandt proved, that KA-radicals of associative near-rings differ from KA-radicals of rings in a remarkable way: There are plenty of KA-radicals of (associative) near-rings with non-hereditary semisimple classes. Hence the famous theorem of Anderson-Divinsky-Sulinsky[11 does not carry over to KA-radicals of near-rings. But while in the class of associative near-rings we still have both hereditary and non-hereditary semisimple classes, the situation may be rather bad, if one drops associativity: VELDSMAN shows, that in the class of all not necessarily associative zero-symmetric near-rings with commutative addition the only KA-radicals with hereditary semisimple class are the two trivial radical classes. The "classical" radicals of near-rings are not necessarily KA-radicals. In fact, among the Jacobson KAtype radicals J, the radicals J, and J, are KA-radicals, while the radicals J, and J, are radicals. (See 111; p. 143/144] for some details and references.) Recently (1986) Kaarli and Kriis proved, that the prime radical of near-rings is also not a KA-radical[9]. Hence one might ask, whether the KA-radicals are an appropriate type of absaact radicals in the case of near-rings. This is the background of the paper by MLITZ. He investigates, whether the Jacobson type radicals of near-rings might fit into the more general concept of M-radicals.

Matrix near-rings over arbitrary near-rings were introduced recently by Meldrum and van der Walt. The purpose of van der WALT'Spaper is, to give a detailed account of the correspondence between the twwsided ideals of the base near-ring and those in the matrix near-ring. MEYER and van der WALT construct a matrix near-ring over an infinite near-field, which is a 2-primitive non-ring with identiy and with a certain additional property. The existence of such a non-ring solves a problem posed by the editor in 1971.

X

Introduction

One may expect that the matrix near-rings of Meldrum and van der Walt will provide quite an effective machinery for constructingcrucial examples or counterexamples. Of course, a substantial portion of the papers in this volume may not be subsumed under a precise headline, but have to be listed under

The papers by FONG and MELDRUM and by MAHMOOD and MELDRUM are devoted to certain distributively generated near-rings - since many years an attractive subject in near-ring theory. CLAY also deals with d.g. near-rings. He proves, that certain near-rings are in fact commutative rings with identity; the structure of these rings and the structure of their group of units is determined. Two notes by DE STEFAN0 and DI SIENO deal with distributive near-rings. The paper by MELDRUM and van der WALT is devoted to the important subject of Krull dimension and tame near-rings. (For the definition of a tame near-ring see [l 1; 8 9gI.) The medial near-rings studied in the contribution of PELLEGRINI MANARA have a subdirect structure which is very close to that of strongly IFP-near-rings (see [l 1; 8 9al). Reduced near-rings (= near-rings without non-zero nilpotent elements) are investigated by RAMAKOTAIAH and SAMBASIVARAO. Result: The set of all idempotents of a reduced near-ring with identity forms a Boolean algebra under a specific partial ordering. The paper by FERRERO CO?TI is devoted to near-rings with the property, that all endomorphisms commute with right (or left) translations. In many cases near-rings of this type have only the identical automorphism. WIEGANDT proves: Let A be a subdirectly irreducible right near-ring. If a left invariant subset of A satisfies a certain permutation identity, then A is a field. The main tool for the proof is a lemma, which is purely semigroup theoretical. BELL and MASON present several commutativity theorems for near-rings admitting suitably constrained derivations. KAUTSCHITSCH studies maximal ideals in certain near-rings, the smctrue of which is not too far from polynomial and power series near-rings. BETSCH presents a class of near-rings, which cannot be one-sided ideals in a near-ring with identity. Also, he characterizes those near-rings, which do have an embedding as a one-sided ideal into a near-ring with identity.

The subject of the paper by PILZ are “separable” (non-linear) dynamical systems (Q,A,B,F,G), where Q,A.B are groups (consisting of the states, inputs, outputs, respectively) and the mappings F,G can be “separated” in a certain way. Separable systems with identical input- and output groups fonn a near-ring with respect to parallel and series connections. System-theoretic questions like feedbacks, reachability questions, iovertibility,. .. are studied within this framework. Concerning the paper by HOFER: If the set R of

Introduction

xi

states of an automaton is furnished with the structure of a module, the mappings of R into itself form a near-ring, called the syntactic near-ring of the automaton. The present paper shows how the knowledge of the ideals of the zero-symmetric part of the syntactic near-ring may be applied to determine reachability in the automaton.

Partially and fully ordered seminear-rings and near-rings are studied by WEINERT, while HEBISCH and WEINERT deal with euclidean seminear-rings and near-rings. STEFANESCU presents a ternary interpretation of infra-near-rings - to be more precise: of the addition in infra-near-rings. This iinerpretation explains the relationship between infra-near-rings and prerings. Furthermore, in this way left infra-distributivity may be understood as distributivity of the multiplication with respect to a ternary composition.

[l] Anderson, T. - Divinsky, N. - Sulinski, A., Hereditary radicals in associative and alternative rings. Canad. J. Math. 17 (1965), 594-603. [2] An&, J., M i n e Geumetrien iiber Fastkoxpern. Mitt. Math. Sem. GieDen 114 (1975), 1-99. [3] Betsch, G. - Clay, J.R., Block designs from Frobenius groups and planar near-rings. Proc. Conf. finite groups (Park City, Utah 1973), p. 473-502. Acad. Press 1976. [4] Betsch, G.- Wiegandt, R., Non hereditary semisimple classes of near-rings. Studia Math. Hungar. 17 (1982). 69-75. [5] Dickson, L.E., Definitions of a group and a field by independent postulates. Trans. Amer. Math. Soc.6 (1905), 198-204. [6] Dickson, L.E., On finite algebras. Ges. d. Wiss. zu Gottingen, Nachr. math.-phys. Klasse (19O5), p. 358-393. [7] Ferrero, G., Stems planari e BIB-disegni. Riv. Mat. Univ. Parma (2) 11 (1970), 79-96. [8] Higgins, P.J.,Groups with multiple operators. Proc. London Math. SOC.(3) 6 (1956), 366-416. [9] Kaarli, K.- Kriis, T., Prime radical of near-rings (1986). To appear. [lo] Karzel, H., Bericht iiber projektive Inzidenzgruppen. Jahresber. Dt. Math. Ver. 67 (1965), 58-92. [ 111 Pilz, G., Near-rings. Revised Ed. Amsterdam: North-Holland 1983. [ 121 Veblen, 0. - Maclagan - Wedderburn, J.H., Non desarguesian and nonpascalian geometries. Trans. Amer. Math. Soc.8 (1907), 379-388. [13] Zassenhaus, H., uber endliche Fastkorper. Abh. Math. Sem. Univ. Hamburg 11 (1936), 187-220.

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Xiii

CONTENTS

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of Registered Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

v vi vii

INVITED LECTURES J . AndrC. Non-commutative geometry. near-rings and near-fields . . . . . . . . . . . . . . U. Felgner. Pseudo-fmite near-fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 15

COMMUNICATIONS H.E. Bell and G . Mason, On derivations in near-rings ...................... G. Betsch. Embedding of a near-ring into a near-ring with identity . . . . . . . . . . . . . J.R. Clay. The near-ring of some one-dimensional noncommutative formal group laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S . De Stefano and S . Di Sieno. On the existence of nil ideals in distributive near.rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. De Stefano and S. Di Sieno. Distributive near-rings with minimal square . . . . . . . C. Cotti Ferrero. Near-rings with E-permutable translations . . . . . . . . . . . . . . . . . Y . Fong and J.D.P. Meldrum. Endomorphism near-rings of a direct sum of isomorphic finite simple non-abelian groups . . . . . . . . . . . . . . . . . . . . . . . . . . P . Fuchs. On the ideal structure in ultraproducts of affme near-rings . . . . . . . . . . . C.K. Cerber. Radicals of a-groups by means of elements . . . . . . . . . . . . . . . . . . . N.J. Groenewald. Note on the completely prime radical in near-rings . . . . . . . . . . . Th . Grundhofer. On p-adic near.fie1ds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U . Hebisch and H.J. Weinert. Euclidean seminear-rings and near-rings . . . . . . . . . . . C . Hofer. Ideals and reachability in machines . . . . . . . . . . . . . . . . . . . . . . . . . . . H.Karzel. Couplings and derived structures ............................ H. Kautschitsch. Maximal ideals in near-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . S.J. Mahmood and J.D.P. Meldrum. D.g. near-rings on the infinite dihedral group . . . C.J. Maxson. Near-rings associated with covered groups .................... J.D.P. Meldrum and A.P.J. van der Walt. Krull dimension and tame near-rings . . . . . J.H. Meyer and A.P.J. van der Walt. Solution of an open problem concerning 2-primitive near-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R . Mlitz. Are the Jacobson-radicals of near-rings M.radicals? . . . . . . . . . . . . . . . . . S. Pellegrini Manara. On medial near-rings ............................. G. Pilz. Near-rings and non-linear dynamical systems ...................... D. Ramakotaiah and V . Sambasivarao. Reduced near-rings . . . . . . . . . . . . . . . . . .

31 37 41 53 59 63 73 79 87 97

101 105 123 133 145 151 167 175 185 193 199 211 233

xiv

Contents

M . Sasso.Sant, Non-commutative spaces and near-rings including PBIBD’s planar near-rings and non-commutative geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . R . Scapellato. On geometric near-rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M . Stefhescu. A ternary interpretation of the infra-near rings . . . . . . . . . . . . . . . . A.P.J. van der Walt. On two-sided ideals in matrix near.rings . . . . . . . . . . . . . . . . . S. Veldsman. Some pathology for radicals in non-associative near-rings . . . . . . . . . . H.J. Weinert. Partially and fully ordered seminear-rings and near-rings . . . . . . . . . . R . Wiegandt. On subdirectly irreducible near-rings which are fields. . . . . . . . . . . . .

245 253 255 267 273 277 295

Near-ringsand Near-fElds, C. Betsch (editor) 0 Elsevier Science Publishers B.V.(North-Holland), 1987

1

NON-COMMUTATIVE GEOMETRY. NEAR-RINGS AND NEAR-FIELDS Johannes ANDRE' Fachbereich Mathematik der Universitat des Saarlandes D-6600 Saarbriicken, Federal Republic of Germany Some relations between near-rings (especially near-fields) and spaces with a non-commutative join are considered. 0.

INTRODUCTION Eighty years ago L.E.Dickson (1905) discovered the nearfields thus proving that the both distributive laws are independent o f each other. Veblen, Wedderburn 1907 gave the first geometric application: They constructed projective planes using nearfields. Zassenhaus 1935 rediscovered the nearfields when he considered the structure of finite sharply twofold and threefold transitive permutation groups (see also e.g. Wielandt 1964 and Kerby 1974). In the sixties further geometrical applications of nearfields follow (see e.g. Arnold 1968, Karzel 1968). Not only plane geometries but also higher dimensional spaces were constructed by nearfields. Arnold 1968 e.g. considered the following spaces: Let F be a nearfield (with the distributive law (a+b)c = ac+bc ) and let Fn be the point-set of this space. The lines are defined by (0.1)

XY

:=

F ( ~ - x )+

(x,y EFn 1

X

,

where, as usual, addition and multiplication are defined componentwise. Two different points are incident with exactly one line (but the general Veblen-condition only holds if F is a skewfield). The spaces thus defined are special Sperner-spaces (cf.Sperner 1960) which Arnold was able to characterize purely geometrically. Consider now spaces with the same point-sets: the lines, however, are defined by (0.2)

X

(y-x)F +

y :=

u

X

.

Here the operation is not commutative in general. Historically we thus obtained the first examples of non-commutative spaces (cf. e.g. Andri? 1975). But there are simpler types of such spaces: Consider e.g. the circles of the euclidean plane together with their centres. In this case x u y means the circle through y with

2

J. Andrh

the centre x (circle space). Besides to nearrings and nearfields the non-commutative geometry can be applied (among other things) to groupsI especially permutation groupsI and to graphs and other combinatorial structures (see e.g. Andr6 1984). In the first section the non-commutative spaces are introduced in a way stated by Pfalzgraf 1984,198533 (which is more general than that originally given by the author). Section 2 gives some geometrical closure conditions (SchlieBungsaussagen):The well-known conditions of Veblen and Desargues can be embedded into an infinite hierarchy of simplex-conditions (Pfalzgraf 1985a). The next section brings the first contact to nearfields. The nearfield spaces already will be defined in more detail. Special interindicated in ( 0 . 2 ) ests seem to have the improper structures of such spaces. They lead to special rank-two-nearfields the so-called Biliotti-nearfields considered in section 4. The following section gives an other way to introduce noncommutative spaces over nearfields closely related to sharply twofold transitive groups (RoBler 1985). The last two sections concern spaces over special types of nearrings (the one and the higher dimensional case resp.) which have been considered especially by E.Theobald (1981) and H.Ney (1983). GEOMETRIC SPACES Let X and R be non void sets whose elements are called proper and improper points resp. A geometric space or more briefly space (Pfalzgraf 1984,1985b) is a collection 1.

S

(1.1)

=

(X,Rl)

where <,>

(1.2)

:

x2\i

t,R

,

(x,y) ++ <x,y>ER

(x,y€X

I

x#y)

is a surjective mapping*. The union-line or more briefly line (Linie) xoy of the two different points x,y (in this order) is defined by (1.3)

xoy

:=

XpJ

:=

(xuy)U {<xly>)

where (1.3') IIere

xpy

~x)U~z€Xl<x,y>=<x,z>)

is the proper part of the line

xuy

. and

<x,y>

its

*Sometimes it is advisable to extend this mapping to X' (instead X2\iX), where <x,x> does not depend on x and <x,x> = <x,y> implies x=y. (See also Pfalzgraf 1984,1985b).

3

Non-commutative geometry

improper point. The point x (or xoy), denoted by

is called a basepoint (Aufpunkt) of

XLJY

(1.4)

x

t

w

or

x

f

xoy

Obviously x*L implies xEL but not vice versa. The following two properties on lines are easy to check. (L1)

XfY

(L2)

Z€

x

E

(x

uY

u Y) ' .ix)

x Uy

implies

=

x

z

.

(exchange condition) Two lines

xoy

and

(1.5)

x

x'oy'

are called parallel, denoted by

y I1 x'

0

,

(incidence condition)

0

y'

,

iff they have the same improper point, i.e. iff <x,y> = <x',y'>. Clearly 1 1 is an equivalence relation. Moreover the following condition holds (Pl') Given an xEX and a line L there exists at most one line with a basepoint x (i.e. of the form L' = xGy) -such that This line (if it exists) denoted b~ LIIL' L'

.

(XllL)

(1.6)

.

Obviously (1.7)

if

(XllL) = L

x

*

and

L

(x I I (y IlL)) = (x IIL)

provided that these lines do exist. A refinement of Euclidean parallel-condition (Pl)

(Pl') is the

g line L and any x<X

there exists exactly one line xtL' In this case (x l l ~ ) thus always exists. EXAMPLES OF GEOMETRIC SPACES Define (i) Group spaces. Let G be a group acting on X L' I IL

.

with

.

(1.8) Then (1.8')

<xfy> : = (1.3') xUy

G(x,y)

:=

i(gx,gy) IgEG)

.

yields =

Gx{x,y)

=

cx). UGxy

In this case 5 = _V(G) is called a group-space with respect to G. (ii) Distance-spaces. Let X be a metric space and d its disAn example of a distance-space is tance. Define <x,y> := d(x,y) the circle space stated in the introduction. Also any connected graph leads to a distance-space in a natural way. (iii) Let be X = R = IR and put <x,y> := y-x In this case we have x u = {xfy) = y p

.

.

.

J. Andrk

4

5

DEFINITION 1.1. The geometric space perly determined* (&) iff

x

(1.9)

uy

x'

=

u

y'

implies

= (X,R,<,>) is called

<x,y> = <x',y'>

pro-

I

i.e. iff the lines are uniquely determined by their proper parts (c.f. Pfalzgraf 198513). Inthis case it makes sense to say that two proper parts of lines are parallel, i.e.

x

(1.9')

u

y II x' 1_1 Y'

DEFINITION 1.2.

<x,y> = <x',y'>

A space is called self-adjoint or symmetric iff

(1.10)

<x,y>

holds for all

means

x,y€X

=



.

REMARK. There exist spaces not being p.d. (Pfalzgraf 1985131, see e.g. example (iii) However

.

PROPOSITION 1.1. Any self-adjoint space 2 properly determined. Then <x,y> = <x',y'> is obviously PROOF. Assume x u = x ' w ' Applying (L2) twice we obtrue if x = x'. Assume now x # x' tain x x' = x y = x' y' - x' x Hence <x,y> =<x,x'>

.

u

u

= <x*,x> = <x',y'>

joint.

.

by

u

.

(1.3') and because the space is selfad-

0

DEFINITION 1 . 3 . Let be = (X,R,<,>) a space. Then U 5 X is The space called a subspace if x,y,zEU imply ( x ~ J y u z5) U is called primitive if it contains only the trivial subspaces 0,

.

{x)

(x€X) and

X

,

otherwise imprimitive.

2.

PFALZGRAF'S HIERARCHY (Pfalzgraf 1985a) The geometric spaces defined above are still too general to obtain interesting properties. It may be reasonable to require a hierarchy of geometrical closure properties (SchlieOungsaussagen). The following q-simplex-condition Sim (qEN) is due to J.Pfalz9 graf (1985a): Given xo,x,, xq,x;,xi E X with ~ x o l x l ~ = ~ x ~ , x ~ ~ Then there exist X~,...,X; E X such that <X~,X!>=<X.,X.>for --I 1 1 i,j E I O , I , . , q 3 , i#j If additionally x; = xo holds we obtain the diminished (verb jiingte) q-simplex-condition Sim q REMARKS. (i) Sim, holds in any space. Sim2 is the Tamaschkecondition (see e.g. Andr6 1981). Spaces with Sim2 are called skewaffine. A skewaffine space additionally with xoy = yox for all

...,

..

.

.

::,yEX 1-G m erc ah n:

is an affine space in the usual sense. The Veblen and the bestimmt

.

S

Non-commutative geometry b

b

Desargues condition are exactly Sim2 and Sirn3 resp. (ii) For any qEN there exist spaces with S m (and hence 9 Sim, for r5q) but not with Simq+l (Pfalzgraf 1985a) The following proposition gives a purely geome ric characterization of finite group-spaces.

.

THEOREM 2.1. A finite geometric space is group-space (cf. example (i) in Section 1) iff Simq hold for all q€N

.

PROOF. Obviously all Sim hold in any group-space (not necessaq be a finite space with rily finite). Conversely let S ( X , R , < , > ) Sim Assume X = ~ x o I x , , x 1 and xi,x;EX with <xo,xl> = q 4 <x;),x;>. Then Sim implies the existence of xi ,...,XI with 4 4 <xi,x.> = <x!,x!>. A s a consequence we have xi # x! for i # j, 1 1 1 7 hence X = 4xA ,...,x;) Let g be the permutation on X with gxi = x! for all i€{O,l,...,q). The permutation group G generated by all these g is transitve on all pairs (x,x') with a given <x,x'>ER This implies = ! (GI 0

.

...,

.

.

.

REMARK. Theorem 2.1 is not true for infinite spaces (cf. Pfalzgraf 1985a). The problem of obtaining a purely qeoemtric characterization for arbitrary group-spaces thus remains unsolved. 3.

SKEWAFFINE, NEARAFFINE AND NEARFIELD-SPACES

DEFINITION 3.1.

A line in a geometric space is called a straight all of its proper points are basepoints. Hence a L is straight iff for any two of its proper points x,y y x#y we have L = x

line (Gerade) if line with

REMARK. If thena line basepoints. is straight

.

the space is skewaffine (cf. sect.2, remark (i)) L is already straight if it possesses two different Moreover, if L is straight and L' l1L then also L' (Andre 1981, Satz 1.1).

PROPOSITION 3.1. Let 2 k g skewaffine space in which any line has at least three proper points. Then is p.d. ----

.

PROOF. Let be L = x u y = x'u y' The case x = x' is clear. Assume x f x' Due to the last remark L = x 1_1 x' = x'u x is straight. Because of lLl2 3 there exists a zEL~{x,y~ Repeated application of (L2) yield ~ ~ , y ~ = ~ ~ , ~ ' > = ~ ~ , ~ ~ = ~ x ' , z ~ = ~ x '

.

.

DEFINITION 3 . 2 . A selfadjoint skewaffine space S = (X,R,<,>) is called nearaffine if (i) straight line meets any line different from it in at

J. Andre

6

most one --

proper point, 3 two proper points can be connected & a finite chain (ii) of straight lines. It is called regular if given two straight lines L and L' with LIIL' = {x) , then there exist yEL\{x) and y'EL'\{xi such that y y' is straight (Andri? 1975). Examples of regular nearaffine spaces are the nearfield-spaces: Given a nearfield F (with (a+b)c = ac+bc ) and an index-set I define (3.1) X : = F('):=

~ ( x ~ ) ~ ~ ~ I x ~ for € F ,only x ~ # finitely O many i€Il,

(3.2) R := :=

(y-x)F

I

for all

x,yEF

.

(Here addition and multiplication are defined in the usual way.) Easy computation yields (3.4)

X

uy

=

(y-x)F +

X

By definition = (X,R,<,>) is a nearfield-space. Such a space b is nearaffim, desarguesian (i.e. Sim3 holds) and regular. Conversely any regular and desarguesian nearaffine space 2 is a nearfield-space S ( F ( ' ) ) and III and F are uniquely determined by S up to isomorphism (Andri! 1975*). Moreover 5 is a group-space with respect to the group of all transformations of the form x * xa + b (x,bEX, aEF\[O]). Let S(F(I)) be a nearfield-space over F Then 0 u is straight iff for all a,bEF there is a cEF such that ua+ub=uc (Andri! 1975,II,Satz 1.2). An improper point uF with such a u f 0 is called internal, otherwise external (see also Denbowski 1971). Let be p : = uF an internal and q : = vF # p an arbitrary improper point. The set of all improper points WF such that w = ua+vb for suitable a,bEF with (a,b) # (0,O) is an improper-line. It is called internal if q can be chosen as an internal point otherwise external. The set R together with the improper lines is called the improper space to S(F(')) and is denoted by ;(F(')) (see also Andr&

.

u

1984, Chap.II,sect.3). 4.

BILIOTTI-NEARFIELDS It would be desirable to get further informations about the structure of J ( F ( * ) ) . If 111 = 3 (in this case we write F3 instead *In that paper the other distributive law is assumed for a nearfield.

Non-commutative geometry

7

of F(') ) then analogous to the desarguesian case J(F3) should possess a plane geometric structure. Indeed, one can show that J(F') becomes a projective plane and, more precisely, a generalized Hughes-plane iff the follwowing configuration holds in S(F') (cf. Andre 1981, Satz 5.2): Given two different lines L and L ' , both not being straight, -with x I then there exist yEL\{x) and - the common - basepoint ~ y'EL'\{x$ such that y y' 2 straight (Biliotti-configuration)?. We are now looking for an algebraic equivalent for F Let K be the kernel of F defined by

u

.

(4.1)

K := K(F) :=

{cEFlc(x+y)=cx+cy

for all

x,yEF)

.

Obviously K is a skewfield and F a left K-vectorspace. The diof F If F is a nearmension of F over K is called the field of rank 2 then any set {l,e) with e€F\K is a basis of F over K , hence

m

(4.2)

F

=

K+Ke

.

DEFINITION 4.1. A rank-2-nearfield nearfield (cf. Biliotti 1979) iff (4.3) for

eEF\K

F

=

.

F = K+Ke

is called a Biliotti-

K+eK

holds.

REMARK. If (4.3) holds for a special eEF\K then it is easy to see that it holds for such e (Biliotti 1979). PROPOSITION 4.1. Let F g nearfield. Then the Biliotti-condiS (F3) iff F g Biliotti-nearfield. tion holds in 0 PROOF. Andre 1981, Satz 5.3**). REMARK. H.Zassenhaus (1986) recently pointed out that there do exist rank-2-nearfields not being Biliotti. The following two propositions, however, show that many rank2-nearfields usually ocurring are Biliotti. PROPOSITION 4.2. & rank-2-nearfield F Biliotti if one of the following conditions hold in F : (i) F is finite. (ii) Centre and kernel of F coincide. PROOF. (i) by counting arguments, (ii) is trivial (see also Andre 1981, Satz 4.1).

.

$ In Andr6 1981,p.458, called DrL It is analogous to the regularity condition noted in sect.3. **A Biliotti-nearfield is there called "Fastkizirper mit Umkehrbed."

J. Andrd

8

Let IH be the skewfield of quaternions (with the standard-basis Define a new multiplication o on IH by l,i,j,k) and tER

.

a o b

(4.4)

:=

{

IbI

I b (-itb

b # O b = O

t

.

Then Bt : = ( H , + , o ) becomes a rank-2-nearfield (with kernel C = IR +,hi if t # 0), the Xalscheuer-nearfield (cf. Kalscheuer 1 9 4 0 , where he proved that any nearfield of finite rank over R with a continuous addition and multiplication is isomorphic to an iHt with a uniquely determined t). PROPOSITION 4.3. Kalscheuer-nearfield Biliotti. Because Q: + j o C PROOF. We must show C + C o j = d: + j o C c C + Q: o j is trivial it suffices to prove that for a given cEC\[Ql the equation

.

(4.5)

c o j

j o w

=

-

.

has a solution wtC Due to jc = cj takes the form equation ( 4 . 5 ) (4.5') c = Iwl 2it ; This yields hence

Icl = Iwl

w = clcl 2it

for all

, consequently c

is a solution of

=

(4.5)

lcl

CCC and

2it

(4.4)

the

; I

n

THEOREM 4 . 1 . F 5 Biliotti-nearfield then the improper space J(F') of the nearfield-space Z(F') is a generalized Hughes-plane K is the kernel of F then J(K') is the and vice versa. If --(uniquely determined) desarguesian Baer-subplane of J ( F ' ) PROOF. Andr6 1 9 8 1 , s 5 . 0 ~

-

.

4

REMARK. Recently E.Scheid ( 1 9 8 4 ) considered the improper space J(F if F is a Biliotti-nearfield. Besides the (internal and external) improper points and lines also internal and external improper planes occur. The internal planes are (pairwise isomorphic) generalized Hughes-planes. Every external plane contains exactly one internal line and only external points outside this line. Only little is known about the structure of external' planes. All internal points, lines and planes form a (desarguesian) three-dimensional projective space over K(F) analogous to the desarguesian Baer-subplane of J It might be fruitful to search for further geometric pro-( F 3 ) 4 perties of J(F ) or, more generally, of z(F")

.

.

9

Non-commutative geometry

5.

OTHER SPACES OVER NEARFIELDS The following construction is due to RBBler 1 9 8 5 . Given a nearfield F (it is possible to extend this construction to neardomains, cf. e.g. Karzel 1 9 6 8 ) and a subgroup U * of (F\[O / . ) , put u : = U*U{O). Then define a space

S

(5.1)

= S(F,U) =

(FIR,<,>)

with (5.1')

R :=

,

{UxIxEF\
<X,y> : =

U(y-X

A simple computation gives

(5.1

"

x

)

uy

=

.

U(y-x) + x

The space S(F,U) is closely connected to the sharply twofold transitive groups of operations X I + xa + b (x,a,bEF, a#O) on F. It is, however, no group-space (in the sense of sect.1, example (i)) in general. ROBler 1 9 8 5 stated some sufficient conditions for this property. PROPOSITION 5.1. and U 5 K(F) -

The space S(F,U) U* is normal in

a

group-space if lU, 1 (F\[O],-) 0

3

.

We close this section with some further properties. PROPOSITION 5 . 2 (RoRler 1 9 8 5 ) . The following properties hold in S(F,U). (i) S(F,U) selfadjoint iff -1 E U (ii) S(F,U) commutative (i.e. x u y = y 0 x,yEF) iff U 2 U 5 U

.

ux

holds for all

.

6.

ONE-DIMENSIONAL SPACES OVER NEARRINGS 6.1. Consider a (at first arbitrary) nearring N (with (a+b)c = ac+bc ) and try to generalize the situation stated in sect.3 after Definition 3 . 2 , at first for 111: 1. (This case is of course trivial if N is a nearfield.) We obtain (6.1) X=N

,

,

R : = {uNluEN\{O))

<x,y> : =

(y-x)N

.

In the space S(N) = (N,R,<,>) defined in that way we thus have (6.2')

due to

X

uy

= {X)U{zENI

(y-X)N = (Z-X)N)

(1.3').

THEOREM 6.1. In the space g(N) takes the form --16.3)

x

uy

=

(y-X)N +

X

defined

&

(6.1)

the line

XD

10

J. AndrC

iff N a geometric nearring (Theobald 1981) & N has the following properties (NO) N 5 zero-symmetric, & Oa=aO = 0 for all aEN (cf. Pilz 1983). wkh ala = a (Nl) For any aEN there exists a unit la (N2) aEbNx[O] implies aN = bN PROOF. This follows by a simple computation.

_ .

.

.

We now state some properties of the space S(N) nearring N

for a geometric

.

'THEOREM 6.2. Let N be a geometric has the following properties:

nearring. Then the space S(N)

.

(i) The euclidean condition (PI) holds in S(N) (ii) S(N) selfadjoint (cf. Def.l.2) iff N symmetric, = -aN holds for all aEN i.e. aN (iii) Sim (cf.sect.2) holds for any qEN Hence _ _ _ -S(N) & 5 9 group-space if N is finite (cf. Theorem 2.1). In this case the group G consists of all transformations x I+ xa+b with x,a,bEN and ya # 0 for all yEN\. (iv) S(N) properly determined iff aN = bN+c implies aN = bN for all a,b,cEN *.

.

PROOF.

.

(i) and (ii) are obvious.

(iii) Let be xo,...,xq,xb;x; E N with (xl-xO)N= (x{ -xi ) N r hence x' = (x,-x0 )a + xb for a suitable aEN. Define x! := fx.-x )a+ xb for i€{l,-.. , q & , Then xl-x! = (x.-x.)a for i,j E 1 0 1 1 1 7 {O,l, q), hence (x;-x!)N = (x.-x.)N , i.e. the first part of 3 1 7 (iii). Thus S(N) is a group-space if N is finite (Theorem 2.1).

...,

A simple computation yields the second part of (iii), cf. also

Theobald 1981. (iv) S(N) p.d. means: (y-x)N+x = (y'-x')N+x' implies (y-x)N = (y'-x')N , i.e. (RE). The converse follows similarly. 6.2. Examples of geometric nearrings are the finite planar nearrings (Pilz 1983,§8, Theobald 1981). In this case the lines of S(N) are the blocks of a suitable block-design (Ferrero 1970, Pilz 1983). M.Sasso-Sant ( 1 9 8 6 ) proved that any integral planar nearring is geometric. Other examples are the derivations of a nearfield discovered by Theobald 1981: Let F be a nearfield and let u:F+F be an idempotent endomorphism of (F,.). Define a new multiplication o by *Theobald 1981b denoted this condition by (RE), the direction uniqueness condition (Richtungseindeutigkeit).

11

Non-commutative geometry

Then F : = ( F , + , . ) b zero-mapping. If F

is a geometric nearring except if

is a skewfield then

F

b

is the is also planar (Theo-

bald 1981). 6.3. We close with some further properties of S(N) PROPOSITION 6.1. Let 5") 5 space over a geometric nearring Then U 5 N 5 subspace (cf.Def.l.3) going through 0 iff N

.

.

.

5 U and UN 5 U Especially if (N,+) & abelian, U subspace through 0 iff it is a right ideal of N

U+U

.

&

a

PROPOSITION 6.2. Let S(N) be a space over a geometric nearring N with abelian addition. Then the parallel-closure-condition holds in S(N) , i .e. x,y,zEN , pairwise different,

+

(Yiixfi) n ( Z I I X ~ )8

.

0

REMARK. If a geometric nearring N with abelian addition has no nontrivial right ideals then the space S(N) is primitive. 7.

HIGHER DIMENSIONAL SPACES OVER NEARRINGS Let N be a geometric nearring, I an index-set. Consider the space S(N(I)) = (XIRI<,>) where X I R and <,> are defined analogously to ( 3 . 1 ) l (3.2) and ( 3 . 3 ) . Then (7.1)

X

y

=

(y-x)N + x

only holds if all units l a occurring in the definition of a geometric nearrings coincide, i.e. N contains a right unit 1 (i.e. xl = x for all xEN). I call such nearrings strongly geometric. (Examples are the derivation rings and the finite planar nearrings.) We are now able to state a structure theorem,analogous to that for nearfield-spaces S ( F ( ' ) ) , discovered by H.Ney. THEOREM 7 . 1 (Ney 1983). Let N d strongly geometric nearring with abelian addition and no nontrivial right ideal. Then the space S ) has the following properties: (i) (N(I)) is a properly determined desarguesian skewaffine space with the parallel-closure-condition and at least three proper points on every line. (ii) Q subspace meets any line not contained in it in at most one point. (iii) Any two points can be connected & a finite chain of primitive subspaces.

-

s

-

-

12

J. AndrC

(iv) Given two primitive subspaces U U' with UnU' = fx3, there exist yEU\ix) y'EU'\{x) and a primitive subspace going through y @ y' (regularity condition). Conversely, any space 5 with these properties (i) to (iv) is of the form S ( N ( ' ) ) for a suitable nearring N with the pro_-._perties stated obove. 111 uniquely determined & 2 0

.

REMARKS. (i) If the primitive subspaces are straight lines then N becomes a nearfield and the space S(N(')) is nearaffine in the sensse of sect.3. (ii) The cardinal number I I I is sometimes called the dimension of Z(N(')). It is possible to generalize the dimension-theory of H-Tecklenburg 1978 (see also Misfeld, Tecklenburg 1979) to such spaces. REFERENCES Andre , J. 1975 Affine Geometrien uber Fastkorpern. Mitt.Math.Sem.Giel3en 114, 1-99. 1981 Nichtkommutative Geometrie und verallgemeinerte HughesEbenen. Math.2. 177, 449-462. 1984 Noncommutative geometry and combinatorics. In: Combinatorics and Applications. Proceed.Seminar on Combinatorics and Applications in honour of Prof.S.S.Shrikande, ed. by K.S.Vijayan and N.M.Singhi, Calcutta, pp.21-39. Arnold, H.J. 1968 Algebraische und geometrische Kennzeichnung der schwachaffinen Vektorraume uber Fastkorpern. Abh.Math.Sem.Univ. Hamburg 32, 73-88. Biliotti, M. 1979 Su una generalizzazione di Dembowski dei piani di Hughes. 674-693. Boll.Un.Mat.Ita1. ( 5 ) Dembowski, P. 1971 Generalized Hughes-planes. CanJ.Math. 2 , 481-494. Dickson, L .E. 1905 On finite algebras. Nachr.kgl.Ges.Wiss. 1905, 358-393. Ferrero, G. 1970 Stems planari e BIB-disegni. Riv.Mat.Univ.Parma ( 2 ) 11, 79-96. Kalscheuer, F. 1940 Die Bestimmung aller stetigen Fastkorper. Abh.Math.Sem. Univ.Hamburg 13, 413-435. Karzel , H. 1968 Zusammenhange zwischen Fastbereichen, scharf 2-fach transitiven Permutationsgruppen und 2-Strukturen mit Rechtecksaxiom. Abh.Math.Sem.Univ.Hamburg 2,191-206. Kerby, W. 1974 On infinite sharply multiply transitive groups. Vandenhoeck & Ruprecht, Gottingen.

s,

Non-commutative geometry

13

Misfeld, J. , Tecklenburg, Helga 1979 Dimension of nearaffine spaces. In: Proceed.Conf.Geometry and Differential Geometry, Haifa. Springer-Verlag, Lecture Notes in Mathematics 792, 97-109. Ney, H. 1983 Planar nearrings and their relations to some noncommutative spaces. Proceed.Conf. Near-Rings and Near-Fields, Harrisonburg, Va., p.47. Pfalzgraf, J. 1984 Uber ein Model1 fur nichtkommutative geometrische Raume. Diss. Saarbrucken. 1985a A hierarchy of geometric configurations generalizing the axioms of Desargues and Tamaschke. Preprint: Faculdad de Ciencias, Universidad de Chile, Santiago de Chile. 1985b On a model for non commutative geometric spaces. Journ. Geometry 2,147-163. Pilz, G. 1983 Near-Rings. The theory and its applications. Revised ed., North-Holland, Amsterdam, New York, Oxford. RoBler, K.P. 1985 Gruppenpaarraume zu scharf transitiven Permutationsgruppen. Diplomarbeit, Saarbrucken. Sasso-Sant, M. 1986 Nichtkommutative Raume und Fastringe. Diplomarbeit, Saarbriicken. Scheid, E. 1984 Fernstrukturen zu vierdimensionalen Fastkorperraumen uber Biliotti-Fastkorpern. Diplomarbeit, Saarbriicken. Sperner, E. 1960 Affine Raume mit schwacher Inzidenz und zugehorige algebraische Strukturen. Journ.r.u.a.Math. 204, 205-215. Tecklenburg, Helga 1978 Zur Dimension fastaffiner Raume. Mitt.Inst.f.Math. TU Hannover 92, 1-25. Theobald, E. 1981a Nichtkommutative Geometrie uber Fastringen. Diplomarbeit, Saarbrucken. 1981b Near-rings and noncommutative geometry. In: Proceed.Conf. on near-rings and near-fields, ed.by G.Ferrero and C. Ferrero-Cotti, Sa Benedetto del Trento, pp.211-218. Veblen, 0. , Wedderburn, J.H.M. 1907 Non-Desarguesian and non-Pascalian geometries. Transact. Amer.Math.Soc. g, 379-388. Wielandt, H. 1964 Finite permutation groups. Academic Press, New York,London. Zassenhaus, H. 1935 Uber endliche Fastkorper. Abh.Math.Sem.Univ.Hamburg IJ, 187-220 1986 Uber Fastkorper 11: Existenz nicht-Dicksonscher Fastkorper endlichen Ranges iiber Grundkorpern unendlichen Transzendenzgrades. Manuscript.

This Page Intentionally Left Blank

Near-rings and Near-fields, C. Betrh (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1987

P S E U D O - F IN I T E

N E A R - F I E L D S

Ulrich Felgner Mathematisches Institut der Universitxt, Auf der Morgenstelle 10, D - 74 Tiibingen

Q 1 . Introduction The theory of finite near-fields is developed up to a high degree of perfection. Starting from Zassenhaus' fundamental classlfication of all finite near-fields ( 1 9 3 6 ) a large number of the most important problems in this area have been solved in the last two decades, most prominently by S.Dancs-Groves, H.Karze1 and H. Liineburg

.

In the case of infinite near-fields there are only sporadic results known. Most of these results are quite impressive and substantial, yet a general structure theory for infinite near-fields is still missing. Under the assumption of suitable finiteness conditions however some authors had been able to develop a satisfactory structure theory. Let me mention the locally-finite near-fields (S.Dancs Groves) and the locally compact near-fields (Kalscheuer, GrundhUfer). Pseudo-finiteness is another interesting finiteness condition and there is some hope that the class of all pseudo-finite near-fields might have a reasonable structure theory. It is the aim of this paper to do a few initial steps towardssuch a structure theory. The notion of a pseudo-finite commutative field is due to J.P. Serre L14I , chap.XIII, 52 A (commutative) field D is called quasi-finite, if D is perfect and has precisely one extension of each finite degree. A quasi-finite field D is called pseudo-finite if every absolutely irreducible variety defined over D has a Dvalued point. J . A x [ l ] proved that the pseudo-finite fields are precisely the infinite models of the first-order theory of finite fields. Thus we are led to define analogously

.

Definition. A near-field F is called pseudo-finite if F is an infinite model of the first-order theory of finite near-fields.

16

L! Felgner

Thus, a near-field F is pseudo-finite if F is infinite and elementarily equivalent to an ultraproduct of finite near-fields. 52.

remarks on finite near-fields

A (left distributive) near-field is a structure P=(F, + ,0 ) which satisfies all the axioms for a skew-field except possibly right distributivity (cf. Wlhling [16] , pp.42-43). Z ( P ) denotes the center and K(P) the kernel of P , where

K ( F ) = {x€F ;

+

\dy,zEF: (y+z)ox = yox

zox

).

We need a brief outline of the Dickson-Zassenhaus construction. For a detailed description see Zassenhaus 11 71 or Ellers-Karzel171. q=

Let

{

(*I

2

be a power of a prime p and let n

f o r a l l primes and

II :

if if

.In 4 In

then then

be such that

rlq-1 41q-1

.

-

Then I (qi 1 ) / (q - 1 ) ; O < i< n 1 is a complete set of residues modulo n . Let wbe a generator of the multiplicative group of GF(qn) and let R = <wn> be the subgroup generated by w n To each a EGF(qn) , a # 0, we associate an automorphism of the Galois field GF(qn) such that i - I)/(q - 1) a E JIa = ( x w xq ; X G GF(qn))

.

%

Finally we introduce on the set GF(qn) as follows: i f a=O aob = a.lla(b) i f a # O .

t

.

~

(

~

a new multiplication

0

i"

It can be shown that the set F = GF(qn) equipped with the addition of the Galois field GF(qn) and the new multiplication 0 is a near-field IF= (F, + , O ) such that Z(P) = K(P) GF(q). The isomorphism type of I!? depends on q , n and w The near-fields which are obtained by the Dickson-Zassenhaus construction are called f i n i t e D i c k e o n n e a r - f i e l d s . The fundamental theorem of Zassenhaus [17] states that up to seven exceptions all finite near-fields are either Galois fields or finite Dickson near-fields. The orders of the seven exceptions are 52, 72, 1 1 2 , 2321 2g2 and 2 59 The subnear-field structure of a finite near-field is analogous to the subfield structure of finite fields. More precisely, the following holds.

.

.

11

Pseudo-finite near-fields

THEOREM (S.Dancs Groves L 4 ] , [5] 1 : L e t P b e a f i n i t e D i c k s o n f o r some p r i m e p . n e a r - f i e l d , IF1 = qn , Z l P ) " = C F l q ) , where q=p' ( i ) I f B is a s u b n e a r - f i e l d i f P , t h e n H = p h w i t h hl 1.n . ( i i l Conversely, i f h l 4 . n , t h e n F has a unique s u b n e a r - f i e l d XI of power p h . Moreover B is a D i c k s o n n e a r - f i e l d , and i f z = g c d l j e , h ) , where O < j s n and j G l q n - l ) / ( p h - l ) (mod n l , t h e n Z t B ) = K l M I "= G F ( p z ) . COROLLARY (S.Dancs Groves): L e t P b e a f i n i t e D i c k s o n n e a r - f i e l d n of power q where Z l P ) C F l q l and q = 'p L e t fl b e a s u b n e a r - f i e l d of P of power qh Then Z(PI s B and Z l B l r C F ( q Z ) where a = g c d l n / h , h 1 .

.

.

Using these results we prove: THEOREM 2 . 1 : Let B b e a f i n i t e D i c k s o n n e a r - f i e l d of power qn G F ( q l where q is a power o f a p r i m e . L e t 1 b e such t h a t Zip1 Among a l l c o m m u t a t i v e s u b f i e l d s D of maximal? s u c h t h a t A21n lF w i t h ZlPI C_B t h e r e i s a u n i q u e maximal o n e , c a l l e d M l F ) , and we haue M I P I 2 CFlq' )

.

.

Proof. By the theorem of Dancs-Groves P has precisely one subClearly, Z ( P ) Z B It also follows near-field B of power q from gcd( n/l , A = A by the above corollary that B is commutative. If, on the other hand, D is any commutative subfield of F such that Z ( P )C _ ID , then \ID/= gd for some d Since d IDrGF(q ) = Z ( I D ) it follows from the above corollary that d = 2 = gcd( d , n/d). Hence d In and since A is maximal we obtain that dlX Since D is the only subnear-field of power gd and since B has a subnear-field of power gd it follows that D is contained in B We define M(F) = B 0

'.

.

.

.

.

.

Notation. M ( F ) = the unique maximal commutative subfield of the finite Dickson near-field P which contains the center of F. In contrast to Z ( P ) and K ( P ) cription of M ( F ) is known.

no 'internal' first-order des-

open problem.

Is there a first-order formula a ( x ) in the language of fields such that for any finite Dickson near-field P we have M(F)

= ( a E F : +(a)

1

?

(I. Felgner

18

We shall need the following lemma on near-field extensions. LEMMA 2 . 2 : L e t

IF b e a D i c k s o n n e a r - f i e l d of power qn s u c h t h a t Z ( P ) Y G F ( q ) , where q i s a power of a p r i m e . F o r any t E 1 (mod n ) t h e r e i s a D i c k s o n n e a r - f i e l d ID w h i c h c o n t a i n s B a s a s u b n e a r f i e l d such t h a t Z f P ) C Z f l D ) % C F l q t I and [Dl= q t n

.

.

Proof. n \t- 1 implies q n - 1 1 qt-' - 1 From (h)it follows that n\qn-1. Hence qtJ qJ (modqn-l)for all j and therefore (I)

(qtj- l)/(qt- 1)

(**)

(qtn- l)/(qn- 1)

(qj- l)/(q- 1) t

3

1

(modqn-l)for all j

(mod n)

Since F is a Dickson near-field there is a generator w of the multiplicative group of the field GF(qn) such that aob = a.Jla(b) for all a,b E GF(qn) with a # 0, where 0 denotes the multiplication in F and * denotes the multiplication in the Galois field a€ -u (qi-l)/(q-l) GF(qn) and JIa = (xxq i ; x E GF(qn)) is a cyclic subgroup of GF(qnt)# of index (u) = GF(qn)* T

= (qnt- l)/(q"- 1).

Select a generator o of

9= that

Er"

(tpln)

:T

GF(qnt)* Clearly

.

such that

a

= (G~) C _ 8" and 1 (mod n) hence R*o = %?aT = %* w (qi-l)/(q-l) = $j,(qti-l)/(q t-1) c

ftW

.

and put It follows from (**I This implies * (qti-l)/ (qt-l) oT = w

. - BG

(using ( + ) I . Now we associate to each a € GF(qnt), a # 0, a fieldautomorphism as follows: = ( x H x q ti ; xEGF(qnt)) +==+ a € ti-l ) / (qt-l 6a We introduce a new multiplication 0 on the set D = GF(qnt) as follows: aob=O if a = O , and a 0 b = a - 6 ~ ( b ) if a#O. As we have mentioned in the outset D = is a near-field of power qnt such that Z ( D ) !S GF(qt). By construction JIa is the restric, hence IF is a subnear-field of D . Clearly, Z ( D ) tion of ba contains a subfield of order q. By the uniqueness result of S. Dancs Groves we have hence 2 (P) C _ 2 (D)

$dq

.

...

LEMMA 2 . 3 : Let P I C B z C b e a t o w e r of f i n i t e D i c k s o n Pt n e a r - f i e l d s such t h a t Z I P I ) = Z f F Z )=... = Z f P t l C F ( q ) . If m d e n o t e s t h e number o f d i s t i n c t p o s i t i v e p r i m e s w h i c h d i v i d e q - I , then t 5 m

.

Pseudo-finitenear-fields

19

Proof. We have lFil = qni for suitable ni and nil nj By the corollary of Dancs Groves we have in addition gcd( n./ni 3

, ni

= 1

for all

for iC j.

1s i < jSt.

From this our claim readily follows.

5

3 . The center of pseudo-finite near-fields

the first-order language based on + , - , ,i5 and 0 , 1 , where x% denotes the inverse of x with respect to the multiplication 0 For any near-field F = (F,+,-, 0 ,O,l} let Th(F ) denote the ZNF -theory of F Thus Let

ZNFbe

,* the set of all sentences expressible in zNF which .

Th(F)=

are true in Then Th(fnf) =

n

I Th(F)

.

F. ;

F is a finite near-field}

is the first-order theory of finite near-fields. A near-field F is called pseudo-finite if ?!I is an infinite model of Th(fnf).

LEMMA 3 . 1 : If F is a pseudo-finite near-field, then ( i ) F is planar, ( $ 3 ) ZIP)= K ( E I (iii) Z(E) is a pseudo-finite field o r a finite field.

Proof. Ad (i): A near-field is planar (or projectivej if it satisfies the following -sentence: V x [ x # 1 -# 3 y: xoy = y + 1 ] This sentence belongs to Th(fnf) since all finite near-fields are planar (cf. Zemmer). Ad (ii): Since Z(F) and K ( F ) are first-order definable and since the equality Z(F) = K ( F ) holds in all finite near-fields P of power > 59*, the claim follows. Ad(iii): If F is a pseudo-finite near-field, then there is an ultraproduct Di/ F of finite near-fields Di such that F I (here Z denotes elementary equivalence, and is an ultrafilter on N ) . Then

zNF

.

Di/F

and the claim follows from Ax' characterization of pseudo-finite fields.

4

The planarity of finite near-fields can be proved quite easily. The function f: y M f (y) = ( y + 1 ) 0 ycL is obviously a bijection

20

(1.

Felgner

onto F - 11 1 (since F is finite). For each x E F from F - {O x # 1 there is hence a y E. F , y # 0 , such that f (y)= x, which means that x.y = y + l . 0

,

8

Our next result is a two-cardinal theorem which concerns the possible sizes of the center of a pseudo-finite near-field. THEOREM 3.2: Let p be any prime and 1 C m E hJ and H, arbitrary, T h e n there i8 a where m a 2 is a88umed in the ca8e p € { 2 , 3 1 pseudo-finite Dickson near-field P such that

.

Proof. Case 1: p = 2 and m12. Let n be any prime divisor of p" - 1. Then for each integer i the pair p" , ni fs a Dickson pair and there is a near-field IFi of power pmn such that Z(Pi) GF(pm 1 . Let be a free ultrafilter on the set of IFi/ be the ultraproduct all natural numbers and let D = over all these finite near-fields. Clearly D is infinite and by E o ~ ' theorem, Z ( D ) GF(pm). Notice also that D is a Dickson near-field. Now our claim follows from the Lawenheim-Skolem theorem. Case 2: p = 3 and mz2. If p" - 1 = 3m-1 = 2t for some t I then m = 2 and t = 3 (see Passman 1131 ,lemma 19.3) Hence, if m 3 3, then there is an odd prime n which divides pm - 1 and for each integer i E N I pm , ni is a Dickson pair. If m = 2 then 9,2' is a Dickson pair for eacp i E N Thus there are Dickson near-fields IFi of power pm" such that Z (Pi) GF(pm) N o w proceed as in case 1. Case 3: pz5: and l f m . If p is not a Fermat prime, then there is an odd prime divisor n of pm - 1 and pm , n i is a Dickson pair for each i Zl. If p is a Fermat prime, then 41pm-1 (since 55.1 and hence pm,Zi is a Dickson pair for each i 1. Now proceed as above. 0

.

.

.

It is easy to see (using ultraproducts and the LawenheimSkolem theorem), that for each ordinal q 2 0 there are pseudofinite Dickson near-fields of any prescribed characteristic, F , such that IZ(P,I = I K ( F ) I = \IF1 = N,

.

Hence the problem arises whether there are pseudo-finite nearThe answer is fields IF such that ( Z ( P ) (= No < IF1 = fit, positive as we shall show in the next theorem. I am grateful to

.

Pseudo-finite near-fields

21

D.Mundici (Florence) for calling my attention to Keisler's paper

.

ll2l

THEOREM 3.3:

There $ 8 an uncountable pseudo-finite near-field whose center ie countably infinite.

P

Proof. We consider the language L(Q) which is formed by adding to a new quantifier (Qx) with the interpretation "there are uncountably many x". For any prime p let E be the P following countable set of L(Q)-sentences:

orrJr

= Th(fnf)

E

P

u

{1+1+

...+),=

01

P 1 QX Vy: Xoy = y-x 1 u

u{

3 X1,"#Xk [ A xi#xj

u {Qx:x=x 1 u k

f i ( vy:

xioy= yoxi)] ; keN1. i=1 It follows from theorem 3.2 that each finite subset of E has P a model. Thus by the Fuhrken-Vaught compactness theorem for L(Q) (cf. Keisler [123 , p.27) E has a model F . Clearly, F has P characteristic p , is uncountable and has a countably infinite center. C l A

if1

COROLLARY 3.4:

Assume the generalized continuum hypothesis GCB. For every regutar cardinal Hd there is a pseudo-finite near-field P o f cardinatity whose center has cardinality

Nd

fl

b+¶

.

Proof. This is a direct consequence of theorem 3 . 3 and Chang's Two-Cardinal theorem (cf. Chang-Keisler [ Z ] , theorem 7.2.7).

0

We do not know whether for any pair of infinite cardinals, N/,< H, , there is always a pseudo-finite near-field P of power 8%whose center has power say

P *

5

4.

Locally-finite sub-nearfields

We show that each pseudo-finite near-field P has a unique maximal locally-finite subnear-field, called the "locally-finite socle" of IF. We also prove that each (finite or infinite) locally-finite near-field appears as the locally-finite socle of some pseudo-finite near-field. THEOREM 4.1: Let P be a pseudo-finite near-field of characteristic p , where p # O . Then P contains a unique maximal locally-finite sub near-fi e ld

.

U. Felgner

22

Proof. Let S(F) be the set of all finite subnear-fields of IF. The first-order statement " i f t h e r e a r e more than 5 g 2 e l e m e n t s t h e n t h e r e a r e p e l e m e n t s which form a s u b f i e l d c o n t a i n e d i n t h e c e n t e r " is true in all finite near-fields and hence in F too (this follows from Zassenhaus' classification of all finite near-fields) Thus GF(p) E S ( F ) and S ( F ) is non-empty. Let L(F) denote the subnear-field of P generated by u S ( F ) We claim that S(F) is upwards directed. First step: F is elementarily equivalent to an ultraproduct of finite near-fields, where F is infinite. Hence we may assume without loss of generality that P is elementarily equivalent to an ultraproduct of finite Dickson near-fields. If A E S ( F ) then A is a subnear-field of almost all components of the ultraproduct (by the theorem of Log). It follows from the theorem of Dancs Groves that A is a Dickson near-field. Thus we have proved that all elements of S(F) are finite Dickson near-fields. Second step: By the theorem of S.Dancs Groves the following sentence holds in all finite nearfields: " i f there are p a elements x l ,x2 forming a subnearf i e l d and if t h e r e a r e p b e l e m e n t s y I J y 2 , , . . forming a subn e a r - f i e l d and if c = l c m l a , b ) , t h e n t h e r e a r e p c e l e m e n t s z l , z2,... f o r m i n g a s u b n e a r - f i e l d such t h a t a t 1 t h e p a e l e and a l l t h e p b e l e m e n t s y l , yz,. are m o n g ments xA, x2, t h e s e p e l e m e n t s z l , z2, .I1 ( lcm(a,b) is the least common multiple of a and b) This sentence, hence, belongs to Thffnf) and is therefore true in P. Thus, if A,BES(F) , ( A ( = p a and IB\ = p b and c=lcm(a,b), then there is a C E S ( F ) such that C contains A and B as subnearfields and I C l = pc Thus S(F) is upwards directed and it follows that S ( F ) is a subnear-field, whence L(F) = uS(F).If ID is any locally-finite subnear-field of F then clearly D G L(F) and the theorem is proved. 0

.

.

,...

...

u

...

..

.

Let F

be a pseudo-finite near-field. If F has characteristic 0 then P has no finite subnear-fields. But if F has non-zero characteristic p then GF(p) is a (central) subfield of F . In this case the maximal locally-finite subnear-field of F is non-trivial, finite or countably infinite. Definition. If P is a pseudo-finite near-field of non-zero characteristic, then L(P) denotes the maximal locally-finite sub-

23

Pseudufinitenear-fields

near-field of

IF.

COROLLARY 4 . 2 : Let P I and F 2 be pseudo-finite near-fields of non-zero characteristic. If P I and P 2 are elementarily equivalent, then L I P l ) 2 L I P 2 ) . Proof. This is a direct consequence of the proof of theorem 4 . 1 . 0 The isomorphism type of a locally-finite near-field can be characterized by socalled Steinitz numbers and sequences of polynomials (cf. Dancs Groves [6] 1. These are first-order characterizations of L(F). THEOREM 4 . 3 : If lL is any locally-finite Dickson near-field, then there is a pseudo-finite Dickson near-field P such that N lL = L / I F I . Proof. Case 1: IL is infinite. Let S ( L ) be the set of all finite subnear-fields of IL. For AES(L) put t(A)=(B€S(IL) ; A G B 1. The set S(L) is upwards directed (see the proof of 4 . 1 ) . Hence, if A,B C S(L) , then there is a C E S(E) such that A S C and B G C and we have t(C) t(A) nt(B), where t(C) # @ Thus ( t(A) ; A € S (IL) ) has the finite intersection property and can be extended to an ultrafilter i%- on the countably infinite set S(IL). Let IF be the ultraproduct of the near-fields in S(L) modulo , i.e.

.

For any a € IL let fa the following function from S(IL) into the Cartesian product of all A € S(L) : fa(A) = a if a E A , and fa (A)= O otherwise. If denotes the equivalence class of fa modulo then a w f a / s is an embedding of IL into F Thus ILC_ L(P). It follows easily from Log' theorem that a finite subnear-field H of F is contained in some A for A€S(L) and hence in L Thus L = L(F)

.

.

.

is finite. We need the following special case of Case 2 : IL Dirichlet's prime number theorem: f#)

For every natural number n there are infinitely many primes p such that p = l (mod nl.

For a direct proof of

(#)

see Estermann

[a] or R.A.Smith

.

24

U. Felgner

Since E i s a finite Dickson near-field we have l l L l = gn Z(L) GF(q) where q ='p is a power of a prime p By ( d ) there are infinitely many primes ni ( i € N ) such that ni = - 1 (mod n) and ni does not divide R By lemma 2.2 there are Dickson near-fields Fi such that

.

and

.

=

qnri = pRnRi

z(Pi) z

I

G F ( ~ ~ ~

such that IL is a subnear-field of Fi and the center of Pi contains the center of L. Now let be a free ultrafilter on the set fbf of all natural numbers and let P be the ultraproduct of all the Fi ( iE ] N ) modulo The theorem of Eo's implies that E is a subnear-field of P I whence L C L ( F ) Now let H be any finite subnear-field of P . Since the existence of H can be expressed by a first-order formula and since H is unique in its order it follows from the theorem of to'; that there are infinitely many i such that M is a subnearfield of Fi Thus, if M Fi and H S F I then by the j theorem of Dancs Groves we have h( 4 n n and h 1 P n n I where \HI= ph It follows from the choice of the primes Ti that h and hence that H is a subnear-field of E This proves that E = L ( P ) 13

.

.

.

.

.

.

Remark. The use of Dirichlet's prime number theorem in the proof of theorem 4 . 3 can be avoided. Let B0 , 8, r..r8nr... be the sequence of all primes different from p and coprime to n . Put ni = ei 'Q'n) I there 'p is Euler's totient function. By Euler's lemma ni 5 1 (mod n) Now proceed as above.

.

Notice that similarly as in theorem 4 . 3 one can prove that f o r each finite or locally-finite field IE there i s a pseudo-finite field M such that lE = L ( K )

.

9 5. Maximal commutative subfields A well-known theorem of P.Hal1, C.R.Kulatilaka and M.I.Kargapolov states that each infinite locally-finite group has an infinite abelian subgroup. We ask whether analogously each pseudofinite near-field has an infinite commutative subfield. Notice that according to theorem 3.2 the center could be finite1

THEOREM 5.1: Each peeudo-finite near-field commutative eubfield.

P has a n infinite

25

Pseudo-finite near-fields

Proof. If F has characteristic 0 then 0,l E Z ( F ) = K ( P ) implies that the field of all rational numbers is contained in Z(F) and hence Z(F) is infinite. Assume from now on that F has characteristic p where p is a prime, pfo. P is an ultraproduct of finite near-fields, Case 1:

If Z ( p ) is infinite then we are done. Assume therefore that Z ( P ) is finite, Z(F)= GF(q), where q ='p for some 1 4E R\J Since F is infinite we may assume w.1.o.g. that all Mi are finite Dickson Since Z ( P ) near-fields of characteristic p Put IM i = qni has q elements we may, by the theorem of Lo's, assume w.1.o.g. that the center of each M i has precisely q elements. Hence zpT( n,) C _ m(q- 1) , where for any natural number k , W(k) denotes the set of prime divisors of k. Let di be the largest natural 2 number such that di Ini Claim: F contains an infinite tower of commutative subfields which all contain Z(F). Proof of the claim: Let M be any subfield of F such that H. By the theorem of .Lo& we may assume that H is a subfield of each Mi such that Z(F)=GF(q) = Z ( M i ) sM . Put IlH\=q h 2 Then h ni for all i E (cf. theorem 2.1). For each W Em(q-1)

.

.

.

.

Z(P)c

.

I

define 2

E n = ( i E N ; (nh)

4

I ni 1

If Err $- for each r r W(q ~ - 1 1, then E* = N UrE, ; rr6 W(q-1) 1 belongs to Let t be the product of all primes in V ( q - 1). Then W ( n i ) & W { q - 1 ) implies ni 1 (th)2 for all i E E * Thus the numbers ni are bounded on a filter set. By the theorem of bo's it would follow that F is finite, a contradiction. There is, hence, a prime n E m(q-1) such that En E Then by theorem 2.1 GF(qh") is a subfield of each M i (for i E E n ) containing the center of Mi. Now En E implies that similarly h GF(q " ) is a subfield of F containing the center of P. Thus M can be extended and our claim follows. It follows from the claim that F contains an infinite locally finite nearfield D such that Z(P)C D , where D is commutative.

.

.

F.

Case 2: F is an arbitrary pseudo-finite near-field. But then is elementarily equivalent to an ultraproduct of finite nearfields and our claim follows easily. c]

F

LI. Felgner

26

As a corollary to the proof of theorem 5.1

we obtain the following

surprising fact. COROLLARY 5 . 2 : Let lF be a pseudo-finite near-field of non-zero characteristic. (i) if Z ( l F ) is finite, then L ( F ) is infinite; (iil if L ( P ) is finite, t h e n Z(P) is infinite.

Proof. (i) follows from the proof of theorem 5.1 and (ii) is just the contraposition of (i). We are now able to extend a result from 5 4 . In theorem 4 . 3 we proved that for any locally-finite Dickson near-field It there is at least one pseudo-finite near-field P of power such that L = L ( F ) A natural question is whether there is only one such F , and if not, how many such F' s are there?

.

THEOREM 5 . 3 :

Let lL be a finite D i c k s o n near-field. I n e a c h uncountable cardinality K there are precisely '2 pairwise nonof power K such that isomorphic pseudo-finite near-fields F lL = LIB).

Proof. By theorem 4 . 3 we know that there is at least one pseudofinite near-field F such that It = L ( F ) By corollary 5.2 Z(P) is infinite and by lemma 3.1 Z(P) is pseudo-finite. Hence Z(P) is not algebraically closed. It follows from a theorem of Cherlin-Shelah [ 3 ] (cf. Felgner 191 , Satz 3 . 1 1 ) that Z ( P ) is not super-stable. It follows readily that also F is not superstable (to see this, work with types over subsets of the center). By a theorem of Shelah (cf. Felgner [g] , Satz 4 . 3 ) there are in each uncountable cardinal K precisely '2 pairwise non-isomorphic structures of power K which are elementarily equivalent to

.

F.n

Pseudo-finite near-fields are not N o - categorical since the multiplicative group is not of bounded exponent (cf. Felgner [9] , Satz 3 . 1 ) . Hence for each finite (or locally-finite) Dickson nearfield lL there are at least two (and by a theorem of Vaught see Felgner 191 ,p.14 - at least three) pseudo-finite countable near-fields F such that L ( F ) = L. The exact number of such countable F ' s is not known to us. We also do not know whether an analogous version of theorem 5 . 3 holds in the case of infinite

21

Pseudo-finite near-fields

locally-finite near-fields E. It seems to be possible to construct for a given locally-finite Dickson near-field E different such that pseudo-finite nearfields F 1 , F 2 , . , with L = L ( F i ) L ( F ~n) Z(P1) = Z ( E ) # L ( F 2 ) n Z(FZ).This would prove that the first-order theory of a pseudo-finite near-field F is not uniquely determined by L ( F ) This would be in contrast to a result of Ax

.

.

[l ] which states that the first-order theory of a commutative pseudo-finite field M is uniquely determined by its subfield L ( X 0 (where L ( M ) = Abs(M) in the notation of Ax).

$ 6.

The

Dickson Property

According to called a Dickson fined on the set and for each a €

JI, = ( x

>

is H.Karze1 [11] a near-field F = { F ,+ , 0 near-field if a binary operation I' " can be deF such that IF' = ( F , + , * ) is a skew-field F - {Ol the mapping

w a-'*(ao x ) : x~ F)

is an automorphism of the skew-field verse of a in the skew-field).

F * (here a-l

is the in-

In many situations it is advantageous to know that the nearfield one is dealing with is a Dickson near-field. It is unknown whether all infinite near-fields are necessarily Dickson nearfields or not. In the sequel we shall discuss the problem whether pseudo-finite near-fields are Dickson near-fields.

zNF

In $ 3 we have introduced the first-order language of d near-field theory which is based on + ,-,0 , , 0 and 1 . In contrast to that language we let ZsF be the first-order language of skew-field theory which is based on + , - , * ,-1 ,O and 1 We introduce the following sets of axioms.

xNF= the usual set of axioms for near-fields 1 1 6 1 ,p.43) formulated in the language

.

(cf.WBhling

zNF .

zSF= the usual set of axioms for skew-fields formulated in the language

zD= the axiom

zsF .

xNFzsF "-".

(formulated in U ) saying that for each a # 0 , x a-1*(aox) is an automorphism with respect to 'I+" and

near-field F = ( F , + , - , 0 ,d,O,l) is hence a Dickson nearfield if a multiplication " * I ' can be defined on F such that

A

28

U.Felgner

(F,+,-I'I-18011}k

csF and

(FI+8-8a,-1,0

I s,081} xD . U zsF - structures xNF. The class of

zNs

Dickson near-fields are hence r e d u c t s of satisfying ZNF v ZSF u ZD to the language all Dickson near-fields is therefore a socalled p r o j e c t i v e or pseudo-elsmentarg class. Since all pseudo-elementary classes are closed under the formation of ultraproducts (cf. Chang-Keisler , p.177) we obtain the following [2] PROPOSITION 6 . 1 : near-fields.

U l t r a p r o d u c t s of D i c k s o n near-fields a r e D i c k s o n

0

It is a difficult and still open problem whether the class of all Dickson near-fields is an elementary class (i.e. ECA) .However the following holds. LEMMA 6.2: T h e c l a s s of a l l near-fields w h i c h are e m b e d d a b l e into D i c k s o n near-fields is an elementary c l a s s fi.e. ECA).

Proof. This class is closed under ultraproducts (use proposition 6.1) and closed under elementary equivalence (if F, 5 F2 where is a subnearfield of the Dickson near-field D1 I then F2 IF1 satisfies all universal -sentences which are provable in can be Z N F u Z S F u CD A compactness argument shows that F2 embedded into a Dickson near-field). 0

.

zNF

THEOREM 6.3: Every pseudo-finite near-field

lF c a n be e m b e d d e d

into a D i c k s o n near-field.

Proof. A pseudo-finite near-field F is elementarily equivalent to an ultraproduct of finite Dickson near-fields. Now apply proposition 6 . 1 and proceed as in the proof of lemma 6.2 to conclude that F can be embedded into a Dickson near-field. 0 We do not know whether all pseudo-finite near-fields are Dickson near-fields or not. However, if F is pseudo-finite nearfield such that F has finite dimension over 2 (P)then F is a Dickson near-field. This follows readily from GrundhCIfer [lo) since the multiplicative group of F is solvable of class 2.

Pseudo-finite near-fields

29

REFERENCES [l]

J.AX: The elementary theory of finite fields. Annals of Math. 88 (1968),pp.239-271.

[2]

- H.J.KEISLER:

C.C.CHANG

Model Theory. (North Holland,Amsterdam

1973). [3] G.CHERLIN

- S.SHELAH:

Superstable fields and groups.Annals of

Math.Logic 18(1980),pp.227-270. [4]

[s] [6]

S.DANCS: The sub-near-field structure of finite near-fields. Bull.Austral.Math.Soc. 5 (1971),pp.275-280. S.DANCS: O n finite Dickson near-fields. Abhandl.Math.Seminar Hamburg 37 (1972),pp.254-257. S.DANCS GROVES: Locally finite near-fields. Abhandl.Math.Seminar Hamburg 48 (1979),pp.89-107.

-

H.KARZEL: Endliche Inzidenzgruppen. Abhandl.Math.Seminar Hamburg 27 (1964),pp.250-264.

[7] E.ELLERS

[a] T.ESTERMA": 8 [g]

(

Note on a paper of A.Rotkiewicz. Acta Arithmetica

1963) ,pp.465-467.

U.FELGNER: Kategorizitdt. Jahresber.d.Dt.Math.Verein.

82 (1980)

pp. 12-32.

[lo] T.GRUNDH6FER: Transitive linear groups and nearfields with solubility condition. To appear in the J.of Algebra. [ll]

H.KARZEL: Unendliche Dicksonsche FastkBrper. Archiv d.Math. 16 (1965),pp.247-256.

[12]

H.J.KEISLER: Logic with the quantifier "there exist uncountably many". Annals of Math.Log1c l(1970) ,pp.l-93.

[13] D.S.PASSMAN: Permutation Groups. (W.A.Benjamin,Inc. ,New YorkAmsterdam 1968). [14) J.P.SERRE:

Corps locaux (Hermann, Paris 1962).

[IS] R.A.SMITH: A note on Dirichlet's (1981) ,pp. 379-380.

theorem. Canad.Math.Bul1. 24

[l6] H.W&HLING: Bericht Pber FastkBrper. 76(1974) ,pp.41-103.

Jahresber.d.Dt.Math.Vereln.

[17], H.ZASSENHAUS: Ubsr endliche Pas tkurper. Abhandl .Math.Semlnar Hamburg 1 1 (1936),pp.187-220.

This Page Intentionally Left Blank

Near-rings and Near-fslds, C . Betsch (editor) 0Elsevier SciencePublishers B.V. (North-Holland),1987

31

ON DERIVATIONS I N NEAR-RINGS HOGlARD E. BELL* and GORDON MASON The l i t e r a t u r e on n e a r - r i n g s c o n t a i n s a number o f theorems a s s e r t i n g t h a t c e r t a i n conditions implying commutativity i n r i n g s imply m u l t i p l i c a t i v e o r a d d i t i v e c o m m u t a t i v i t y i n s p e c i a l c l a s s e s o f nearr i n g s . We s h a l l add t o t h i s body o f r e s u l t s s e v e r a l c o m m u t a t i v i t y theorems f o r n e a r - r i n g s a d m i t t i n o s u i t a b l y - c o n s t r a i n e d d e r i v a t i o n s . 1.

INTRODUCTION Throughout t h i s paper N w i l l denote a zero-symmetric l e f t n e a r - r i n g w i t h

m u l t i p l i c a t i v e c e n t e r 2.

A derivation on N i s d e f i n e d t o be an a d d i t i v e

endomorphism s a t i s f y i n g t h e "product r u l e " D(xy) = xD(y) + D(x)y f o r a l l x , y ~N; elements x o f N f o r which D(x) = 0 a r e c a l l e d constants.

For x,y

E

N,

t h e symbol [x,y] w i l l denote t h e commutator x y - y x , w h i l e t h e symbol (x,y) w i l l denote t h e a d d i t i v e - g r o u p commutator x + y - x - y . The d e r i v a t i o n D w i l l be c a l l e d c o m t i n g i f [x,D(x)] = 0 f o r a l l X E N . F i n a l l y , N w i l l be c a l l e d

prime i f a,bEN and aNb = {O} i m p l y t h a t a = 0 o r b = 0. (Note t h a t t h i s d e f i n i t i o n i m p l i e s t h e usual d e f i n i t i o n o f prime n e a r - r i n g . It does n o t seem t o be known whether t h e two d e f i n i t i o n s a r e e q u i v a l e n t . ) 2.

PRELIMINARY RESULTS

We begin w i t h two q u i t e general and u s e f u l lemmas. LEMMA 1

L e t 0 be an a r b i t r a r y d e r i v a t i o n on t h e n e a r - r i n g N.

Then N

s a t i s f i e s t h e f o l l o w i n o partial d i s t r i b u t i v e law: (aD(b) + D(a)b)c = aD(b)c + D(a)bc f o r a l l a,b,c

E

N.

Proof

Note t h a t D((ab)c) = abD(c) + (aD(b) + D ( a ) b ) c , and t h a t D(a(bc)) = aD(bc) + D(a)bc = a(bD(c) + D(b)c) + D(a)bc = abD(c) + aD(b)c + D(a)bc.

Equating these two expressions f o r D(abc) now y i e l d s t h e announced

p a r t i a l d i s t r i b u t i v e law. LEMMA 2 divisor.

Proof

L e t D be a d e r i v a t i o n on

I f [u,D(u)]

= 0, then (x,u)

N, and suppose u c N i s n o t a l e f t zero i s a c o n s t a n t f o r every x

E

N.

From u ( u + x ) = u2 + ux, we o b t a i n u D ( u + x ) + D ( u ) ( u + x ) =

uD(u) + D(u)u + uD(x) + D(u)x, which reduces t o uD(x) + D(u)u =

D(u)u + uD(x).

Since D ( u ) u = uD(u), t h i s e q u a t i o n i s e x p r e s s i b l e as U ( D ( x ) + D ( u ) - D ( X ) - D ( u ) ) = 0 = uD((x,u)). Thus, D ( ( x , u ) ) = 0.

*Supported by t h e N a t u r a l Sciences and Engineering Research Council o f Canada, Grant No. A3961.

H.E. Bell and C.Mason

32 THEOREM 1

L e t N have no nonzero d i v i s o r s o f zero.

I f N admits a n o n t r i v i a l

commutinq d e r i v a t i o n D, then (N,+) i s a b e l i a n .

Proof L e t c be any a d d i t i v e commutator.

Then c i s a c o n s t a n t by Lemma 2.

Moreover, f o r any W E N , wc i s a l s o an a d d i t i v e commutator, hence a l s o a constant.

Thus, 0 = D(wc)

=

wD(c)

+ D(w)c

Since D(w) f 0 f o r

and D(w)c = 0.

some W E N , we conclude t h a t c = 0. 3.

PRIME NEAR-RINGS

Since prime r i n g s a r e t h e s e t t i n g i n which d e r i v a t i o n s i n r i n g s have been most f r u i t f u l l y studied, we proceed t o c o n s i d e r prime n e a r - r i n g s . LEMMA 3 (i) (ii)

L e t N be a prime near-rincl.

If z t Z \

{OI, then z i s n o t a zero d i v i s o r .

I f Z c o n t a i n s a nonzero element z f o r which z + z

E

Z, then (N,+)

is

abel i a n . ( i i i ) L e t D be a nonzero d e r i v a t i o n on N. and O(N)x = { O } i m p l i e s x = 0.

Then xD(N) = i01 i m p l i e s x = 0,

( i v ) I f N i s 2 - t o r s i o n - f r e e and D i s a d e r i v a t i o n on N such t h a t D2 = 0, then D = 0. Proof ( i ) I f Z E Z \ { O ) and zx = 0, then zNx = {OI, hence x = 0. ( i i ) L e t Z E Z \ I01 be an element such t h a t z + z E Z, and l e t x , y ~N. z + z i s d i s t r i b u t i v e we g e t ( x + y ) ( z + z ) = x ( z + z ) + y ( z + z ) =

Since

xz + xz + y z + yz = z ( x + x + y + y ) .

On t h e o t h e r hand, ( x + y ) ( z + z ) =

(x+y)z + (x+y)z = z(x+y+x+y).

Thus, x + x + y + y = x + y + x + y and

t h e r e f o r e x + y = y + x. (iii)L e t xD(N) = {Ol, and l e t r, s be a r b i t r a r y elements o f N. Then 0 = x D ( r s ) = x r D ( s ) + x D ( r ) s = x r D ( s ) . Thus, xND(N) = { O ] ; and s i n c e D(N) f { O l , x = 0.

A s i m i l a r argument works i f D(N)x = { O ) s i n c e Lemma 1 p r o v i d e s enough

d i s t r i b u t i v i t y t o c a r r y i t through. 2 ( i v ) For a r b i t r a r y x,y E N, we have 0 = D ( x y ) = D(xD(y) + D ( x ) y ) = X D ’ ( ~ ) + D(X)D(Y) + D(X)D(Y) + D ~ ( X ) Y= P D ( ~ ) D ( ~ )Since . N i s 2-torsion-free, D(x)D(N) = I01 f o r each X E N, and ( i i i ) y i e l d s D = 0. THEOREM 2

I f a prime n e a r - r i n g N admits a n o n t r i v i a l d e r i v a t i o n D f o r

which D(N) 5 Z, then (N,+)

i s abelian.

Moreover, i f N i s 2 - t o r s i o n - f r e e ,

then

N i s a commutative r i n g . Proof

L e t c be an a r b i t r a r y constant, and l e t x be a non-constant. Since D ( x )

Z \ {O},

Then

i t follows e a s i l y t h a t

E Z. Z. Since c + c i s a c o n s t a n t f o r a l l c o n s t a n t s c, i t f o l l o w s from Lemna 3 ( i i ) t h a t (N,+) i s a b e l i a n , p r o v i d e d t h a t t h e r e e x i s t s a nonzero constant. Assume, then, t h a t 0 i s t h e o n l y c o n s t a n t . Since D i s o b v i o u s l y commuting,

D(xc) = xD(c) + D ( x ) c = D ( x ) c

C E

E

33

On derivations in near-rings i t f o l l o w s from Lemma 2 t h a t a l l u which a r e n o t zero d i v i s o r s belong t o t h e

center o f (N,+), then f o r a l l y

denoted by C(N).

E

I n particular, i f x

N, we g e t D ( y ) + D ( x )

-

D(y)

-

0, D(x)

E

C(N).

But

D ( x ) = D ( ( y , x ) ) = 0, hence

(Y,X) = 0. We complete t h e p r o o f by assuming t h a t N i s 2 - t o r s i o n - f r e e and showing t h a t N i s comnutative. a,b,cEN;

By Lemma 1, (aD(b)

+

and u s i n g t h e f a c t t h a t D(ab)

aD(b)c + D(a)bc.

D(a)b)c E

= aD(b)c

+ D(a)bc f o r a l l

Z, we g e t caD(b) + cD(a)b

=

Since (N,+) i s a b e l i a n and D(N) 5 Z, t h i s e q u a t i o n can be

rearranqed t o y i e l d D(b)[c,a]

= D(a)[b,c]

f o r a l l a,b,c

E

N.

0 and Suppose now t h a t N i s n o t commutative. Choosino b,cEN w i t h [b,c] 2 l e t t i n g a = D ( x ) , we g e t D (x)[b,c] = 0 f o r a l l X E N; and s i n c e t h e c e n t r a l 2 2 element D ( x ) cannot be a nonzero d i v i s o r o f zero, we conclude t h a t D ( x ) = 0

for a l l x

E

N.

But by Lemma 3 ( i v ) , t h i s cannot happen f o r n o n t r i v i a l 0.

The n e x t theorem extends t o n e a r - r i n a s a theorem o f H e r s t e i n [2]. L e t N be a prime n e a r - r i n g a d m i t t i n g a n o n t r i v i a l d e r i v a t i o n

THEOREM 3

such t h a t [D(x),D(y)]

= 0 for a l l x,y~N.

Then

(N,+) i s a b e l i a n ; and

D

if N

i s 2 - t o r s i o n - f r e e as w e l l , then N i s a commutative r i n g .

Proof The argument used i n t h e p r o o f o f Lemma 3 ( i i ) shows t h a t i f b o t h z and z + z comnute elementwise w i t h D(N), then zD(c) = 0 f o r a l l a d d i t i v e commutators c.

Thus, t a k i n g z = D(x), we g e t D ( x ) D ( c ) = 0 f o r a l l

D(c) = 0 by Lemma 3 ( i i i ) .

X E

N, so

Since wc i s a l s o an a d d i t i v e commutator f o r any

W E N , we have D(wc) = 0 = D(w)c; and another a p p l i c a t i o n o f Lemma 3 ( i i i ) g i v e s c = 0. Assume now t h a t N i s 2 - t o r s i o n - f r e e . By t h e p a r t i a l d i s t r i b u t i v e law, 2 2 D(D(x)y)D(z) = D(x)D(y)D(z) + D ( x ) y D ( z ) f o r a l l x , y , z ~ N; hence D ( x ) y D ( z ) = D ( D ( X ) Y ) D ( Z )- D ( X ) D ( Y ) D ( Z )= D ( z ) ( D ( D ( ~ ) -~ )D ( ~ J D ( A )= D ( Z ) D * ( X ) =~ D2(x)D(z)y. Thus

-

D ~ ( X ) ( Y D ( Z ) ~ ( z ) y )=

o

f o r a l l x,y,z

6

N.

Replacing y by y t , we o b t a i n D2(x)ytD(z) = D2(x)D(z)yt = D2(x)yD(z)t 2 E N , so t h a t D ( x ) N [ t , D ( z ) ] = {Ol f o r a l l x,t,.z

f o r a l l x,y,z,t

E

N.

The

primeness o f N now shows t h a t e i t h e r D2 = 0 o r D ( N ) 5 Z; and s i n c e t h e f i r s t o f these c o n d i t i o n s i s impossible by Lemma 3 ( i v ) , t h e second must h o l d and

N

i s a commutative r i n g by Theorem 2. I n t h e p r o o f o f t h e n e x t r e s u l t we use a c o n s t r u c t i o n from [l],c a l l e d t h e l o c a l i z a t i o n o f N a t Z \ {O}, which a l l o w s an embedding o f N i n a n e a r - r i n g w i t h 1. (x,z) (x,,z,)

N*

S p e c i f i c a l l y , i f Z i s nonzero, c o n s i d e r t h e s e t o f ordered p a i r s

with x N

E

N and z

(x,,z2)

E

Z \ (01.

Obtain an equivalence r e l a t i o n

t o mean t h a t x 1 z 2 = x 2 z 1 .

- by d e f i n i n g

L e t N* be t h e s e t o f equivalence

H,E. Bell and G.Mason

34

classes <x,z> w i t h a d d i t i o n and m u l t i p l i c a t i o n d e f i n e d by <xl,zl>

+ <x2,z2>

=

<xl z2 + x2z1, z1z2> and <x z ><x2,z2> = <x1x2.z1z2>; and embed N i n N* by mappinq x t o <xz,z>. THEOREM 4

x

-

D(x)

E

L e t N be any prime n e a r - r i n q a d m i t t i n q a d e r i v a t i o n D such t h a t

Z for all

X E

N.

Then (N,+)

commutinq and N i s 2 - t o r s i o n - f r e e , Proof

i s a b e l i a n by t h e arqument o f Theorem 2.

0 i s t h e o n l y constant.

-

I f i n addition D i s

I t i s c l e a r t h a t c o n s t a n t s a r e i n Z; hence, i f t h e r e e x i s t s a nonzero

constant, (N,+) x

i s abelian.

then N can be embedded i n a n e a r - f i e l d .

-

D(x), and conclude t h a t x

x - D(x) + x

-

Thus, we assume t h a t

We can then apply Lemma 2 t o any nonzero u o f t h e form

-

D(x) = x + x

D(x)

c(N).

E

D(x+x)

E

I t f o l l o w s t h a t f o r each x

E

N,

Z; hence we g e t (N,+) a b e l i a n by

Lemma 3 ( i i ) once we demonstrate t h a t t h e r e e x i s t s a nonzero element o f t h e form x

-

D(x). Assume, t h e r e f o r e , t h a t x

-

D(x) = 0 f o r a l l x

E

xy = D(xy) = xD(y) + D ( x ) y = xy + xy, so xy = 0. prime n e a r - r i n g , hence (N,+)

N.

Then f o r a l l x , y ~N,

But t h i s i s i m p o s s i b l e i n a

i s abelian.

We now i n t r o d u c e t h e a d d i t i o n a l hypotheses t h a t D i s commuting and N i s 2-torsion-free.

Since Z has been shown t o be nonzero, we can l o c a l i z e a t

Z \ to), embeddinq N i n a n e a r - r i n q N* w i t h 1. <x,z> has a r i q h t i n v e r s e i f t h e r e e x i s t s y o t h e r elements

X E

N exceptional.

Note t h a t i n N*, t h e element

N such t h a t xy

E

Z \ {O}.

Call a l l

Note t h a t i f D i s t h e t r i v i a l d e r i v a t i o n ,

t h e r e a r e no nonzero e x c e p t i o n a l elements, so N* i s a

n e a r - f i e l d ; hence, we

assume t h a t D f 0. 2 I f x i s any e x c e p t i o n a l element, then x - D(x2) = x ( x - D(x) - D ( x ) ) = 0. 2 2 2 2 Thus, f o r e v e r y Y E N , we have x y - D(x y ) = x y ( x D(y) + D ( x 2 ) y ) = 2 2 -x D(y) = x ( - x D ( y ) ) E Z; hence x D(N) = {Ol, and by Lemma 3 ( i i i ) , x 2 = 0. We

-

now g e t D(x2) = 0 = xD(x) xD(x) = 0.

+

xD(x), and the absence o f 2 - t o r s i o n y i e l d s

Thus x ( x - D ( x ) ) = 0; and i f x f 0, t h e f a c t t h a t Z c o n t a i n s no non-

zero d i v i s o r s o f zero f o r c e s x e x c e p t i o n a l also, so xy

-

-

D(x)

D(xy) = xy

=

-

0.

But f o r e v e r y

Y E

N, xy i s

(xD(y) + D ( x ) y ) = -xD(y) = 0;

consequently, xD(N) = COl and t h e r e f o r e x = 0.

Thus, N has no nonzero

e x c e p t i o n a l elements and N* i s a n e a r - f i e l d . COROLLARY 1

L e t N be a prime distributively-qenerated n e a r - r i n g a d m i t t i n g

a d e r i v a t i o n D such t h a t x

-

D(x)

E

Z for a l l x

E

N.

Then N i s a commutative

rinq.

Proof By t h e theorem, (N,+) i s a b e l i a n , which i n t h e s e t t i n a o f d i s t r i b u t i v e l y generated n e a r - r i n g s f o r c e s N t o be a r i n g . I n t h e presence o f d i s t r i b u t i v i t y o u r hypothesis on D i m p l i e s t h a t D i s commuting; hence i f D f 0 we can invoke a r e s u l t o f Posner [ 4 ] t o t h e e f f e c t t h a t a prime r i n g a d m i t t i n g a n o n t r i v i a l commutinq d e r i v a t i o n must be commutative. event t h a t D = 0, C o r o l l a r y 1 i s obvious.

I n the

On derivations in near-rings 4.

35

NEAR-RINGS WITHOUT NILPOTENT ELEMENTS I n our f i n a l s e c t i o n we c a p i t a l i z e on t h e good b e h a v i o r o f a n n i h i l a t o r

ideals w i t h respect t o derivations. LEMMA 4

L e t N have no nonzero n i l p o t e n t elements.

d e r i v a t i o n D, a n n i h i l a t o r s a r e D - i n v a r i a n t .

Then, f o r any

Thus, t h e r e e x i s t s a f a m i l y o f

completely prime i d e a l s { P a l a ~ h )such t h a t N i s a s u b d i r e c t p r o d u c t o f t h e and such t h a t , f o r each a E A , t h e d e f i n i t i o n E ( x + Pa) =

n e a r - r i n g s NIP,

D(x) + Pa y i e l d s a d e r i v a t i o n

on NIPa.

Proof R e c a l l t h a t i n n e a r - r i n g s w i t h o u t n i l p o t e n t elements, t h e r e i s no d i s t i n c t i o n between l e f t and r i q h t a n n i h i l a t o r s . 0 = D(xy) = xD(y)

+ D(x)y.

Suppose t h a t xy = 0.

Then

L e f t - m u l t i p l y i n g by x and n o t i n g t h a t N i s an

IFP-near-rinq, we g e t x 2 D(y) = 0, so t h a t ( x D ( y ) ) * = 0 and hence xD(y) = 0. The remaining c o n c l u s i o n s f o l l o w from t h e f a c t t h a t o f N/P, [3,

f o r a f a m i l y {Pa)

N i s a subdirect product

o f c o m p l e t e l y prime i d e a l s which a r e a n n i h i l a t o r s

9.361. THEOREM 5

L e t N have no nonzero n i l p o t e n t elements, and suppose t h a t N

admits a d e r i v a t i o n D such t h a t x

-

D(x)

E

Z for all

X E

N.

Then ( N , + )

is

abelian.

Proof Use t h e s u b d i r e c t - p r o d u c t r e p r e s e n t a t i o n from Lemma 4, and l e t 1 has no nonzero d i v i s o r s o f zero, be a t y p i c a l f a c t o r n e a r - r i n g N/Pa. Then hence is c e r t a i n l y prime and t h u s

(m,+)

is a b e l i a n by Theorem 4.

REFERENCES

[l] H. E. B e l l , C e r t a i n n e a r - r i n g s a r e r i n g s , J. London Math. SOC. ( 2 ) 4 (1971 1, 264-270. [2] I . N.’Herstein, A n o t e on d e r i v a t i o n s , Canad. Math. B u l l . 21 (1978), 369-370. [3] G. P i l z , Near-rings, 2nd Ed., North-HoTland, Amsterdam, 1983. [4] E. C. Posner, D e r i v a t i o n s i n prime r i n g s , Proc. Amer. Math. SOC. 8 (1957), 1093-1 100.

Howard E. B e l l Mathematics Dept. Brock U n i v e r s i t y S t . Catherines, O n t a r i o Canada L2S 3A1

Gordon Mason Dept. o f Math. U n i v e r s i t y o f New Brunswick F r e d e r i c t o n , New Brunswick Canada

This Page Intentionally Left Blank

Near-ringsand Near-felds, G. Betsch (editor) 0 Elsevier Science Publishen B.V. (North-Holland), 1987

37

EMBEDDING OF A NEAR-RING INTO A NEAR-RING WITH IDENTITY Gerhard BETSCH Universitat Tubingen, Mathematisches I n s t i t u t D-7400 Tubingen, Fed. Rep. Germany

I t i s well known, t h a t an a r b i t r a r y near-ring N may be embedded into a near-ring N w i t h identity. Details and references a r e t o be found, e.g., i n [3; 9 1 , section c l . Also, i t i s very well known, t h a t any ring A i s an ideal of a ring A* w i t h identity [ Z ; p. 111. I n t h i s note we present a c l a s s of near-rings, which cannot be one-sided ideals i n a near-ring w i t h identity. And we characterize those nearrings, which do have an embedding a s a one-sided ideal into a nearring with identity. We follow the terminology and notation of 131, except t h a t we consider l e f t near-rings, and consequently our near-rings a c t from the r i g h t on N-groups. Furthermore, we r e s t r i c t our consideration to zero-symmetric near-rings. Hence, "near-ring" in the sequel always means "zero-symmetric l e f t near-ring". N will always denote a near-ring. W e a r e looking f o r "nice" embeddings of N i n t o a near-ring ?I with identity. I f (r,+) i s a (not necessarily commutative) group, then M o ( r ) denotes the near-ring (with i d e n t i t y ) of a l l mappings of r i n t o i t s e l f which leave the neutral element of r fixed. The additive groups of near-rings N , ?? a r e denoted by N t , ?I+,

,...

...

1. For the sake of convenience we recall a few known f a c t s , a ) If r i s a f a i t h f u l N-group, then N i s embeddable i n t o Mo(r), and will be considered as a subnear-ring of Mo(r). Conversely, i f N i s embedded into a near-ring N w i t h i d e n t i t y , then ?I+ i s a f a i t h f u l N-group. Consequently, embedding a near-ring into a near-ring with i d e n t i t y and finding a f a i t h f u l Ngroup are equivalent (easy) problems. b ) Let ( r , + ) be a group properly containing N+. Then the definition n, i f y E r\N yn := yn, i f y E N turns r into a f a i t h f u l N-group.

i

(This remarkably simple construction was discovered independently by several authors. See [31 f o r references. I learned i t long ago from unpublished manuscripts of H. Wielandt, which date back t o 1937-1952.)

38

G. Betsch

Now any subnear-ring 'fl o f

Mo(r),

which contains N and idr,

i s a near-ring

w i t h i d e n t i t y c o n t a i n i n g N.

-

2. D e f i n i t i o n . cp:N

An i d e a l embedding ( I E ) o f N i s a near-ring monomorphism

8 o f N i n t o a near-ring

W

w i t h i d e n t i t y such t h a t cp(N) i s a two-sided

i d e a l o f 8. Usually, we i d e n t i f y N and cp(N). Right and l e f t i d e a l embeddings (RIE and LIE) a r e defined i n an analogous way.

-3. Remrks. 1) For every group (r,+) t h e n e a r - r i n g Mo(r) has o n l y t r i v i a l [r; Theorem 11 and [ 3 ; Theorem 7.301. Hence the embedding o f N i n t o

ideals

Mo(r) according t o s e c t i o n 1 i s never an i d e a l embedding. 2 ) The m o t i v a t i o n f o r our i n t e r e s t i n I E ' s i s the f o l l o w i n g : We can expect a c l o s e r r e l a t i o n s h i p between the p r o p e r t i e s o f N and W i n the case, where N i s an i d e a l o f 8 than i n the case, where N i s merely a subnear-ring of K. One example: L e t r be a f a i t h f u l N-group o f type 1 . I f N i s an i d e a l of a subnear-ring o f Mo(r) w i t h i d e n t i t y , then r i s a f a i t h f u l N-group o f type 2, and t h e "best" d e n s i t y theorem i s a v a i l a b l e f o r 8. C f . [3; 5 4, sections c )

a

and d ) ]

.

4. Proposition. L e t N be a near-ring w i t h non-commutative a d d i t i v e group Nt and m u l t i p l i c a t i o n given by ab If

W

1

b, i f a # 0

0, i f a = 0.

i s a n e a r - r i n g w i t h i d e n t i t y 1 containing N, then N, cannot be a normal

subgroup o f

~

=

N,.

Consequently, N i s n o t a one-sided i d e a l i n 8.

Proof. We choose n,n' E N such t h a t n t n ' # n ' t n .

Now we consider t := n t l t n We have

- 1.

n t = n + n, n't = ntn'+n-n'.

I f t E N, then t = n t = n ' t , hence n = n ' t n - n u , which i s a c o n t r a d i c t i o n t o

the choice o f n and n ' . Hence t f N, and N+ cannot be a normal subgroup o f

Rt

Remark. The author does n o t know an example o f a n e a r - r i n g w i t h commutative a d d i t i v e group,

-~ 5. Theorem.

which has no LIE and no R I E . L e t N be a near-ring. The f o l l o w i n g statements a r e equivalent:

a ) N has a LIE. b ) There e x i s t s a f a i t h f u l N-group

r

(hence N

5 M0(T))

such t h a t

39

Embedding of a nearring into a nearring with identity

i) t h e mapping x + -1 + x t 1 o f Mo(r) i n t o i t s , e l f induces an automorphism o f N+ (1 = i d e n t i t y o f Mo(r)). ii)( n t 1z)m E N f o r a l l n,m E N and z E Z . (We c o n s i d e r t h e c y c l i c subgroup <1> o f Mo(r),

generated by 1 as

Z-module, which e x p l a i n s Iz).

-

c ) N has an IE.

Proof. a )

b ) . I f 8 i s a n e a r - r i n g w i t h i d e n t i t y c o n t a i n i n g N, then N+ i s a f a i t h f u l N-group as w e l l as a f a i t h f u l 8-group. By s u i t a b l e i d e n t i f i c a t i o n we o b t a i n N 5 8 5 Mo(r), and t h e i d e n t i t i e s o f a and Mo(r)

r

:=

-

c o i n c i d e . Now i f N i s a l e f t i d e a l o f 8, then i)and ii)a r e t r u e . b)

c). Let

r

be a f a i t h f u l N-group w i t h i ) and ii).We have N c - M := Mo(I').

L e t N+ be t h e subgroup o f M+ generated by N and {l}. From i ) we i n f e r , t h a t 'J+ = I n + lz I n E N, z E Z 3 and t h a t N, i s a normal subgroup o f 8., Now i i )

I+ i s m u l t i p l i c a t i v e l y Hence N+ i s t h e a d d i t i v e group

implies, t h a t

closed w i t h respect t o the m u l t i p l i c a t i o n

i n M.

o f a subnear-ring

W

of M c o n t a i n i n g N,

and 1 E 8. Furthermore, N i s a l e f t i d e a l o f 8. This f o l l o w s from i i ) and t h e f a c t , t h a t N+ i s a normal subgroup o f Nt.

Now l e t n,m,x E N and z1,z2 E Z. Repeatedly u s i n g i i ) and t h e f a c t , t h a t N+ i s a normal subgroup of

-El,

we o b t a i n , f o r s u i t a b l e elements n',n"

E N, t h e

equations ( n + m + lzl)(x+ 1z2) = (n+m+

lz 1 )x

-

( m + lzl)(x+ 1z2) Izl)(lz2) - (m+ 1zl)(tz2)

+ (n+m+

1zl)(lz2) - ( m + 1zl)(lz2) = n ' + n" + ( m t 1zl)(lz2) - ( m + lzl)(lz2) = n ' Hence N i s a l s o a r i g h t i d e a l i n N.

-

(mt

1z,)x

= n' + (n+mt

c ) ==+a)

t

n " E N.

i s trivial.

The p r o o f i s completed. C o r o l l a r y . I f N has a LIE, then N a l s o has a R I E . Remark. The a u t h o r does n o t know, whether t h e converse o f t h i s c o r o l l a r y i s t r u e or f a l s e . 6. Theorem. -~

equivalent:

L e t N be a n e a r - r i n g . Then t h e f o l l o w i n g statements a r e

40

C.Betsch

a) N has a RIE. b ) There exists a faithful N-group r (hence N 5 Mo(r)) such that i) the mapping x + -1 + x + 1 of Mo(r) into itself induces an automorphism ( 1 = identity o f Mo(r)). of N, i i ) ( n + l z ) m - ( 1 z ) m E N for all n,m E N and z E Z. The proof of this theorem follows the lines of section 5 and will be omitted.

REFERENCES [l] Berman, G. and Silverman, R.J., Simplicity of near-rings of transformations. Proc. Amer. Math. SOC. lo (1959), 456-459. [21 Jacobson, N., Structure of Rings. AMS Coll. Publ. vol.XXXVI1. 2nd Ed. Providence R.I. 1964. [3] Pilz, G . , Near-rings.2nd Ed. Amsterdam: North-Holland 1983.

Near-rings and Near-fields, C. Betsch (editor) 0 Elsevier Science Publishers B.V.(North-Holland), 1987

41

THE NEAR-RING OF SOME ONE DIMENSIONAL NONCOMMUTATIVE FORMAL GROUP LAWS

James R. C L A Y Department o f Mathematics l l n i v e r s i t y o f Arizona Tucson, A 7 85721

1.

INTRODUCTION

[ll

Hazewinkel shows i n

t h a t , f o r a commutative r i n g

A

with identity, i f

t h e r e i s t o be a one dimensional noncommutative group l a w over

b

must have an element

0

f

In fact, i f

He goes on t o show t h a t t h i s i s s u f f i c i e n t . r i n g w i t h i d e n t i t y , and

i s a nonzero element o f

b

o f f i n i t e order, t h e n t h e r e i s an element and

pc = 0

f o r some prime

p, F(X,

c

A

A

in

r i n g s are p r o v i d e d by and

If

t

A

A

i s a commutative

t h a t i s n i l p o t e n t and

such t h a t

c

f

0,

c2 = 0,

and Y ) = X + Y t cXYP

d e f i n e s a one dimensional noncommutative formal group law. p

A, then

t h a t i s n i l p o t e n t and o f f i n i t e a d d i t i v e order.

A = K[tl/(t2),

where

K

Examples o f such

i s a f i e l d o f characteristic

A = 7 t h e i n t e g e r s modulo p2. P2' d e f i n e s a one dimensional commutative formal group law, t h e n

i s an indeterminant, and by

F(X, Y)

F

t h e endomorphisms o f

form a r i n g .

However, i f

F

i s noncommutative, t h e

sum o f two endomorphisms need n o t be an endomorphism, so t h e r e i s no hope t h a t t h e endomorphisms of

t h e power s e r i e s over (Ao[[Xl],

tF),

(Ao[[X]],

+F,

w i l l be a r i n g .

F

Rut for

d e f i n e d by

A, w i t h zero constant terms, become a group

and when one adds composition

0)

tF

with identity

I(X)

= X.

0,

one gets a n e a r - r i n g

It i s i n t h i s s e t t i n g t h a t t h e

endomorphisms o f a noncommutative group law become i n t e r e s t i n g .

They form a

monoid w i t h respect t o composition, and each endomorphism i s a d i s t r i b u t i v e (Ao[[X]],

element i n t h e n e a r - r i n g w i t h respect t o

tF,

t F , 0).

Hence, t h e endomorphisms generate,

a d i s t r i b u t i v e l y generated (d.g.)

near-ring,

with identity

i ( X ) = X [2,33.

F o r t h e examples o f

A

p r o v i d e d above, these d.g.

n e a r - r i n g s t u r n o u t t o be

commutative r i n g s w i t h i d e n t i t y . It i s t h e purpose o f t h i s paper t o demonstrate t h i s , t o determine t h e s t r u c t u r e o f these r i n g s , and t h e s t r u c t u r e o f t h e i r

J.R. Clay

42 groups o f u n i t s . 2.

PRELIMINARIES

A

F o r a commutative r i n g

F(X, Y)

E

w i t h i d e n t i t y 1, t h e one dimensional formal group

A a r e t h e formal power s e r i e s i n two i n d e t e r m i n a t e s

laws over

Yll

ACCX,

o f t h e form

such t h a t t h e " a s s o c i a t i v e " c o n d i t i o n

i s valid.

If,i n a d d i t i o n ,

Otherwise

F

Let

F(X, Y ) = F ( Y , X ) , we say

A,[[X]]

i s commutative.

i s nonconnnutative.

A,[[X]]

denote a l l power s e r i e s over a ( X ) = al X

So

F

i s j u s t t h e power s e r i e s over

I f one d e f i n e s

on

+F

A,[[Xl]

2

a2 X

t

o f t h e form

A

... .

+

A

w i t h zero f o r t h e constant term.

by

a ( X ) +F B ( X ) = F(a(X), B ( X ) ) , then

(A,[[X]],

+F)

i s a group.

For

f, g

A,[[Xl],

E

if

fogfX) = f(g(X)).

t h e n (A,[[X]],

+F,

0)

i s a r i g h t d i s t r i b u t i v e near-ring w i t h i d e n t i t y

x.

l(X) =

The endomorphisms o f

a r e power s e r i e s

F

Y)) = F(a(X),

a(F(X, If

F

i s commutative, and i f

(End F,

tF,0 )

a E

End F

satisfying

A,[[X]]

af'f)).

denotes t h e endomorphisms o f

i s a ring with identity.

However, i f

F

and i f hom(F, F ) denotes t h e endomorphisms o f F, t h e n f o r t h e power s e r i e s o f course. of

(Ao"XI3,

So

hom(F, F)

generates, under

+F,

then

E

hom(F, F ) ,

a subgroup

A,[[Xll,

(Hom(F, F), +F)

+F).

( f +F g) = ( a o f ) (hom(F, F ) , o ) , under

ring

a, B

need not be i n hom(F, F ) , though i t i s i n

a tF f3

Since t h e semigroup a o

F,

i s noncommutative,

(hom(F, F), tF ( a

(Hom(F, F). +F, o),

+F,

o g)

0)

c o n s i s t s o f elements

for all

f,

g

E

A,CCX13,

a

that satisfy

t h e semigroup

generates a d i s t r i b u t i v e l y generated (d.g.)

a sub-near-ring of

(A,[[X!l,

tF, 0).

near-

43

The near-ring o f some noncommutative formal group laws These o b s e r v a t i o n s a r e e a s i l y proven.

THE ELEMENTS OF hom(F, F )

3.

A

Throughout t h e r e s t o f t h i s paper, gers modulo teristic

p2,

p,

A = Z 2,

If

p

a prime, t

and

then

or K [ t ] / ( t 2 ) ,

where

K

i s a f i e l d of c h a r a c -

i s an i n d e t e r m i n a n t . F

P

w i l l denote t h e noncommutative formal group l a w

+ pXYp.

F(X, Y) = X + Y

(1) A = K[tl/(t2),

and i f

w i l l denote e i t h e r t h e r i n g of i n t e -

then

F

w i l l denote t h e noncommutative formal group

1aw F(X, Y) = X + Y + tXYp.

(11)

I n t h i s l a t t e r case, t h e r e i s no harm i n t h i n k i n g o f t h e elements o f = b + Bt,

R

of t h e form

where

b, 8

E

and where

K,

t 2 = 0.

A

t o he

We have done

t h i s a l r e a d y i n (11). We want t o determine t h e elements o f and i n (11).

hom(F, F )

for

F

as d e f i n e d i n ( I )

Since t h e r e s u l t s a r e very c l o s e f o r each o f t h e two laws

( I ) and

( I I ) , and s i n c e t h e p r o o f s a r e very n e a r l y p a r a l l e l , we s h a l l develop our r e s u l t s concurrently. hom(F, F ) f o r ( I ) a r e e x a c t l y t h e elements

We s h a l l see t h a t t h e elments of o f t h e form

k

m

where

a

+ p

a(X) = ax

(1-a)

{O, 1) and each ak

E

E

5

1

k =O = {O,l,

a k XP

,

..., p-11.

F o r (11), t h e elements o f hom(F, F) w i l l be shown t o be e x a c t l y t h e elements o f t h e form (11-a) where

a

{O, 1) and each

E

ak

E

K.

Towards t h i s goal, we s h a l l see f i r s t t h a t i n each case, i f

a E hom(F, F ) ,

then

where

ai

E

A.

When we c o n s i d e r

a E hom(F, F )

a ( x ) = al However, f o r

F

x

as i n ( I ) . we s h a l l w r i t e

and

F

+ a2

x2 +

as i n ( I I ) , we s h a l l w r i t e

... .

J. R. Clay

44

.

2 a(X) = B1 X + R 2 X +...

I n each case, we study t h e i m p l i c a t i o n s o f t h e c o n d i t i o n

m

Now

m

1

a(F(X, Y)) =

ak F(X, Y ) k

or

1

a(F(X. Y)) =

k=l k

2,

2

Bk F(X, Y)k.

For

k=l

we have (X

+

Y)k

+

(X

+

Y)k

+ k(X + Y)k”(tXYP),

k(X

+

Y)k-l(pXYp),

for

(I),

for

(11).

F(X, Y I k =

So we get m

a(F(X, Y)) =

(I-b)

1

m

ak(X + Y)k

+

1

(pXYp)

k=l

kak(X + Y)k-l

k=l

and m

m

(11-b)

a(F(X, Y ) ) =

1

Rk(X

+

1

Y)k + (tXYP)

k=l

kBk(X + Ylk-’.

k=l

The r i g h t hand s i d e o f ( 2 ) y i e l d s

j>l

and m

(11-C)

F(a(X), a ( Y ) ) =

1

1 Bi

Bk(Xk + Yk) + t

i>l

k=l

BP Xi

Yip.

J

j>1 1 < i < p.

Suppose B i = 0.

and

Bp+i = 0.

C o l l e c t i n g terms o f degree

C o l l e c t i n g terms o f degree pap = 0

Also

and

i, we conclude

p + i, we conclude

tBp = t(bp + t B ) = t b

P

P’

By a r a t h e r messy i n d u c t i o n argument, one shows t h a t

= 0

if

1 < i < p,

This g i v e s described by

a

and t h a t

(I-a)

and

sketch t h e steps f o r involved.

pakp = 0

and

t h e form described i n (1). (11-a)

(I-a).

akp+i = 0

and

and Rkp+i

tRkp = tbkp. However, t o get

a

takes c o n s i d e r a b l e c a l c u l a t i o n .

The proceedure f o r (11-a)

i n t h e form We s h a l l

i s p a r a l l e l b u t more

For t h a t case, we s h a l l p o i n t out a few adjustments t h a t must he

made.

m

+

Xkp, where each a k = l kp i n g t h a t ( 2 ) be v a l i d , t h e e q u a t i o n With

ai = 0

ap+i = 0

a(X) = a1 X

pakp = 0, we get, f r o m i n s i s t -

45

The near-ring of some noncommutative formal group laws

Using

pakp = 0,

which f o r c e s k = ko pS-l

we get

kakp = 0 .

akp

So, i f

(ko, p) = 1,

with

f

then

0,

p

divides

k,

k o = 1 and so

one shows t h a t

p(k.

Setting

k = pS-l.

T h i s i n t u r n makes k a k ~ P k=l p

.

m

a(X) = al

x + 1

R e t u r n i n g our a t t e n t i o n t o (31, we now c o n s i d e r t h e consequence o f k

m

pal = pa!”.

C e r t a i n l y t h i s can be i f

and so

pal = 0

a(X) =

1

a

Xp

k=O p pa k = 0. A l t e r n a t i v e l y , i f pal # 0, we express al = mp + i w i t h P 2 1. So pal = p i and a! E (mp + i ) p E i p mod p ( i , p) = 1, 1 < i < p

with

.

-

Hence

pa!”

p = pip,

= ( p i ) i p = pip”.

which makes

T h i s means

i p = 1 and so

p i = pip+1

and consequently

i = np + 1 f o r some m

n. Thus k Xp , where

al = (m + n ) p + 1. which i n t u r n makes a(X) = X + 1 a k=O pk pa k = 0 f o r each k. P T h i s completes t h e proof of h a l f o f t h e f o l l o w i n g theorem, and t h e proof o f t h e other h a l f i s d i r e c t . THEOREM 1-1.

For

F

( I ) , we have

d e f i n e d by m

hom(F, F ) = {ax + p

1

ak

xp

k l a E { O , 11,

ak

E

Zpl.

k =O We w i l l get a s i m i l a r theorem f o r (11). al

= (m

+ n)p + 1 i f

B1 t = R f

+1

t,

separately, w i t h

that

pal

f

we a l s o get from t h e analogous c o n d i t i o n

0,

R1 = 1 + Blt

R1 = Blt

or

1 < i < p,

R1 = B l t .

conclude t h a t

Rip

= 0,

i s valid.

and

c o n d i t i o n s f o r an i n d u c t i o n : f i r s t segment-

Consider these two cases

considered f i r s t .

I n t h i s f i r s t case, assume t h a t ( 2 )

For

Above, where we saw t h a t

po < i < p l implfes

B

P

We a r e f o r c e d t o have

= BPt.

R~ = 0,

Set up t h e f o l l o w i n g

J.R. C b y

46

p1 < i < p2

second segment-

implies

= 0,

Ri

B1=Blt; Assume

P

P

k t h segment-

p k - l < i < pk

( k + l ) t h segment-

pk < i < p

k +1

B1 = 1 + Blt,

I n t h e case

For

implies

Bi

= 0,

0,

F

So we a l s o have

defined by ( I I ) , we have

THE ELEMENTS OF Hom(F, F ) . The elements o f

a

=

one uses c a l c u l a t i o n s p a r a l l e l t o those used

f o r ( I ) t o get t h e d e s i r e d r e s u l t s .

4.

R.1

B k = B k t . P P B1 = Bit, we g e t

So, i n t h e case

THEOREM 11-1.

implies

E

So

hom(F, F )

form a monoid w i t h respect t o

0,

and i f

hom(F, F )

and

hom(F. F)

i s a semigroup o f d i s t r i b u t i v e elements i n t h e n e a r - r i n g

(Ao[[X]],

+F,

0).

a group (Hom(F,

f, g

E

A,[[Xll,

then

which i s closed w i t h r e s p e c t t o

a d i s t r i b u t i v e l y generated (d.g.1 t o i d e n t i f y t h e elements o f For

a, B

E

hom(F, F),

9).

t h e semigroup hom(F, F ) w i l l generate

+F,

With respect t o

F), +F)

a o ( f +F g) = ( a o f ) +F ( a o

n e a r - r i n g (Hom(F, F), +Fs

0, 0).

thereby y i e l d i n g We now proceed

Hom(F, F). we s e t m k a = a X + ~1 a k x P , k=O

and

k bkXP,

OD

B=bX+a

1

k =O

A i s either p or ( I I ) , respectively.

where

(4)

or

t,

depending upon whether

A* = 0

I n each case, a +F B = a

+

F

and we get

.

+ AabXP+1

i s of form ( I ) , o r

47

The neawing of some noncommutative formal group laws I n d e r i v i n g t h i s , we use t h e f a c t t h a t

b

a +F B = 6 t

0)

making (Hom(F, F), +F,

a

F S i m i 1a r l y ,

{O, 11 and so bp = b. So a r i n g w i t h i d e n t i t y t ( X ) = X.

E

k

OI

1 (abk

a o B = abX +

(5)

k=O

(Hom(F, F),

+Fy

THEOREM 2 .

For

0,

a commutative r i n g w i t h i d e n t i t y .

t ( X ) = X)

i s a commutative r i n g w i t h i d e n t i t y Equation ( 4 ) shows t h a t a E Hom(F, F),

+,

T h i s g i v e s us

d e f i n e d hy ( I ) o r ( 1 1 ) , we have t h a t

F

(Hom(F. F), +F,

For

,

i s a commutative monoid, t h u s making

so (hom(F, F), o y t ( X ) = X)

usual

+ bak)XP

0,

t ( X ) = X)

t ( X ) = X.

hom(F, F )

i s properly contained i n

let

ma

be t h e sum of

m

a = [(m

-

a = ma

+

and l e t

I)

-

m

Hom(F, F ) .

w i t h respect t o t h e

a's

a] +F a.

One e a s i l y sees t h a t m

(6)

order

2,

So, i f

a e hom(F, F ) .

f o r any p

.

w i t h respect t o

Consider t h e case w i t h (m

(7)

-

i s an odd prime, each

+F, b u t f o r 4

a = 0, and has o r d e r

if

p

xaXP+'

p

a E hom(F, F )

has o r d e r

One e a s i l y gets k

m

B ) = (ma + nb)X + X

a ) +F ( n

has

a = 1.

if

2.

f

p = 2 , each

a E hom(F, F )

1 (mak

+ nbk)XP

k=O

+ [-m(n-1) a + n(n-1) b + mnab],lXP+l. Recall t h a t

a, b

E

11,

{O,

m(m-1) That i s , i f

[s(s

- 1)/2]X

s + Xu

ma + nh = s

so

7 P'

and

n ( n r- 1 ) b +mnah =- s ( s - I ) 2 .

a +

i s the c o e f f i c i e n t o f

XP+'

i s the coefficient of

THEOREM 3. For an odd prime

E

X

for

for

y.

y E Hom(F, F),

then

So we have

p, t h e r i n g Hom(F, F) c o n s i s t s of a l l power s e r i e s

o f t h e form

where

m

A, = p,

E

Z

P'

ai E K

and

respectively.

i

m

Take

or

y = maX

+

1

i=o

aiXP

+

a.

Z

E

P'

1

[m(m

-

depending upon whether

l)/21aXXP+1

and

X

= t

or

J.R. Clay

48 m

+

6 = nbX

- 1)/2]bAXp+1

i

1 biXP +

X

[n(n

i=O a, b

{O, 11.

E

a r b i t r a r i l y from

Then

8) y o 6 = mnabX + X

T

i

- 1)/23abAXP+1.

+ [mn(mn

+ mabi)XP

(nbai

i=O T h i s a g a i n shows t h a t

i s commutative i n

0

Now c o n s i d e r t h e case where 2

Hom(F, F ) .

p = 2.

For

m

k

akXek

+ AbXP+l

+

a = aX

A

1 a X p + XaXP+'. k=O c o n s i s t s o f a l l power s e r i e s o f t h e f o r m

a = XaXPtl

Hom(F, F )

Hom(F, F), where

3

and

a = aX

+

T

+x

aX

A

1

akXP , we g e t k =O One e a s i l y g e t s t h a t

k=O

a, bc Z2,

where A = 2,

and

ak

I(X)

If

E

depending upon w h e t h e r

Z2,

Hom(F, F )

M,

x

i s a unitary ( l e f t ) Ri s a r i n g on t h e s e t

where (r, m) + ( r ' , m ' ) = ( r

(9) ( r , m) Now

M

RXM,

module, t h e n t h e i d e a l i z a t i o n o f M, denoted by R

i s a commutative r i n g w i t h

0)

i s a commutative r i n g w i t h i d e n t i t y and

R

or

A = t

= X.

THE STRUCTURE OF THE R I N G

5.

ak

Again (Hom(F, F), +F,

respectively.

identity

K or

E

(R x M,

units

+,

of

U

0

-

+ r ' , m + m')

( r ' , m ' ) = (rr', r m ' + r'm).

i s a commutative r i n g w i t h i d e n t i t y (1, 0).

)

c o n s i s t s o f t h e elements (r,

RXM

m) where

r

The g r o u p o f

i s a unit of

R,

(r, m ) - l = (r-', -r-2m).

and

A = ((r, O)((r, 0)

Let

An B = ((1, O ) } ,

A

and

(r, m) = (r, 0 ) ( 1 , r - h ) .

and

11)

E

B

= ((1, m ) ( m

So

U

For

(r, m )

E

t h e elements o f

R

M,

x

J(R)

x

(r, m)k = ( r k , k r k - l m ) where

M,

M).

J(R)

Then

It = AR, s i n c e

11, and

i s a d i r e c t product o f

8, as a m u l t i p l i c a t i v e group, i s i s o m o r p h i c t o

that

E

a r e normal subgroups o f

R

and

A

Notice

R.

M, as an a d d i t i v e group.

so t h e n i l p o t e n t elements a r e

i s the radical o f

R.

J(R)

So

x

M

i s t h e r a d i c a l o f RXM. It i s easy t o see t h a t t h e i d e a l s o f RXM a r e t h e

an i d e a l o f R, and R'M

M'

i s a submodule o f

M

R ' x M',

where

R'

is

with the additional property that

- M'. C

Our r i n g s m

a = mX + X

(Hom(F, F),

1 aiXP

i=o

i

+ [m(m

+=,

-

0)

have t h i s s t r u c t i v e i f

1)/23XXp+1

from

Hom(F, F ) .

p # 2.

Take

It i s formed f r o m

49

The near-ring of some noncommutative formal group laws ( m y (ai))

E

N = {0,1,2,

Zp

THEOREM 4.

D e f i n e t h e map

M = KN

if

X = t.

Z*

product

P

Zpinodule of

J

0)

Here

Note t h a t

II.

M

by

i s isomorphic, v i a

M y where

M = 7N i f A = p, and P F ) i s isomorphic t o t h e d i r e c t

Hom(F, F )

i s a l l endomorphisms o f t h e

i

1 aiXP , i s a maximal i d e a l , and i=O Hom(F, F) are t h e isomorphic images, v i a form

to

$:7 XM + Hom(F, F) P

The group o f u n i t s o f Hom(F, The r a d i c a l

M.

x

m

if a. = t.

M = KNy

p, t h e r i n g (Hom(F, F ) , +F,

For an odd prime

t o the i d e a l i a t i o n o f the

J,

and

P'

U v denotes a l l mappings from V

Z module. P

i s a unitary

if X = p, M = IN

where

My

x

...) and

4,

p = 2

A t t h e end o f Section 3, t h e case where

='

The i d e a l s o f P' of t h e submodules o f M.

Hom(F, F ) / J

A

7

was considered.

An element

m

a

Hom(F, F )

E

i s o f t h e form

a = aX

+

X

1

k =O ak

E

Z2

if

x

= 2 , and

ak

E

K

if A = t.

a XZk k

+ XbX3, where a, b

Take

6 = cX + A

p

E

Z2,

ckXZk + XdX 3

k=l Then we can i d e n t i f y a <->

[(a,

a +F 6 <-> a

So, f o r Zqinodule

0

6 <->

B <->

b ) , (a,)].

[(c, d),

(c,)l,

[ ( a + c, b + d + ac), (ak f c,)] [(ac,

ad + bc), (cak + a c k ) l .

p = 2 , (Hom(F, F), +F, o ) i s isomorphic t o t h e i d e a l i z a t i o n o f t h e N M, where M = Z2 i f X = 2 , and M = KN i f A = t.

Since J ( Z 4 ) = {O, 2} i s a proper i d e a l o f

Z4, i t f o l l o w s t h a t t h e r a d i c a l m

1 akXPk + XbX3, k=O we again have t h e submodules

J o f Hom(F, F ) i s a l l power s e r i e s o f t h e form X Hom(F, F ) / J = Z2. of

M

Since

J(Z4)M = {O},

and M'

i d e n t i f i e d w i t h i d e a l s o f Hom(F, F), b u t i n t h i s case, one a l s o g e t s

i d e a l s from t h e i d e a l s o f t h e form

i s isomorphic t o

Z2

x

M.

J(Z4) x M'

in

Z4XM.

The group o f u n i t s

.

J.R. Clay

50 6.

AN INTERESTING EXTENSION. We have seen i n theorem 4 t h a t (Hom(F, F),

X = p,

Z

and each ai E Z if P' P A l l t h e powers of X a r e o f t h e form E

pi

and each

except f o r

c o e f f i c i e n t i s d e r i v e d from t h e c o e f f i c i e n t

t h a t t h e elements

+

a = kX

1

ai Xp

1

Xp+I,

+

A = t.

and i t s

X.

Hom(F, F )

i

0

X

if

a. E K

+ XaO o f

k

p = 2, we have seen t h e elements o f

If

2,

f

a r e o f t h e form

o f Hom(F, F )

k

motivates t h e idea o f

0)

We a l s o n o t i c e d , if p

i d e a l i z a t i o n of an Rmodule M.

where

+F,

t o be o f t h e form

mXXP+'

i=O

7 and m does n o t depend upon t h e c o e f f i c i e n t o f P' m o t i v a t e s l o o k i n g a t H, t h e s e t o f a l l power s e r i e s o f t h e form

where

k, m

where

k, in E Z

E

i

m

For

P' 8 = UX

+

ai Xp

+

This

and

XmXP+l

i =O

i

m

bi X p

A

1

a = kX + X

X.

+ XvXp'l,

we get

i=O

and

i

m

a o 6 = kuX +

1 (uai

i=O

+ kbi)XP

2 p+l + (kv + mu )AX a <->

Taking d i r e c t i o n from S e c t i o n 5, we have t h e h i j e c t i o n 6

[v,

<--)

(u, b ) ]

where

a +F 6 <->

a = (ai),

b = (bi)

E

M.

[m + v + ku, (k + u, a

. [m,

(k, a)],

Then

+

b)]

and o 6

<->

[kv + mu

2

, (ku,

kb + ua)].

This, as i n Theorem 4, m o t i v a t e s t h e f o l l o w i n g c o n s t r u c t i o n . Let

R

be a commutative r i n g w i t h i d e n t i t y , and l e t

M

be a u n i t a r y R -

51

The near-ring o f some noncommutative formal group laws module.

As before,

R x (RXM).

RXM i s t h e i d e a l i z a t i o n o f

M.

Consider t h e set

Taking guidance from t h e l a s t paragraph, d e f i n e (b, m ) ] + [c,

[a,

(d, n ) ] = [a

+ c + bd, (b + d, m + n ) ]

and [a,

-

(b, m ) l

(d, n ) ] = [bc + ad

[c,

2

,

(bd, bn + dm)].

It i s d i r e c t t o prove

(R x (RXM), +, - )

THEOREM 5.

i s a n e a r - r i n g w i t h i d e n t i t y [O, (1, O)]

in

which t h e l e f t d i s t r i b u t i v e law i s n o t v a l i d . COROLLARY 6.

(H, +F,

x.

l(X) =

THEOREM 7.

The u n i t s

u and

i s a r i g h t d i s t r i b u t i v e near-ring with i d e n t i t y

0)

= {[a,

( u s m)]"

[a,

I1 o f

R

(u, m)])a,

= [-u3a,

(RXM)

x

u

i s the set

R, u i s a u n i t o f R, m E MI,

E

-u-2m)].

(u-',

I n d i s c u s s i n g group s t r u c t u r e , we s h a l l use t h e n o t a t i o n and terminology

[41.

found i n

It i s easy t o see t h a t t h e a d d i t i v e group

of

R+

by t h e a d d i t i v e group o f RXM r e a l i z i n g

f o r each e(b, m) = lR

(b, rn) E RXM,

f:(RXM)+

d e f i n e d by

(RXM)' + R +

x

(R

i s an e x t e n s i o n x (RXM), +) + e:(RXM)+ + Aut R , where

and w i t h f a c t o r s e t

f[(b,

m),

(d, n ) ] = bd.

What i s n o t so obvious i s t h a t t h e group o f u n i t s ,

t h e r i n g R,

- by

where

U(RXM),

Here,

T

which i s isomorphic t o t h e a d d i t i v e group

t h e subgroup T = {LO, U(R) that

Observing t h a t

[a,

T

i s a subgroup o f

where

by

S

+

+

U, and t h a t

(1, O)][O, ( u s m)],

(u, m ) ] = [au'*,

0 + R

R+xe U(RXM),

(u, m )

E

U(RXM) = Il(R)

T.

S

MI,

i s a normal

Sn T = {[O, (1, O)]}. we get

II = ST,

We can w r i t e t h e e x t e n s i o n hy

I J + U(RXM) + 1,

of

R+ x

R, r e s p e c t i v e l y .

It i s d i r e c t t o see t h a t

U(RXM).

i s a s e m i d i r e c t product o f

so U

(u, m)]/

a r e t h e u n i t s of t h e r i n g s RXW and

i s isomorphic t o

subgroup o f IJ, U

-

(1, O)]la E R},

S = {[a,

o f a near-ring

U,

i s a s e m i d i r e c t product o f t h e normal subgroup

R x (RXM),

so

J.R. Clay

52 (a,

(u. m ) )

(b, (v, n ) ) = (a + (u. m)b, ( u s m)(v = (a + u’lb,

The isomorphism i s d e f i n e d by

F[a,

( u s m ) l = (au-*,

, n))

(uv, un t v m ) ) . (u, m)).

ACKNOWLEDGEMENT The a u t h o r wishes t o thank P r o f e s s o r A. F r i h l i c h ,

who i n i t i a l l y suggested

t h i s l i n e o f research, and w i t h whom t h e a u t h o r has had many s t i m u l a t i n g discussions. REFERENCES

[l]

[2] [3] [4]

Hazewinkel, N., Formal Groups and A p p l i c a t i o n s (Academic press, New York, 1978). P i l z , G., Near-rings (North-liolland, Amsterdam, 1983). Frb’hlich, A., D i s t r i b u t i v e l y Generated Near-rings I. I d e a l Theory, Proc. London Math. SOC. 8 (1958), 76-94. Rotman, J. J., Theory o f Groups ( A l l y n and Bacon, Boston, 1984).

Near-ringsand Near-fields,G.Betach (editor) 0 Elsevier Science Publishers B.V. (North-Holland),1987

53

ON THE EXISTENCE OF N I L IDEALS I N D I S T R I B U T I V E NEAR-RINGS

S.De Stefano and S.Di Sieno*

ABSTRACT

In this paper we show first that the sum of the nil right ideals of a distributive near-ring N and the sum of the nil left ideals of N have the same two-sided annihilator. This ideal - here denoted by a ( N ) proves useful in studying the existence of nil non-nilpotent ideals when t h e near-ring is weakly semiprime. In particular, we show that N has a nil non-nilpotent left ideal if and only if a ( N ) is properly contained in N and that every one-sided non-nilpotent ideal contains a nil non-nilpotent one-sided ideal if and only if a ( N ) coincides with the (two-sided) annihilator of N . Finally we prove that N has a twosided nil non-nilpotent ideal if and only if the ring N / a ( N ) has one.

1.

PRELIMINARIES

Throughout this paper N will be a distributive near-ring; standard terminology

and relative notations can be found in C41.

If it is not stated otherwise,

it will be understood that definitions and results relative to left structures hold also for the corresponding right ones.

L of N is calLed nilpotent if there is a positive inte-

A left non-zero ideal

; more in general, L is called niZ if for every x in L

ger k such that Lk=(0)

h there exists a positive integer h such that x =O ; finally, L is called maximal

n i l Left ideaz if it is maximal in the family of all nil left ideals of N . By Zorn's lemma, every nil left ideal is contained in a maximal nil left ideal: in particular, this is true of the (two-sided) annihilator of N A ( N ) = In< N 1 nx = xn =

o V X CN 1 .

Moreover every maximal nil left ideal of N is the inverse image of a maximal nil left ideal of N / A ( N ) under the canonical epimorphism 'TI: N + / A ( N )

and hence con-

tains kern. This proves REMARK 1 : Every naaxifnaZ

nil l e f t i d e a l of N contains t h e annihilator of N.

Now, if the near-ring N is a non-ring, A ( N ) is different from zero (since A ( N )

contains the conmutator subgroup of the additive group of N ) : thus every non-ring N has a (trivial) nil non-zero one-sided ideal.

Our interest is in those nil

(one-sided) ideals which are n o t contained in A ( N ) .

*

Both authors are members of the G . N . S . A . G . A .

of the Italian C . N . R .

S. De Stefan0 and S. Di Sieno

54

THEOREM 2 : The near-ring N has a n i l l e f t i d e al not contained i n A(N) i f and

only i f it has a n i l r i g h t idea2 not contained i n A(N). Precisely, i f L i s a n i l l e f t i d e a l of N not contained i n A(N),euery r i g h t ideal generated by an e le ment of L which i s not in^(^) w i l l work; and s im ilarly on the other side . PROOF : Let L be a n i l l e f t i d e a l n o t contained i n A(N) and c o n s i d e r t h e r i g h t i d e a l I (x) generated by a n element x of L which does n o t belong t o A(N). 2 By commutativity of t h e sum i n N , t h e square y2 of any element y of I r ( x )

tie

S

y

=

t(-m.+z.x+xn.+m.) i=1 1 1 1 1

where

S E N , zicZ,

mi.nic N

Since nx belongs t o t h e n i l l e f t i d e a l L , (nx)

has t h e form xn, w i t h n i n N .

h

i s zero f o r a s u i t a b l e p o s i t i v e i n t e g e r h ; t h i s i m p l i e s y2(h+1)=0, i . e . I (x) i s nil.

On t h e o t h e r s i d e , I (x) i s n o t contained i n A(N) and i s t h e r e f o r e an i-

d e a l of t h e r e q u i r e d kind. I n t h e same way one proves t h e " i f " p a r t of t h e s t a t e m e n t . From now on i t w i l l be supposed t h a t N has a l e f t i d e a l n o t c o n t a i n e d i n A(N).

2. THE NIL-ANNIHILATOR L e t L be t h e family of t h e maximal n i l l e f t i d e a l s of N, and R b e t h e family of t h e maximal n i l r i g h t i d e a l s of N .

Evidently t h e sum of t h e i d e a l s of L ( r e -

s p e c t i v e l y of R ) c o i n c i d e s with t h e sum of a l l t h e n i l l e f t ( r e s p e c t i v e l y r i g h t ) i d e a l s of N. The following r e s u l t h o l d s :

of the i d e a l s of L coincides v i t h the l e f t annihilator of the swn of the i d e a l s o f R. THEOREM 3 : The l e f t annihilator of t h e swn

PROOF : Denote t h e l e f t a n n i h i l a t o r of a s e t X by A ( X ) and 9. t y - by S9. t h e l e f t a n n i h i l a t o r of t h e sum of t h e i d e a l s o f a n n i h i l a t o r of t h e sum of t h e i d e a l s of R . S,

= nA9.(L)

-

f o r s a k e of b r e v i -

L

and by S r t h e l e f t

Clearly

and

Sr = nA9.(R)

where L ranges i n L and R i n R . I n o r d e r t o prove t h a t S

is contained i n S9., t a k e L i n 1 and x i n L .

By the-

orem 2 t h e r i g h t i d e a l I (x) is n i l even i f x does n o t belong t o A(N): t h u s t h e r e

is a maximal r i g h t i d e a l R of R c o n t a i n i n g I ( x ) . A9.(x) 2 A9.(Rx); t h e r e f o r e , i f x ranges a l l over L,

Since R

c o n t a i n s x, one g e t s

A9.(L) 2 nA9.(Rx). being A9.(L) = n A (x). 9.

Now l e t L vary i n

s e t o f ) R , so t h a t one can deduce

L

and x i n L : then R

ranges i n (a sub-

55

Nil ideals in distributive near-rings

The same kind of arguments a p p l i e d t o an i d e a l R of R shows t h a t t h e o p p o s i t e i n c l u s i o n h o l d s too: t h a t is, S

c o i n c i d e s with S e'

I n a completely analogous way one proves t h a t t h e r i g h t a n n i h i l a t o r Dr of t h e sum of t h e i d e a l s of R c o i n c i d e s w i t h t h e r i g h t a n n i h i l a t o r Dk o f t h e sum of t h e i d e a l s of

L.

This i m p l i e s t h a t t h e sum of t h e i d e a l s belonging t o t h e f a m i l y of t h e i d e a l s belonging t o t h e family R have t h e same (two-sided) which w i l l be c a l l e d nil-annihilator of N and denoted by a ( N ) . ed i d e a l o f N and can be d e s c r i b e d as t h e i n t e r s e c t i o n of S l e n t l y , as

Sk n D

sr

nDr,

sr

a.

L

and t h e sum

annihilator,

I t i s a two-sid-

and D

or, equiva-

nDL.

It i s u s e f u l t o p o i n t o u t t h e f o l l o w i n g

REMARK 4 : Let B be a ni2 l e f t ideal of N. I f there e x i s t s a p o s i t i v e i n t e g e r k i s contained i n a(N), then B i s n i l p o t e n t of class ( k + l ) . k

k such t h a t B

Indeed, t h e (two-sided) a n n i h i l a t o r of B c o n t a i n s a(N) and hence B

RELATIONS BETWEEN THE NIL-ANNIHILATOR

3.

.

AND THE ANNIHILATOR OF N

F i r s t of a l l n o t e t h a t , under t h e assumption (made a t t h e end of s e c t i o n 1) t h a t N has a n i l l e f t i d e a l n o t contained i n A(N), t h e i d e a l a(N) i s d i f f e r e n t from N. =

NL

I n f a c t , i f a(N) = N, f o r every n i l l e f t i d e a l L i t happens t h a t

= (0).

LN =

t h a t i s , L i s c o n t a i n e d i n A(N).

On t h e o t h e r s i d e , i f N i s a non-ring,

then a(N) i s d i f f e r e n t from (0) t o o ,

s i n c e it c o n t a i n s A ( N ) . In general t h e i n c l u s i o n a(N) 3 A(N) i s proper. t h e c l a s s o f weakly s e m i p r i m e n e a r - r i n g s ,

Restricting our a t t e n t i o n t o

i t is p o s s i b l e t o c h a r a c t e r i z e t h e

near-rings N f o r which a(N) and A ( N ) c o i n c i d e . R e c a l l t h a t , i f N i s weakly semiprime, A(N) c o n t a i n s two-sided i d e a l s o f N

-

-

besides the n i l p o t e n t

a l l t h e n i l p o t e n t one-sided i d e a l s o f N ( s e e C11): s o

t h e assumption t h a t N has a n i l l e f t i d e a l which i s n o t contained i n A(N) i s eq u i v a l e n t t o t h e assumption t h a t N has a n i l l e f t i d e a l which i s n o t n i l p o t e n t . L e t now prove

a weakly semiprime near-ring. The f o l ~ o w i n gare equivalent: i ) every non-nilpotent l e f t ideal contains a n i l non-nilpotent l e f t ideal;

THEOREM 5 : Let N be

i i l A(N) PROOF :

=

a(N).

i ) 3 i i ) . I f a(N)

+ A(N),

t h e i d e a l a(N) i s non-nilpotent and, by r e -

mark 4 , a(N) cannot c o n t a i n n i l non-nilpotent

l e f t i d e a l s . s o t h a t i ) is f a l s e .

S. De Stefan0 and S. Di Sieno

56

ii) => i J . Suppose A(N) nilpotent.

=

a(N) and l e t B be a l e f t i d e a l of N which i s n o t k

Then B2 i s n o t contained i n A(N) (and mare i n g e n e r a l no power B

of B i s c o n t a i n e d i n A(N)): indeed i t w i l l be shown t h a t t h e r e e x i s t s a maximal 2

n i l r i g h t i d e a l R of N such t h a t RB SC(0).

In o r d e r t o prove t h i s r e s u l t . f i r s t of a l l observe t h a t i n t h e semiprime r i n g N ' = N/A(N) t h e r i g h t a n n i h i l a t o r Dr(N') c o i n c i d e s with a(") and t h a t t h e image of D (N) under t h e n a t u r a l epimorphism

( s e e C21)

n: N -

N'

i s con-

tained i n D (N'). Conseque:tly,

i f i t were t r u e t h a t RB2-(0) f o r e v e r y maximal n i l r i g h t i d e a l 2 2 t h u s , f o r every x ' E n(B ) and

R, then t h e set n(B ) would be contained i n a ( N ' ) ;

f o r every maximal n i l r i g h t i d e a l R ' of N ' ,

i t would r e s u l t x'R'=(O).

But t h i s 2

would imply t h a t f o r every maximal n i l r i g h t i d e a l R of N t h e product B R

.

is

con-

t a i n e d i n A(N), because every maximal n i l r i g h t i d e a l R of N i s t h e i n v e r s e imTI of a maximal n i l r i g h t i d e a l R ' of N ' . Hence, f o r every maximal n i l r i g h t i d e a l R of N , i t would be B 3R=(O)=RB3 , and t h e r e f o r e B 3 would be contained

age under

i n a(N) = A(N), c o n t r a r y t o t h e i n i t i a l remark. Now l e t

E

be a maximal n i l r i g h t i d e a l such t h a t kB2P(0): then EB has an e l e -

ment x such t h a t xN#(O).

Since

k 5

is contained i n

n B , t h e l e f t i d e a l generated

by x i s c o n t a i n e d i n B , i s n i l (by theorem 2 a p p l i e d t o ( a s x does not belong t o A ( N ) ) ,

4.

k) and

i s non-nilpotent

t h a t is I (x) i s t h e r e q u i r e d i d e a l .

e

TWO-SIDED NIL IDEALS OF N If N i s a weakly semiprime n e a r - r i n g w i t h a(N) = N , everyone of i t s n i l i d e a l s

i s n i l p o t e n t and two-sided,

s i n c e i t i s c o n t a i n e d i n A(N).

On t h e c o n t r a r y , i f a(N) i s a proper i d e a l of N , then N has c e r t a i n l y a n i l

non-nilpotent one-sided i d e a l , b u t i t i s n o t known i f t h i s f a c t i m p l i e s t h a t N has n i l non-nilpotent two-sided i d e a l s .

An answer t o t h i s q u e s t i o n w i l l come

down from t h e following

THEOREM 6 : Let N be a weakly semiprime near-ring.

N has

a nil non-nilpotent

two-sided ideal if and only if N / ~ ( N )has one. PROOF : C a l l $ t h e n a t u r a l epimorphism of N onto N/a(N). Suppose t h a t N has a n i l non-nilpotent two-sided i d e a l I! s i n c e I i s n o t cont a i n e d i n a(N) ( s e e remark 4 ) , $(I) i s a n i l two-sided i d e a l of N/a(N) d i f f e r e n t from z e r o .

A c t u a l l y i t i s t h e r e q u i r e d i d e a l a s i t is n o t n i l p o t e n t , by remark 4 .

Suppose now t h a t N/a(N) has a n i l non-nilpotent two-sided i d e a l I ' and t a k e 2 I $-'(If). Since I i s n o t contained i n a(N), t h e r e i s e i t h e r a maximal n i l 2 r i g h t i d e a l R of N such t h a t I Rg6(0) o r a maximal l e f t i d e a l L of N such t h a t 5

LI2#(0) * I n t h e f i r s t c a s e I R i s d i f f e r e n t from zero and g e n e r a t e s a two-sided i d e a l B

51

Nil ideals in distributive near-rings which i s contained i n I.

Since I ' i s n i l , f o r every x of B t h e r e i s a p o s i t i v e

i n t e g e r k such t h a t xk l i e s i n a ( N ) :

k

of t h e sum i n N 2 , Bx = ( O ) .

t h e r e f o r e IRxk=(0) and, by t h e commutativity

I t follows t h a t every element x of B i s n i l p o t e n t .

2

On t h e o t h e r s i d e , B i s not n i l p o t e n t i t s e l f s i n c e . b e i n g I R#(O),

t h e product

I B i s d i f f e r e n t from zero, which i m p l i e s t h a t B i s n o t c o n t a i n e d i n A(N).

There-

fore B is the required ideal. I n t h e second c a s e an analogous argument proves t h a t t h e two-sided i d e a l gene r a t e d by L I i s n i l and n o n - n i l p o t e n t . Now, i f N has a n i l non-nilpotent l e f t i d e a l L, then a l s o the r i n g N/a(N) has a l e f t i d e a l of t h e same kind ( f o r i n s t a n c e t h e image

$(L) of L under t h e n a t u r a l

epimorphism) and v i c e v e r s a ( s e e t h e i n i t i a l remark of t h i s s e c t i o n ) .

Hence, t h e

problem s t a t e d a t t h e beginning of t h i s s e c t i o n admits a p o s i t i v e answer i f and o n l y i f every r i n g having a n i l non-nilpotent one-sided i d e a l has a n i l non-nilp o t e n t two-sided one ( i . e .

t h e KUthe's c o n j e c t u r e f o r r i n g s ( s e e C31) i s t r u e ) .

REFERENCES [ l ] D e S t e f a n 0 S. and D i S i e n o S . , A n e l l i e q u a s i - a n e l l i debolmente semiprimi, A t t i Ace. Sc. T o r i m 1 1 7 (1983).

[ 2 1 D i Franco F . , On t h e a n n i h i l a t o r of one-sided maximal n i l i d e a l s i n semiprime r i n g s , BOLL. Un. Mat. ItaL. ( 6 ) , 4 - A (1985). 65-70. 131 H e r s t e i n I.N., c a , 1968).

Non commutative rings (The Mathematical A s s o c i a t i o n of A m e r i -

141 P i l z G . , Near-rings (North-Holland, Amsterdam. 1983).

Dipartimento d i Matematica Universit2 degli Studi Via S a l d i n i , 50 20133 MILAN0 - ITALY

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Near-rings and Near-fslds, G. Betsch (editor) 0 Elsevier Science Pubbhen B.V.(North-Holland), 1987

59

DISTRIBUTIVE NEAR-RINGS WITH MINLMAL SQUARE S.De S t e f a n o and S . D i Sieno

*

ABSTRACT There e x i s t many examples of d i s t r i b u t i v e near-rings such t h a t t h e i r s q u a r e g e n e r a t e s a minimal l e f t N-subgroup of N. Here i t i s shown t h a t a near-ring belongs t o t h i s c l a s s i f and only i f i t i s t h e semi-direct sum of a z e r o near-ring and a skew-field,

1.

PRELIMINARIES L e t N be a d i s t r i b u t i v e near-ring:

terminology and n o t a t i o n s are t h o s e of L21.

I n p a r t i c u l a r , denote w i t h E(N) t h e r i g h t near-ring which is d i s t r i b u t i v e l y gene r a t e d by t h e set End(N) of t h e endomorphisms of t h e a d d i t i v e group o f N and with C (N) t h e c e n t r a l i z e r of N when i t i s regarded as a ( l e f t ) N-group: N

CN(N) =

{$t:

End(N)

I $(xy)

=

x$(y) f o r a l l x , y ~ N l .

The c e n t r a l i z e r of N is a monoid, w i t h r e s p e c t t o composition, which g e n e r a t e s ( d i s t r i b u t i v e l y ) a subnear-ring of E(N), which w i l l be denoted w i t h C(N). Every r i g h t product by an element n of N (which w i l l be denoted w i t h h ) i s an element of CN(N) and t h e r e f o r e of C(N); n e v e r t h e l e s s , i f N i s a non-ring,

the s e t

n E N ) does n o t exhaustC(N) nor C (N), s i n c e i t does n o t c o n t a i n any N automorphism of N. The set Ass(N) i s a s u b r i n g o f C(N) which w i l l be c a l l e d t h e

Ass(N) = {h,,

associated ring of N .

An easy computation shows t h a t Ass(N) i s anti-isomorphic

t o t h e q u o t i e n t r i n g of N by i t s r i g h t a n n i h i l a t o r

2.

THE R I N G "SQUARE OF N" 2 L e t S be t h e subgroup of t h e a d d i t i v e group of N generated by N : i t i s e a s i l y

seen t h a t S i s

a r i n g with r e s p e c t t o t h e product d e f i n e d i n N , which w i l l be

c a l l e d r i n g square of N.

Therefore E(S) c o i n c i d e s w i t h t h e s e t End(S) of i t s

generators. Moreover, S i s both a l e f t and a r i g h t N-subgroup of N , s u r e l y d i f f e r e n t from

*

Both a u t h o r s a r e members of t h e G.N.S.A.G.A.

of t h e I t a l i a n C.N.R.

S. Be Stefan0 and S.Di Sieno

60

zero i f N i s n o t a n i l p o t e n t near-ring. From now on i t w i l l be supposed t h a t N i s non-nilpotent and S i s a minimal ( l e f t ) N-subgroup. Under t h e s e assumptions, t h e r i g h t a n n i h i l a t o r A ( S ) of t h e r i n g S i s z e r o ,

I n f a c t A (S) i s a ( l e f t ) N-subgroup of S S

-

s i n c e i t is a n i d e a l of t h e N-group

- and t h e r e f o r e i s zero, because S i s minimal and non-nilpotent. Hence t h e r i n g A s s ( S ) i s anti-isomorphic t o S , which i m p l i e s t h a t any two d i s -

t i n c t elements s and s ' of S g i v e r i s e t o d i s t i n c t products h

and h s l .

Moreover,

s i n c e t h e minimal N-subgroup S i s idempotent, t h e r e e x i s t s i n S an idempotent e l ement e which a c t s i n S as a r i g h t i d e n t i t y ( s e e C11 proposizione 1.1).

I t fol-

lows t h a t every endomorphism of t h e ( l e f t ) N-group S i s a r i g h t product by an e l ement of S , t h a t is C(S)

-

CN(S) = Ass(S).

This proves PROPOSITION 1 : Let S be a minimal ( ' l e f t ) N-subgroup of N.

The c e ntralize r of S coincides with the ring associated with S and i s anti-isomorphic t o S.

3.

RELATIONS B E W E N THE CENTRALIZERS OF N AND S. Let now $ be an element of C(N).

=

For every element s of S i t h o l d s

$(s)

=

$ ( s e ) = s $ ( e ) , which means t h a t $(S) i s contained i n S .

THEOREM 2 : L e t th e ring S be a minimal ( l e f t ) N-subgroup of N .

The correspond-

ence a which associates w i t h each element $ o f C(N) i t s r e s t r i c t i o n to S i s an epimorphism of C(N) over CN (S), which induces a ring isomorphism between Ass(N) and CN(S). The kernel of ci i s formed by those $ of C(N) whose images l i e i n the r i g h t annihilator of N. PROOF : The r e s t r i c t i o n a($) of $ t o S is an endomorphism of t h e a d d i t i v e group of S s i n c e , f o r any two elements s and s ' of S, $(s+s')

=

(s+s')$(e) = $(s)+$(s')

and obviously a($) is also an element of C (S). N The correspondence a i s e v i d e n t l y a n e a r - r i n g homomorphism; a c t u a l l y i t i s an epimorphism, s i n c e CN(S) c o i n c i d e s w i t h A s s ( S ) . Moreover, f o r every hn i n Ass(N) and every x i n N i t r e s u l t s hn(x)

-

xn = (xn)e = x(ne) = hne(x).

Hence t h e correspondence between t h e elements h of C (S) i s one-to-one,

N

of Ass(N) and t h e elements h

ne

t h a t i s , a i s a n isomorphism between t h e s e two r i n g s .

F i n a l l y , i f $ belongs t o k e r a , then n$(N) = $(nN) = ( 0 ) , f o r every n i n N, so t h a t t h e i n c l u s i o n $(N)

5 Ar(N)

holds.

Conversely, t h i s i n c l u s i o n i m p l i e s t h a t

61

Distributive near-rings with minimal square g ( s ) i s z e r o f o r every s i n S , q . e . d .

COROLLARY 3 : Let the ring S be a rfrinzhaZ ( l e f t ) Fl-subgroup of N .

The near-ring

C(N) generated by the centralizer of N is the semidirect sum of k e r a and a sub-

ring isomorphic t o CN(S). I t coincides with the subnear-ring C of those elements $

of E ( N ) such t h a t $(xy) = x$(y) f o r any t#o x,y belonging t o N .

PROOF : The near-ring C(N) c o n t a i n s t h e s u b r i n g H of r i g h t products by e l e m e n t s o f

s,

which i s isomorphic t o CN(S).

Obviously, C(N) = kera+H.

Furthermore, t h e in-

t e r s e c t i o n of k e r a w i t h H is z e r o , because, i f t h e N-subgroup hs(N) = N s i s cont a i n e d i n Ar(N), then i t i s zero ( s i n c e S i s minimal and N i s non-nilpotent)

and

t h i s implies s=O. The second p a r t of t h e s t a t e m e n t follows from a p r o p o s i t i o n similar t o theorem 2 , concerning t h e n e a r - r i n g C i n s t e a d of C(N).

4.

A CHARACTERIZATION OF NEAR-RINGS WITH MINIMAL SQUARE

The image by a of t h e i d e n t i c a l endomorphism 1 of N i s t h e r i g h t product by t h e r i g h t i d e n t i t y e of S, b e i n g

a ( l ) ( s ) = s = s e = h (s) T h i s remark and preceding theorem 2 imply t h a t e i s also a

f o r every s i n S .

Left i d e n t i t y i n S .

I n f a c t , i f $I is an element of CN(N) such t h a t a ( $ ) = ht

(with t i n S ), f o r every s of S

Hence

and consequently

et = t.

Since a i s s u r j e c t i v e , such an e q u a l i t y h o l d s f o r e-

very t i n S , which means t h a t e is a l e f t i d e n t i t y i n S . Moreover, every element s of S d i f f e r e n t from zero has an i n v e r s e because t h e N-subgroup Ss c o i n c i d e s with S .

This proves

PROPOSITION 4 : If S i s a minimal ( l e f t ) N-subgroup,

then S i s a (possibly skeul

field. A s a consequence, S i s minimal as a r i g h t N-subgroup t o o .

Moreover, under t h e i n i t i a l assumptions, t h e f o l l o w i n g s t a t e m e n t h o l d s : PROPOSITION 5 : The near-ring N i s the semidirect swn of Ar(N) and S . PROOF : The r i n g N/Ar(N) i s anti-isomorphic t o Ass(N), which i n t u r n i s anti-

isomorphic t o S ( s e e p r o p o s i t i o n I and theorem 2 ) . proves t h e s t a t e m e n t .

Since Ar(N) n S i s z e r o , t h i s

62

S. D e Stefano and S. Di Sieno

Propositions 4 and 5 show that every near-ring N which is non-nilpotent and has minimal square is a semidirect sum of a zero near-ring and a (skew) field which is both a left and a right N-subgroup of N.

This is a characterization

of near-rings with minimal square, since the following result holds: PROPOSITION 6 : If t h e d i s t r i b u t i v e near-ring N i s a semidirect sun of a mrml subgroup B which i s a zeronear-ring and of a two-sided N-subgroup T which i s a

skew-field, then N i s non-nilpotent and has a square which i s minimal and coincident with T. PROOF : Since T is a skew-field, it has an identity e which is an idempotent element of N: therefore N is non-nilpotent.

Furthermore, for any two elements

b +t and n2 = b +t of N, the product n1n2 = tlb2+blt2+tlt2is an element 1 1 2 2 of the two-sided N-subgroup T; thus S is a two-sided non-zero ideal of the skewn1

=

field T and hence S coincides with T.

This proves also that S is a minimal (left)

N-subgroup.

FINAL REMARKS In spite of corollary 3 and proposition 5, one cannot prove that C(N) is the semidirect sum of the centralizer of S and C(A (N))

=

the distributive near-ring over the symmetric group S

E(A (N)).

See,for instance,

3'

This same exampleshows also that the near-ring generated by the centralizer is not left distributive even if the distributive near-ring has minimal square.

REFERENCES c11 De Stefano S. and Di Sieno S., Anelli e quasi-anelli debolmente semiprimi, A t t i Ace. Sc. T o r i m 117 (1983). 121 Pilz G . . Near-rings

(North-Holland, Amsterdam, 1983).

Dipartimento di Matematica Universitz degli Studi Via Saldini, 50 20133 MILAN0 - ITALY

Near-rings and Near-fulds, G . Betsch (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1987

63

NEAR-RINGS WITH E-PERMUTABLE TRANSLATIONS Celestina COTTI FERRERO Dipartimento di Matematica, Universith degli Studi, 43100 Parma, Italy *

Abstract. Our purpose is to study rings and near-rings N such that their endomorphisms are permutable with the left (right) translations. This situations is related to rigidity questions, and we prove that in a lot of interesting cases such an N is &-rigid. We obtaincharacterizationsof such near-rings with further conditions too.

1.

INTRODUCTION It is well known that the

study on rings

with given group of automor-

phisms has been very intensive.Lately, the rigid rings (see f.e. (131 , [ 141, [ 151

),

[ 8 ], [ g ] , 1

that are the ones whose only endomorphisms are the trivial

ones, have been studied. In [ 4 ] we find a work on P -rings, i .e. on rings h R that have a function h (of R in R) such that xh(y)=h(x)y, and 2 -rings h in wich h(x) belongs to the centre of R. Here we study,with the name ELT(ERT)-near-rings,the near-rings whose endomorphisms are permutable with left(right) translations.In particular,we state that a reduced ELT-ring is ft-rigid,and it is either rigid o r a direct sum of reduced ELT-rings.For reduced artinian ELT-rings we obtain a characterization as direct

sum

of complete local rings.Moreover,if a reduced ELT-ring has a

zero-square endomorphism,then it

is a P -ring and a Zh-ring,in the sense h of[4],and it turns out to be a ring with involution if and only if it is anticommutative and the characteristic of its annihilator is two. At last we completely characterize the ELT-rings such that the set of their nilpotent elements is a minimal ideal. 2.

GENERAL REMARKS Let N be a left near-ring;concerning notations and elementary results we

ofter. refer to [ll]wlthout any explicit remark. Also,we denote by

A (M), A (M),A(M) respectively the left,right,two-sided S

d

_.

*This research was partially supported by italian MPI.

C

64

Cotti Ferrero

of N ; by s

annihilator of the subset M

the left translation (

x -ax)

defined by the element a belonging to N, and by d the right translation a defined by a. Moreover N+ denotes the additive group of N, and N o its multiplicative semigroup; as in ring theory we will calla near-ring without non-zero nilpotent elements nediiced

.

Let u s remember that a near-ring N is said to be n i g i d (see [ a ] ) if the only endomorphisms are the trivial ones; it is called A-n-igcd

(see [12]) if the

only automorphism of N is the identity ;we will call an endomorphism f of N p a u p < if KerffO and ImffN. A proper endomorphism f of N is said to be a nean-pnv&ectiun of

ists a subnear-ring H of N such that Im f c H

N if there ex-

and f is the identity

H. Let I be an ideal of N;thenudi.cal of I is the set

fiof

on

the elements of

N a power of which belongs to I. As

will denote the subnear-ring of N generated by the products

usual, N2

of two elements of N.

Oe{ini.tiun InL9h.t)

A.

A nean-niny

t/2an4lativn4 c m d e w i M

an ELT+-nean-ning

Near-rings

with

i t 4

i 4

if

Let

cuded t

3t w i d

endvmv@um4. i t 4

N ~d

be

i 4

a

if

IERl=nean-aingl

le{t(ni&t)

i t 4

be

legt caUed

i / z a n A l a . t i u n ~ commute

addiiive youp.

"E-permutable left(right) translations

near-rings. The nean-ning N

D e j l n i t i v n 13.

i t 4

t&RT+-nean-ning)

w i 2 h t h e endwnunphcm4 v{

[ o N; {

an U i - n e a n - n i n 9

i 4

'I

&RT+ i { eveizy endvmonphcm v[ N+

Lett

nean-ning,

and

let

{

are ELT(ERT)-

i 4

one v{ N

N'

be an endvmonphcm

I t l - p m u . t a b l e i{ { commute w i t h a d M e l e { t / n i g h & )

tnandla(atiun4 u{ N.

The following propositions are obvious consequences of definitions A , B .

Pnopv4itivn

1.

3[ f

N, M e n xfIyl=fIxl{Iyl Pnvpo4ition 2.

i 4

a t It I 4 d

-pernutable

e n d u m v q h i m v{

a new-ning

I [(xly=~Ix/[Iyl I , V X , ~ E N .

L e t N be a ~ L t n i b u t i v enean-ning and l e t

endvmunphim v{ N. Then x-ftxl

E

A I3m {l and [l[Iylxl=y[Ixl,

be a t - p m u t a b l e A

V x , y ~N.

65

Nearrings with E-permu table translations T n o p o 4 i t i u n 3.

yxd; Men and [

i 4

[ox

L e t N be a di4tnibuz?ive

on

7m f l and on N3m

From Prop.2 follows x-f(x)

E

4uch M a 2 xy=O h @ e 4

e n d m o n p h u n f of N,

each .t -penmutable

M e identi%

nean-nq

x-flx)

e A U m i!)

[.

A (Im f),and so x-f(x) E

A(Im

f) since now xy=O 2

implies yx=O.We have the result observing that f(f(x)y)-f(x)y=(f (x)-f(x) )f(y)+ 2 +f(x)( f(y)-y ) and also f ( yf(x)1-yf(x)=f(y) ( f (x)-f(x)) + ( f( y -y ) f (x)

.

From now,the zero symmetric part (the constant part) of a near-ring N will be denoted by No(Nc).

be a t -penmutable endurnonphim of a nean-ning N; Men d x-tlxl E Ad13m [ I and ,Llx[lyll=[lxly, V x , y e N ;

P n o p o ~ i t i u n 4. 1.

2.

3.

L e t a!,

Ken [ E A (3rn [le No; d Ncc 3m and [ i 4 t h e & e n t i t y

c

it

4.

No

i4

on N C' '

a neduced nem-ning,.then

{on a U x , y , n

E

N,,xy=O

h p & e ~ yu=O

and my-4.

1.

Immediate from definitions A,B.

2.

Sincey-f(y)EA ( I m f ) , v y s N , i f y E K e r f t h e n y E A (Imf). d d Let Y E N then Oy=y and f(O)f(y)=f(y):it follow that f(y)=f(O)f(y)=f(O)y=

3.

C'

=Oy=y and our assertionis shown. We note that No (in consequence of our hypothesis) is a zero-symmetric

4.

reduced near-ring;the assertion follow from [ 2 1 lemma 1.

3.

NEAR-RINGS WITH E+-PERMUTABLE LEFT TRANSLATIOS

A near-ring is called weakly cummutative (see [ll] Such

near-ring is obviously me&&

a

(see [ 101

)

if,vx,y,zEN,xyz=yxz.

, and it is also an

?Fp-nem-

/Ling.

P n o p o ~ i t i o n5 , L e t N be an <+-nean-ning; then

7. 2.

3.

N

i 4

weakly cummufaLive;

i 4

if N

a d d i t i v e l y penmutable w i t h AdlNI; i 4

d i 4 t n i b u t i v e Men

?I

i 4

an i d e a l u[ N contained in .the centne of N'

For 1. observe that left translations commute within themselves.For 2. ob-

C. Cotti Ferrero

66

serve that left translations commute with the inner automorphisms defined by the elements of A (N).To prove the third assertion it is enough to remember d that the distributivity of N implies that N2 is a ring.

Theomn 1. Let N be on <+-nea/r-aing and l e t e be

M

idempotent element o{ N.

Then N=A lel+eN,whene A l e l i 4 M i d e a l o{ N , w i t h d A l e l = O , d i 4 a l e f t i d e a l d d d and N-oubpoup o{ N and A ( e l n eN=O.Aho,A l e l f O i { e i 4 n o t a l e f t i d e n t i t g . d d If e i s not a left identity of N,then there exists an x

E

N such that ex

N A (e).Furthemore A (e) is an #x;it follow that A (e)+O,in fact ~ X E ex-xE d d d ideal of N because (Prop.5) N is an IFP-near-ring.We see immediately that eNA (e)=O and that eNnA (e)=O.From Prop.5 it follows that NeNceN,and eNNc_eN, d d Now obviously,x=x-extex,and N=A (e)+eN.Let us prove,now,that eN is a normal d + subgroup of N :indeed,since now s commutes with the inner automorphisms e induced by the elements of A (e),we have that va E A (el a+en=en+a.It follows d d thatvxtN,since we can write x as a+en',we have -x+en+x~eN.HenceeN is a left ideal of N;it is obviouslyan N-subgroup ofN,too,and N+ turns out to be directly reducible.

C onollmy 1.

L e t N be a &lT+-necv-niq w i t h o u t .Le,C2 idenfitity,and l e i e be

idempotent element o{ /;then N domonphim {

of

c4 4m

o{ ELT-nem-ning4

M

and t h m e exi4t.1 an en-

N 4 u c h that A lel=Ken { and d=%{. d

NEAR-RINGS WITH E-PERMUTABLE LEFT TRANSLATIONS

4.

We start our study considering ( a s a particular case) the ELT-ring.

Theonem 2. 4m

A neduced <-ning

W

i4

A -nipid and it

i4

e i t h m a i p i d on a d b e c t

o{ aeduced <-ning.

Via lemma 1 of [ 2 ] we already know that R is an IFP-ring.Let f be a non-zero endomorphism of R.Then A(Im

f)=Ker f: in fact

(Prop.3) Ker f C A(Im

f) and

A(Im f)c Ker f because if zEA(1m f) then ,from Prop.2,f(z)E A(Im f) and fC(z)E 2 €A(Im f).It follow that f (z)f(y)=O=f(z)y, v y E R (PropS3),andso f(z)=A(R)==O: i.e. z ~ K e rf.From this it derives that R is A-rigid;moreover R does not have non-identical monomorphisms or epim0rphisms:indeed if Ker f=O then A(Im f)=O and

V X ER

x=f(x) (Prop.2)..If Im f=R then A(Im f)=A(R)=O,and f is the identity

67

Near-rings with E-permu table translations even in this case.

Now,if R is not rigid,let f be a proper endomorphism of R,then (Prop.3),Im f is an ideal of R and Ker f n Im f=O,being R a reduced ring:therefore R=Ker f@

Im f.Now Ker f and Im f are reduced rings and ELT-rings because any endomorphism of theirs can be extended in a trivial way to an endomorphism of R ; the same for their left translations. There exists obvious examples of reduced non-rigid ELT-rings.

A /reduced ELT-/ring

ConoUany 2.

ltipd if,and o n 4 i f , i . t

i 4

i4

di/re&lg

hneducible.

The d i n e c t 4 m a n d 4

ConoUany 3 .

neh

of

the p o p m dmonph,irn4

p o & e d o n A of

ConoUany 4. only it4

i f

of

it

a lteduced ELT-ning R m e e x a t l g .the k u of

R upon m i t a b l e idea.&

R . B e ~ i d e 4 , f h e endomoaphim4 v { R m e

of

R.

A neduced ning R ha4 a pltopm t -pe/unu-table endomo/rphim 4

i4

dine+

neducible and f

a p a o i e c t i o n of

i4

R

c i f and

on one

of

&neck 4 m a n d . J . A n o n - n i p d a d i n i n n neduced ELT-/ring

Theonm 3 .

i4

a d i n e d 4um

of

{ini.telg

man9 complete local n i n p .

If R is non-rigid,then (Th.2),it possess at least one proper endomorphism, and therefore it is a direct sum

of two reduced ELT-rings.Finally ,R being

artinian it is a direct sum of finitely many reduced rigid rings which are artinian,too.Thus,from th.3.1 of 1 8 1 ,R is direct sum of complete local rings.

If the ring R has non-zero nilpotent elements then we let Q(R) denote the set of the nilpotent elements of R.

Let R be

Theonem 4.

u

/ring 4uch t h a t : O # Q l R ) T R,xg=O impie4

luithouf non-zeno nilptent endomo/rphim4.The .in9 one

p/rvpm

non-pw

endornoapturn, Q ( R )

i n n e d u u b l e , i c and v n l g if it

i 4

a &/reed

minimal own

of

R

i4

and

p=O

a <-ning uith R'=R/QIRI

ond W af

i4

lea42

4ubcLu~ecUy

a n i l 4imple ELT-ning and a

4ubdinectly inneducible a i p d i n t e p a l domain. Since xy=O implies yx=O,Q(R) is a proper ideal of R and R/Q(R) is a reduced ring.Let f be a non-zero and non nilpotent proper endomorphism of R,and let R'=R/Q(R)

be subdirectly irreducible,with Q(R) minimal.It is easy to see that

C.Cotti Ferrero

68

the endomorphism f' induced by f on R' is t -permutable.If f'was proper then R' would be directly reducib1e;therefor-e it must be either Ker f'=O or I m f'=O. Recall now that (Prop.3) Im f is an ideal of R.If Ker f'=O then (Th.2) f'is the identity on R' and therefore v x cQ(R):indeed if f(y)=O then y

E

E

R x-f(x) E Q(R).From here we get Kerf

Q(R);hence it must be Ker f=Q(R),as now it

or

cannot be Ker f=O:in fact,if it is so,then it would be either rmfnQ(R)=O I m f2Q(R).But

if Im fiQ(R),then from x-f(x)E Q(R) we would get N c I m f,and f

would turn out to be an automorphism,but this is excluded by o u r hypothesis. Then,let Ker f=Q(R);from x-f(x)

E

Q(R)=Ker

2

f we get f (x)=f(x), v x

E

R,and so

R=Ker f @ I m f.It follows that I m f is a reduced ELT-ring,subdirectly irreducible,and thus

it is rigid (Th.2) and

it is an integral domain by

lemma 3 of [ 21,while Ker f is a nil simple ELT-ring. At

last,let I m

f'=O,and therefore Im

f c Q(R).At

because ~ X E Rthere exists a natural number m=m(x) m

thus x

t

Ker f.

Since

Im f f 0

,

we

get

v

once we have R= Ker f m

m

such that f(x) =f(x ) S O : I m f=Q(R) and Ker ff0,

because otherwise R would be ni1,in contradiction to what is supposed.Moreover, A(R)

being

c

Q(R),we have A(R)=O

or A(R)=Q(R).

If A(R)=O

then Ker f=

A ( I m f) (similary to what we did in Th.2), and R=Im f @ Ker f again, because

Ker f n Im f=O, since otherwise it should be I m fc_ Ker f, and f would be nilpotent,in contradictionto what it is supposed to.Finally if Im f=Q(R)=A(R), always with A(R)

fl

Ker f=O,we get

f(xy)=xf(y)=O, therefore R

Ker f is maximal, since Im f is now simple. any case, N is a direct sum

L Ker ~ f, and

Again R=A(R) @ Ker f. In

of a simple nil ELT-ring and of a subdirectly

irreducible reduced ELT-ring:

thus it

is a subdirectly irreducible rigid

(Th.2) integral domain (see [ 21). Vice versa, let R=A @ B, where A is a simple nil ELT-ring and B a subdirectly irriducible rigid integral domain. Let f be a non-zero endomorphism of R: f'

it induces an endomorphism f' in R/A

B. As now B is rigid, then

is either the identity or the zero endomorphism. In the first case

x-f(x)

E

A, and Ker f E A.Therefore Ker f =O or Ker f =A: if Ker f=A, then

Im f = B and R=Ker f @ Im f, with fL(x)=f(x),

and f is t -permutable; if

Ker f=O, we have I m f=R, and f is an automorphism. With easy calculation we show that f is t -permutable. Now, let f' be the zero-endomorphism: then Im f=A. Since here f cannot be nilpotent, we get A n Ker f=O, and f induces the identity on A.

Then

69

Nearrings with E-permutable translations we have YXE R , f(x)

2

E

A, and so f (x)=f(x). It follows that x-f(x)E Ker f, so

R=Im f @ Ker f , hence f is t -permutable. Moreover A is minimal, and contains all the nilpotent elements of R, and R has at least one proper endomorphism: for instance the second projection.

We recall that if f is a function of a ring to itself, usually S denotes the set of symmetric elements, i.e. the x E R such that f(x)=x, and K denotes the set of skew-symmetric elements, i.e. the y E R such that f(y)=-y.

Fheo/zem 5.

Let R be a /ring 4uch t h a t xg=O impbe4 yx=O, and le.t O f Q I R l f R

be minimal in 8; Men R ha4 a ~ e n o - 4 q u m e t -penmutable

it and o n l y ije M a e exi4f4 an automoaphiom 9 ble,

and w h w e

4ek of

4keu,+4yun&ic

A{

6

R* t h a t

elmd4

Yn p f i c u l m , g i n an i n v o l u t i o n

QIRIJ?.

OF

endomoaphim

i4

c4 2 and t -pwunutad an i d e a l which c o n t a i n 4

and only i; 8

i 4

anticonunutakive

and chm QlRl=Z.

Indeed, now, QCR) is an ideal of R. and it is minimal because of our hypothesis.

Let

f

of R. Then Im f x-f(x)

E

c

be

a

zero-square non-zero

t -permutable endomorphism

Ker f , and furthermore A(R) E Q(R).

From Prop.3 we get

A(Im f), and so f(x)E A(Im f). It follows that Im f i s a zero-ring,

and we get at once Im f=Q(R): since v x , y ~R it is xf(y)=f(xy)=f(x)f(y)=O, we have Im f E A(R) C Q(R)

and

Im f=A(R)=Q(R):

SO

hence Ker f is maximal

in R. Let us consider the function g=f-i: at once we get g(xy)=-xy, g(xy)=xg(y) =g(x)y, and g(x)=-x,

'Jx

E

A(R).

Now g is an additive homomorphism ( P r o p .

3 ) ; Im g i s an ideal of R, and obviously Ker f G Im g .Then we have Img=R,

R , that is excluded by our hypotesis.

because Ker f=Im g implies f(x)=O, v x

E

On the other. hand, Ker g=O since x

Ker g i f and only i f f(x)=x, i.e. iff

x

E

E

A(R) C Ker f. At last the set of skew-symmetrical elements of g is Ker f .

Vice versa, let g be an additive automorphism of R as it was said in the statement, and

let f=g+i: f is the required endomorphism.

Now it is sufficient to recall th.3 of [5]in order to complete the proof.

Let N be a near-ring; by C ( N ) we will denote the ideal of N generated by the elements xy-yx. In the near-ring

case we have

C Cotti Ferrero

70

L e t N be a nem-ning

Theonem 6. fie

comuta.ton

e i ~

rm N A K ~

CINI.

be

QlNl

i 4

an idea4 w h i c h o n t &

t -pmutable

a pnopm

endomonphim.

Then

4-

+=,

on N=ym

Indeed now N'=N/Q(N) the

L

Let

4uch M a t

w i t h ym

L~QINI.

fn+e/i

is reduced and commutative, hence it is a ring;

f on N'

function f' induced by

is a t -permutable endomorphism of S

f'=O, or f'=i, or f' is proper.

its, since f is it.By Th.2 we have

in the second one f is an epimorphism,

In the first case we obtain that is excluded. At

last we note that

N. Finally we obtaint/xER, x-f(x)

E

Ker f,

By J2 we will denote the (Betsch) radical (see [ll]

of the near-ring

N (concerning the distributive case, see also [ 611, and we will call N a ra[12] ) an dical near-ring if it concides with its radical J .Moreover (see 2 ideal I of a near-ring N is called prime of type 0 if for all K,J ideals of N, KJ

c

I implies either K

if xNy c_ I implies x

c

I or J

I; it is called prime of type 1

I.

Finally, by P (N) and P (N) ( s e e [12]) 1 we will denote the prime radical or type 0 and 1, respectively.

Pnopo4ition

6.

Let

E

I

N

y

07

be

E

a

non-nadical

d i o t n i b u t i v e <-nean-ning

with

N f Q I N l f O ; t h e n we have 1.

2.

1.

=P =QtNI; o r N / z I N ) i a a 4ubdinect 4wn

'T

2

Q&

&ie-ld.J.

We start proving that every prime ideal of type 0 is of type 1. Let

I be a proper ideal of N of type 0, and let xNyc I; N being we have xNyN G I . A l s o

zero-symmetric,

xN and yN are ideals o f N: indeed x N c N

2

is contained 2

in the centre of N+ (Prop. 5 ) , and N being weakly commutative it is NxNcxN c_

c xN, too. It follows that either xN c I or yV

C_

I. If xN

c I,

let < x

be

the ideal of N generated by x, since N is distributive and N2 is a ring, then < x zN way we Of

xN 5 I, hence < x I C I: therefore we obtain x

E

I. In the same

can proceed in the case yN 5 I. So every prime ideal of type 0 is

type 1: at once we get P (N)=P (N). Since now (Prop. 5 ) N is an IFP-near0

1

ring, th. 3.9 of [12] confirms our statement. 2.

From the previous point, Q(N) turns out to be an ideal of N, and N/Q(N)

71

Nearrings with E-permutable translations

is a reduced commutative ring; it follows that N/J (N) is semisimple and 2

commutative, and so it is subdirect sum of fields.

We note that in the case of rings or distributive near-rings, the conditions of

permutability

with

endomorphisms of right or left translations, give

results absolutely similar.

Theonem 7.

and

.4-ni@d

L e t N be an &RT-nean-ning, e i M e n it

g e c t i o n of N t o one of

Let

i 4

it4

n i F d on

and

lei

No be neduced; -?hen N

a n y pnopen endorno/cphim V R N

i 4 a

i 4

nem-pno-

N-4ubgnoup.

f be a non-zero endomorphism of N; we see immediately that

Ker f=A (Im f), and this implies (Prop. 4 ) that f is idempotent, and that d

f cannot be neither an epimorphism

nor a monomorphism different from the

identity. Consequently N is &-rigid,

and if f is proper, then f =f implies

2

that f is a near-projection of N on Im f,which now is an N-subgroup of the near-ring N. REFERENCES Abu-Khuzman, H. and Yaqub, A., Structure and commutativity of rings with constraints on nilpotent elements, 11, in: Math. J . Oklayoma Univ. 21 (1979), pp. 165-166. Bell, H.E., Near-rings in which each element is a power of itself, in: Bull. Austral. Math. SOC. 2 (1970), pp. 363-368. Bergen, J., Automorphisms with unipotent values, in: Rend. Circ. Mat. Palermo, (2) 30 (1982), pp. 225-232. Blass, A.R and Stanojevic C . V . , On certain classes of associative rings, in: Math. Balcanica 1 (1971). pp 19-21. Ferrero Cotti, C. and Suppa Modena, A., Sugli stems con involuzione, in: Riv. Mat. Univ. Parma (4) 7 (1981). De Stefano, S. and Di Sieno, S . , S u l radicale di Jacobson di un quasianello distributivo, in: Rend. 1st. Lombard0 (A) 112 (1978), pp 192-204. Malone, J.J., More on groups in which each element commutes with its endomorphic images, in: Proc. Amer. Math. SOC. 65 (1977), pp.209-214. Maxon, C.J., Rigid rings, in: Proc. Edinburgh Math. SOC. (1978), 21 PP. 95-101. McLean, K.R., Rigid artinian rings, i n : Proc. Edinburgh Math. SOC. (1982) 25, pp. 97-99. Pellegrini Manara, S., On Medial near-rings, this volume. Pilz, G., Near-rings (North-Holland, N.Y. 1977). Ramakotaiah, D. and Koteswara Rao, G., IFP near-rings, in: J. Austral. Math. SOC. (series A) 27 (1979), pp. 365-370. Suppa Modena, A., Sui quasi-anelli distributivi A-rigidi, in print. Suppa Modena, A,, Sugli anelli q-rigidi, in print. 115) Suppa Modena, A., Sugli anelli i-rigidi, in print.

72

C. Cotti Ferrero

SUMARY S i 4fudiano m e U L e q u a d - a n e k N fut& i w,i endwnoa&mi 4ono p m u t a bcLi con l e L a m l a y o n i . 4 i n i 4 h e ( d e 4 ~ e I . Queota 4 i t u a s i o n e 6 r o d e g a t a con gue4tioni d i nigcditc?, e o i pnova che in m o l t i cadi n o t e v o h (uz t a l e N h A-nigido. S i ottengonu c a / r a t t a i ~ ~ a & o n idc f a & q u a ~ i - a n e l l c o u t t o u & a i o a i condcyoni.

Near-rings and Near-fElds,G. Betsch (editor) 0Elsevier Science Publishers B.V. (North-Holland), 1987

13

ENDOMORPHISM NEAR-RINGS OF A DIRECT SUM OF ISOMORPHIC FINITE

SIMPLE NON-ABELIAN GROUPS Y. FONG and J.

D. P. MELDRUM Department of Mathematics, National Cheng Kung University, TAINAN, TAIWAN.

Department of Mathematics, University of Edinburgh, Mayfield Road, EDINBURGH EH9 352. SCOTLAND.

Let G be an arbitrary finite simple non-abelian group and H = $" G the external direct sum of n copies of the given group G. In this paper we study the structure of E(H). the distributively generated near-ring which is generated additively by all the endomorphisms of H. Here we show that E(H) en Mo(G) where Mo(G) = { f G + G; Of=O}. We also prove the key result that E(H) is a 2-primitive near-ring.

2

1. INTRODUCTION. A near-ring i s a set R with two binary operations + and

such that (R,+) is a not

necessarily abelian group with identity 0, (R,*) is a semigroup and x(y+z) = xy + xz for all

In general the extra axiom Ox = 0 for all x E R i s imposed to give a

x,y,z E R.

zero-symmetric near-ring. An element s E R is called distributive if (x+y)s = xs

+

ys for all

x,y E R. If there exists a multiplicative semigroup of distributive elements S such that R =

G p < S > , i. 8. R i s additively generated by S as a group, we say that R is a distributively generated near-ring, in short a d. g. near-ring. Let

R

-

be a near-ring and G an R-module. We call G monogenic if there exists g E G

such that gR = G, that is G

{gr; r E R}.

Let G be a monogenic R-module.

Following [I],

we say that G is an R-module of type 0 if G is simple, that is it has no non-trivial proper R-ideals; it is an R-module of type 1 if G is simple and for all g E G, either gR = G or gR = {O}; it i s an R-module of type 2 if G has no non-trivial proper R-submodules. If has an identity it is clear that type 1 and type 2 modules coincide.

R

For v = 0,l.Z. a

near-ring R Is called v-primitive on G if G is a faithful R-module of type v. The near-ring

R is called v-primitive if it is v-primitive on some R-module G. Examples of near-rings:

(1) If we let M(G) = { f G

+

G} where G is an arbitrary group (not necessarily abelian)

and define the product f o g of the two mappings f,g € M(G) by the rule x(f*g) = ( x f ) ~for all

x E G and the sum f + g by the rule x(f+g) = xf

+ xg for all x €

0, then (M(G),+;)

forms a

near-ring. (2)Let Mo(G) = {f E M(G); Of = 0). Then (Mo(G),+;) The

dlstrlbutive

elements

of

Mo(G) are

End

Is a G,

zero symmetric near-ring. the

semlgroup

of

ail

the

endomorphisms of G. Here in this paper, E(G) the d. g. near-ring generated by End G, is of

Y. Fong and J.D.P. Meldrum

74

special interest. More precisely, E(G) is called the endomorphism near-ring of the group G. For general results and definitions in near-rings, we refer to Pilz [lo].

We use l e f t

near-rings where he uses right near-rings, but otherwise the notations and definitions are similar.

2. THE STRUCTURE OF End H. Here we quote the following theorem which can easily be found in any standard text on group theory.

Theorem 2.1.

Let {GI; i E I} be a family of groups

*

l equal to the internal direct sum C i ~ Gi i i : Gi

+

@ i ~Gi. l

*

where Gi

The external direct sum 9 ~ Gi 1 is

is the image of the canonical injection

Conversely the internal direct sum E i ~ Hi l of a family of normal

subgroups of a group is isomorphic to the external direct sum Bicl Hi.

*

In view of theorem 2.1, we shall agree to identify x in G i with x i i in Gi , so that Gi = Gi* and the internal direct sum and external direct sum coincide. So readers should keep in mind that we are going to change notations between external and internal direct sums from time to time for the sake of convenience.

Now we are going to determine a l l

the endomorphisms of $" G. Let G be a finite simple non-abelian group and F = (Gi; Gi Write H = G1 @ Gg @ . . . $ G, or $?!1 Gi. Thus we tacitly assume G i

Lemma - 2.2.

G, i E {1,2,.

5 H for a l l

If G is a finite simple non-abelian group then End G = (0)

. . ,n)>.

i E I.

u Aut G

where 0 is the zero endomorphism of G and Aut G is the automorphism group of G .

Proof. Elementary. Take

any

(gi(l),i,gi(2),2,

arbitrary

element

. . .,gi(,,~,~) E H

a E

End

H

and

a

typical

element

h

=

where the second suffix indicates which copy of Gt and the

first suffix indicates which element of Gt. Thus gi(t1.t E Gt for all t E I1.2,

. . .,n}.

Then

Here we note that for all t E I, gi(t),ta E H. Therefore

where gi(t),t

-B

g't,j is an endomorphism of G (since Gt

then gi(t),titalrj is in Gj where

it: Gt +

G for all t E I). For if gi(t),t E Gt,

H is the canonical injection and nj: H

+

Gj is the

canonical projection. Thus itan, is a homomorphism from Gt to Gj for all t,j E 1. Gt Z G, 2 G, so itani is just equivalent to an element of End G.

itan,.

-

Since

Here we write at,,

Thus we have

lemma - 2.3. gi(t),t(at,i+at,z+

With the assumptions and notations as above, we have gi(t),tU =

. . . +at,n)

for allgi(t),t E Gt, t E 1.

proof. The proof is immediate from the fact gi(t),tat,j = gi(t),titanj equation (1).

g't,j and

75

Endomorphism near-rings of a direct sum of groups Thus we have Theorem 2.4.

Let G be a finite simple non-abelian group and H the external direct

sum o f n copies ofG. Then End H = {(a@; ai,, E End G, 1 5 i,j

n} is the Set

Of

a// n x n

matrices which have at most one non-zero entry in each column

proof.

For all h = (gi(l),~,gi(2),2, . . .,gi(n),n) E H and a E End H. by repeatedly applying

lemma 2.3 and theorem 2.1, we have

.....

I

an,l

Suppose that for some i E I, there exists every Sj.gk E G

Hence g,ai,i + gkak,i = gkak,i + g,ai,i for all gj,gk E G. Since G i s finite simple non-abelian and aj,i

0, ak,i

0, so a,,i,ak,i E Aut G. Fix gk

with gkc4k.i. But g,a,,i

+ 0,

then for every gi E G, gjaj,i commutes

runs through the whole of G since al,i E Aut G. Therefore gkak,i is

in the centre of G which is a contradiction because G has no centre. Hence result. Since End H contains all the endomorphisms of H, it forms a multiplicative semigroup under composition of mappings. As those enodomorphisms in End H have just been given in the form of matrices as in theorem 2.4, so the product of any two endomorphisms can then be carried out in the form of matrix multiplication. The next result gives a complete multiplication table of End H. Theorem 2.5.

n

Y

Ths correspondence given in theorem 2.4 between End H and a set of

n matrices is an isomorphism of multiplicative semlgroups

Proof. The proof is just a matter of simple routine checking. Here one immediately realises that addition for End H is not closed in general because G is non-abelian.

As in general near-ring theory, one knows that End H can

generate an endomorphism near-ring E(H) additively. With this, we attempt to investigate

Y.Fong and J. D.P. Meldrum

76

the structure of E(H) in the following section.

3. THE STRUCTURE OF E(H). Here I(G), (A(G).E(G))

denotes the

near-ring

generated additively

by

inner

all

automorphisms (automorphisms, endomorphisms) of a given group G. We now quote the following result which is due to A. Fr’dhlich [S]. Theorem 3.1.

If G is a finite simple non-abelian group, then I(G) = A(G) = E(G) =

Mo(G). Here for a l l i,j E (1.2. . .

. ,n},

we define E E to be the matrix with a E End G a t the

intersection of the ith row and jth column and the zero mappings elsewhere. Then these n x n matrices E E (i,j E (1.2. . . . ,n}) s t i l l sit inside End H. Thus the following corollary i s obviously a special case of theorem 2.5.

Corollary

3.2.

for a//ES,E!,

E End H (1

L i,j.s,t 5 n),

. . . ,n}, the near-rings 5 n, are subnear-rings

Thus it is easy to verify that CEZ; a E End G> for all i E (12, which are additively generated by E a where a E End G and 1 5 i

of the endomorphism near-ring E(H). In fact all <€a; a E End G> are isomorphic to E(G). Hence we have Theorem 3.3. Let G be a finite simple non-abelian group. Then cEZ;

a E End G > Y

Mo(G) for a l l i E I. Heres denotes “is near-ring isomorphic to’:

Proof.

G and

Immediate from the above remark, corollary 3.2, the simplicity of

theorem 3.1.

We have been tacitly identifying Mo(G) with its isomorphic copies in E(H). notational convenience, we denote by diag(a1, down

the

diag(a1,.

main

. . ,an)

(€2; a E End G}

diagonal,

and

E End H, diag(a1,.

n {EZ;

zeros

. . . , an)

elsewhere.

. . ,an) = Ey/l)

a E End G} = (0).

the n x n matrix with a l ,

.

Here

for

+ Efd2)

we +

remark

...+

that

and so if i

Now we claim that Gp
For

. . ,an

all

+

j

. . . ,an);

ai E End G > i s just the direct sum of its normal subgroups GpCEU; a E End G> where 15 i

5 n.

. . . ,an); ai E End G>, then there exists a finite number . . . , diag(ay.1, . . . ,a:,) E (diag(a1, . . . ,an); ai E End G} + . . . + diag(a:,,, . . . ,a:,n) = diag(E;,, such that x = diag(a;,,, . . . ,a,!,J diag(C;xl aj,,>O, . . . ,0) + + diag(0, . . .,O,EY=, ai,,). Thus Gp = Gp<Ey,; a E End G > + . . , + Gp<E:,; a E End G>. Again by the above For if x E Gp
of elements diag(al,,,

,a:,),

remark, one can easily see t h a t Gp<Ea; a E End G> and

all the

subgroups

Gp<Ea;

a

f End G>

ai E End G> are in fact normal. Hence Gp
7 f Gp<E$ a E 5 i 5 n of

1

. . . ,an);

End G > = (0) if i Gp
ai E End G> =

f: j

. . ,a,); Gp<Ea;

a E End G > . Thus we have Theorem 3.4.

Let G be a finite simple non-abelian group. Then
17

Endomoqhism near-rings of a direct sum of groups ui E End G > .

the near-ring generated additively by (diag(a1,

. . . ,an);

ai

E End G}

is

isomorphic to a direct sum of n copies of the transformation near-ring Mo(G), i. e.

. . . ,an); ai E


Proof. The

diag(61,l..

2.5,

diag(al,lBl,l,.

,

G > 2 fBn Mo(G).

only thing which really needs proof is the multiplicative closure.

. .,an,-,),

diag(al,l,. theorem

End

. . ,Bn,n) E

we

. ,an,nBn,n)

diag(al,l,

E
. . . ,an); ai E


have

..

. . ,an); ai E

End G>.

. .Un,n)diag(81,1.

End G>.

Let

Applying

. . . .5n,-,)

--

The rest follows from

theorems 3.1, 3.3 and the above remarks. Here we come to an important result. Then we

Let G be a finite simple non-abelian group and H = Bn G.

Theorem 3.5. have E(H) 2

an MO(G).

Proof. Immediate from theorem 3.4 and the fact that

. . . ,an; a, E

Cdiag(a1,

End G >

is a subnear-ring of E(H). Denote by antidi(a1, 1 5 i 5 n,

and

0

. . . ,a,)

. . . ,an); ai E

Gp
the n x n matrix with ai in the (n+l-i,i)th position for

elsewhere.

Here

we

wish

to

point

out

that

End G> is only group isomorphic to M+o(G) +

copies). GpKantidi(a1, . . . ,a,);

in

...+

ai E End G> itself does not form a near-ring.

general M+o(G) (n Take the

special case when n = 2: if antidi(a,B),antidi(y,6) E Gp, we have antidi(a,f3)antidi(y,&J = diag(By,a6) which does not lie inside Gp
a,B E

End Gr.

Now we come to our key result.

.6. Theorem 3

Let G be a finite simple non-abelian group and H = Bn G.

Then H is an

E(H)- module of type 2 and E(H) is 2-primitive

-

It suffices t o show that for every g E H

Proof.

g = (gi(l),~,gi(2),2, . . . ,gi(n),n) E H

- (0).

{O},

gE(H) = H. Take any

NOW we only need t o show that for every

g,* E Gi*, j E 1, there exists B E E(H) such that g 6 = g,*.

Then the theorem follows

immediately from the result on the external direct sum of groups. Take a = (aid) E End H,

0

g = (Bi(1),lpgi(2),2,

(projection):

H

+

Choose as,l = as,2 =

(0,

,

. . . .gi(n),n) E H. Assume gi(s),s E Gs - (0). . . . ,O,gi(s),s,O, . . . ,O). Thus

Gs* that sends g into (0,

. . . = as,j-l

= as,i+l =

. . ,O,gi(s),s,O, . . . ,O)Q

= (0,

...

-

as," = 0. Thus

. . . .O,gi(s),saS,j,O,

where gi(s),sas,j is in the jth component. Since a,,

Gs E 0, for a l l sJ E

Here we have ns we have

,

. . ,O)

is an endomorphism from Gs into Gj,

I and E(G) = Mo(G), and hence there exists a map

sends gi(s),s into 91. Here, take h = € l a 1 + E242 + and write ai* = (at!) with

...+

A: 0,

+ Gj that

Erar where ai E End G, E i = f.1

Y. Fong and J.D.P. Meldrum

78 a[\

=

ai if k 0

= is, j =

s,

otherwise.

ACKNOWLEDGEMENT. The first author would like t o acknowledge financial support from the National Science Council, Republic of China for visiting the Mathematics Department of Edinburgh University for the year 1984-85.

REFERENCES

ill]

Betsch. G. Primitive near-rings Math. 2. 130 (1973). 351-361. Fong, Y. The endomorphism near-rings of symmetric groups Ph. D. thesis, Edinburgh University, 1979. Fong, Y. Endomorphism near-rings of a direct sum of isomorphic finite simple non-abelian groups Technical Report for National Science Council, Republic of China, 1984-1985. Fong, Y. and Meldrum, J. D. P. The endomorphism near-rings of symmetric groups of degree at least five J. Austral. Math. SOC.30A (1980). 37-49. Fong, Y. and Meldrum, J. D. P. The endomorphism near-ring of the symmetric group of degree fouc Tamkang J . Math. 12 (1981), 193-203. Frahlich, A. The near-ring generated by the inner automorphisms of a finite simple group J. London Math. SOC.33 (1958). 95-107. McCoy, N. H. The theory of rings (Macmillan, New York. 1964). Meldrum. J. 0 . P. On the structure of morphism near-rings Proc. Royal SOC. Edinburgh 81A (1978), 287-298. Meldrum, J. D. P. Near-rings and their links with groups (Pitrnan Research Notes No. 134, London, 1985). Pilz, G. Near-rings (North-Holland, Amsterdam, 1977 (Revised Edition 1983)). Scott, W. R. Group Theory. (Prentice-Hall, Englewood Cliffs, New Jersey 1964).

Near-ringsand Near-fslds, G.Betsch (editor) 0 Elsevier Science Publishers B.V.(North-Holland), 1987

79

ON THE IDEAL STRUCTURE I N ULTRAPRODUCTS OF AFFINE NEAR-RINGS

P e t e r Fuchs S e c t i o n 1 c o n t a i n s some g e n e r a l r e s u l t s a b o u t i d e a l s i n u l t r a p r o d u c t s of 0-groups. We s h a l l o b t a i n n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s s u c h t h a t a n u l t r a p r o d u c t of Q-groups h a s t h e a c c o r d c c on i d e a l s and g i v e a n example t o show t h a t t h i s s i t u a t i o n does a c t u a l l y a r i s e . In s e c t i o n I1 we t h e n i n v e s t i g a t e t h e i d e a l s t r u c t u r e i n u l t r a p r o d u c t s of endomorphism r i n g s and a f f i n e near-rings.

1.

ULTRAPRODUCTS WHICH HAVE ASCENDING CHAIN CONDITION ( a c c ) OR DESCENOXNG CHAIN CONDITION ( d c c ) ON IDEALS. I t is w e l l known t h a t d i r e c t p r o d u c t s o v e r a n i n f i n i t e i n d e x s e t d o

n o t f u l f i l l any c h a in c o n d itio n s.

The f o l l o w i n g example however shows

t h a t t h e r e e x i s t u l t r a p r o d u c t s which d o have c h a i n c o n d i t i o n s . Example:

1.1

each

Let G,

automorphisms on

G,.

U

be a n i n f i n i t e set,

S

let

0 E S

be a g r o u p and

a n u l t r a f i l t e r on

For

S.

be a f i x e d p o i n t f r e e g r o u p of

A,

F o l l o w i n g t h e n o t a t i o n i n [ 6 ] w e d e n o t e by M o (G,) A0

t h e n e a r - r i n g of t r a n s f o r m a t i o n s on

and by dim

G,

(G,)

t h e number o f

A,

%.

o r b i t s u n d e r t h e a c t i o n of

G := II G,

Define

and

U

H := {h: G

G13(h,)

easy t o see t h a t

141

2 in

N :=

v((n,)/~,(Y,)/~)

H

u

5 AutN(G).

V(Y,)/u

i s a f.p.f.

M'

(G,)

is 2-primitive on

Au t M O (Ga)(G,,)

Let

in [6]

Conversely i f

g

E

=

M;;(G).

ml E

[6]

3n

f o r some

N = MQ(G) H

and

E

n.

m N

N:

(G,p)

+

where

Mi(G),

i s dense i n

(h,)/U

by 7.13 in [ 6 ] .

Suppose

N

It i s

G.

By Theorem

p:

N

X

G

+

G,

h((n,)/U)((Y,)/U) MiutN(G)(G).

Clearly

t h e n by Theorem 2 i n [ 4 ]

AutN(G)

ALl

in

h:

Im(h)

f o r some

= A,

= (h,(Y,))/UJ.

h((Y,)/U)

G:

E

g r o u p of automorphisms on

:= (n,(Y,))/U.

(h,(Y,))/U

g((Y,)/U)

A,

H

By 4.52

(n,(Y,))/U.

:=

n

E I[

E

Thus

n

u

AutMo ( G ) ( G , ) . A, a g E H

and

But

is dense

Im(h)

I

n) E U. Then { a dirn%(GLl) (G,) A' Now we c a n show t h a t dimH(G) = m. By 4.60 i n {,(dim

i s simple.

P. Fuchs

80

I n o u r main r e s u l t of t h i s s e c t i o n we s h a l l c h a r a c t e r i z e a l l t h o s e u l t r a p r o d u c t s of or

dcc

%groups w i t h c o u n t a b l y many o p e r a t i o n s which have

on i d e a l s .

V

n-groups.

a polynomial

i s a n e q u i v a l e n c e c l a s s of terms ( w o r d s ) .

If

s u r v e y of p o l y n o m i a l s and t e r m s s e e [ 5 ] .

[ 6 ] we l e t of

A.

AV[x]

+

If

-

P(A)

:=

For a d e t a i l e d

d e n o t e t h e s e t of a l l p o l y n o m i a l s i n t h e v a r i a b l e

o u t t o be a near-ring,

where

A

-

A

+

A p(a)

{PIP

E

A"[x]}

(A [x],+,o)

x turns To

, x ) ] E A [ X I we can a s s o c i a t e a p o l y n o m i a l

= w(a l...an,a)

then

V

then

i s t h e c o m p o s i t i o n of p o l y n o m i a l s . V

o

p = [w(a l...a

p:

function

V , then

E

Following t h e terminology i n

denotes t h e group operation i n

each polynomial

A

is a v a r i e t y of a - g r o u p s ,

about i d e a l s i n p

acc

We s t a r t by r e c a l l i n g some d e f i n i t i o n s and r e s u l t s

Va

(P(A),+,o)

A.

E

If

o

i s a l s o a n e a r - r i n g where

t h e u s u a l c o m p o s i t i o n of f u n c t i o n s . For a s u b s e t

B

cA

is

let (B) denote

B.

t h e i d e a l g e n e r a t e d by

The f o l l o w i n g theorem shows t h a t i d e a l s of

a-groups can be d e s c r i b e d

by p o l y n o m i a l f u n c t i o n s .

1.2

([6])

Theorem: 1)

( a ) = {?(a)(;

2)

VB

5 A:

A

Let

b e a n &group,

A.

a E

E (P(A)),)

1

(B) =

(b)

bE B

V

I n a l l of t h e f o l l o w i n g l e t many o p e r a t i o n s ,

va E

b e a v a r i e t y of

U

be a s e t ,

S

an u l t r a f i l t e r on

s. Suppose

p E

I f we r e p l a c e (aa)/U J

by

in

A,)"[x],

p = [w((aT)/u

...

U

w((ay)/U

(az)/U,x)

th e n w e g e t a term

a;

( I %)'[XI

a

, w (al

...

a , pa = [w ( a ,

polynomial shows haw

(n

by

w U

ak,x)

in

uz

V, 0 E S.

and e a c h

A,

and a

The f o l l o w i n g p r o p o s i t i o n

V ll A,[x].

is r e l a t e d t o

U

h:

(n

A,)"[xI

+

n

U b)

x:

P ( n A,)

Notation:

Ai[x],

U +

U

1.4

E

Proposition: a)

let

and A,,

( a z ) / ~ , x ) ] . Let

...

V

S

...

each

a i , x ) ] E A,[x].

U 1.3

%groups w i t h countably

Let

A

I P(A,) U E

V

h ( p ) := ( p a ) / u

-h (-p )

i s a n . r homomorphism.

:= ( p a ) / u

with operations

6l

i s a n . r homomorphism.

{w,(m c Nf.

For

n' C

81

The ideal structure in ultraproducts o f affine neawings 1)

:= { [ w l

Pn'

w

If

If For

in

A' A'

P':

Vn E ti:

2 A,

n

Pn

in

for

then

a(w)

-<

I

n Pn.

w

are

s h a l l denote

n~

we c o n s i d e r ( i f e x i s t e n t ) a s e q u e n c e

N,

E

OCCUK

W.

P*' a ( w )

:= {[w] E

n we w r i t e

9' =

V

i s a t e r m i n t h e l a n g u a g e of

t h e number o f symbols i n

3)

which

n')

e l e m e n t s of 2)

v

( A [ ~ l ) ~ ~ wa El lSI

E

xl...

xk...

such t h a t

W e say t h a t

A'

5A

fulfills

.... 3

Cn,k,

xl...x

steps.

does n o t f u l f i l l

If

A'

in t h e c o n d i t i o n s

Cn,k

1.5

If

Proposition:

in

F :=

C

n,k

IN

( d e n o t e d by

we w r i t e

t Cn,k}

{O(A,

A,

We a r e now r e a d y t o s t a t e

Jl A, U

E

A ' C Cn,k)

k

The i n t e r e s t

Cn,k.

A'

if

s t e m s from t h e f o l l o w i n g r e s u l t .

e v e r y c h a i n of i d e a l s i n

fact that

n,k

A ' a s d e f i n e d above s t o p s a f t e r a t most

e v e r y sequence

E

has length OUT

is H 1 - s a t u r a t e d

U

f o r some

n,k

N, then

5 k.

characterization result. i f

&

is a n

Using t h e

U-incomplete u l t r a f i l t e r

(see 121, pg. 305) o n e c a n p r o v e :

1.6

Theorem:

1)

If a) b) c) d)

U

i s U-incomplete

lI A, U ll A, U II A, U

3k

t h e following statements a r e equivalent:

has the acc f o r i d e a l s has t h e dcc f o r i d e a l s h a s a composition s e r i e s E

steps

N:

Every c h a i n of ideals i n

It A, U

s t o p s a f t e r a t most

k

P. Fuchs

a2 3n.k

e)

2)

U

If

{,(A,

H:

fulfills

is w-complete

n

a)

E

A,

C

n, k

1E U

then

f u l f i l l s t h e a c c f o r i d e a l s <=$ { o ( A ,

f u l f i l l s the acc

U f o r ideals}

II A,

b)

E

f u l f i l l s t h e dcc f o r i d e a l s

<+{ , ( A ,

f u l f i l l s t h e dcc

U f o r ideals}

11 A, U

c)

E

<+IO I A,

h a s a composition s e r i e s

has a composition

U

series1

It s h o u l d b e p o i n t e d o u t t h a t a s i m i l a r r e s u l t c a n a l s o b e o b t a i n e d f o r

u n i v e r s a l a l g e b r a s w i t h c o u n t a b l y many o p e r a t i o n s . Let

U

U-incomplete u l t r a f i l t e r o n

he a n

does n o t f u l f i l l

acc

e x i s t e n c e of a c h a i n of i d e a l s which is a n a0

2

t h e r e f o r e h a s c a r d i n a l i t y a t least

1.7 Theorem:

Let

U

s u c h t h a t e a c h c h a i n of i d e a l s i n

n

s e t (see [2],

Ill

.

a-incomplete.

be

U

I n t h i s c a s e i t is p o s s i b l e t o show t h e

dcc.

OK

2.

11 Maff(V,) U

THE IDEAL STRUCTURE OF

Let

V

S.

h

C

V

+

V,

m6(v) := 6

HomF(V), 6

E

the vector space ultraproducts

AND

(Ho%(V),+,o)

endomorphism r i n g of t h e v e c t o r s p a c e

m6:

V} V)

V).

u

n

Hom

F.

If

V

+

Fa

(V,)

8

o

FOR AN

denotes t h e (The

6

E

and

V

Vlf = him6

f o r some

(The a f f i n e n e a r - r i n g

of

In t h i s s e c t i o n we p r e s e n t t h e i d e a l s o f (V,),

Fa

n

Maff(V,)

i n some d e t a i l and a p p l y r e s u l t s

U

from t h e p r e v i o u s s e c t i o n .

D e t a i l e d p r o o f s c a n b e found i n 131.

r e m a i n d e r of t h e p a p e r l e t

S

F,V,

E

set.

Moreover, i f

Maff(V) := { f :

k

steps o r there

is a r i n g .

is a n e a r - r i n g w . K . ~ . + , o .

11 Hom

u

then

n1

which is a n

be a vector space over a f i e l d

c o m p o s i t i o n of f u n c t i o n s t h e n


stops a f t e r

U 11 A, U

pg 264) and

Then t h e r e e x i s t s e i t h e r

A,

e x i s t s a c h a i n of i d e a l s i n

ULTRAFILTER 11 ON

II A,

and s u p p o s e t h a t

S

a n u l t r a f i l t e r on

be a set,

be a v e c t o r space o v e r t h e f i e l d

F,

for a l l 0

r e s i i l t shows t h a t i t s u f f i c e s t o look a t i d e a l s of

S.

E

II Hom

u

Fa

(V,)

For t h e S

and

The n e x t

i n order

83

The ideal structure in ultraproducts o f affine near-rings

t o f i n d a l l i d e a l s of [ 7 ] and is omitted.

2.1

[31

Proposition:

I = I.

To s t u d y i d e a l s i n

The p r o o f is s i m i l a r t o a r e s u l t i n

Maff(V,).

U

{Oj

Let

+ n U

Is II Maff(V,).

f

Mc(Vo)

f o r some

I 4 ll Hom

n

HornF (V,)

u

o

be t h e c l a s s of a l l c a r d i n a l s

we i n t r o d u c e t h e f o l l o w i n g c o n c e p t .

u

a n u l t r a f i l t e r on

such t h a t

T

2.2

Definition:

is called

T'

the

5 T)

{,IT,

nonempty a n d h a s t h e r e f o r e a l e a s t e l e m e n t

d e n o t e d by

(V,)

F,

0 - u

be a s e q u e n c e o f c a r d i n a l s and

(To)oEs

Then

E

S.

u.

Let

Let

K

is

K

T'.

U-limit

of t h e s e q u e n c e

(To)oEs,

= lim(T,).

T'

U l i m (T,)

We n o t e t h a t

i s always a l i m i t c a r d i n a l i f

U

a l l cardinals

If

T.

FV

is a vector-space

= TI $

{,IT,,

t h e n t h e i d e a l s of

for

HomF(V)

are w e l l known.

2.3

[l]

Theorem:

A

each o r d i n a l

Let

be a v e c t o r s p a c e o v e r a f i e l d

FV

Ti

let

:= {h E HomF(V)ldim I m ( h )

a)

I i s a n o n z e r o i d e a l of o r d i n a l A.

b)

Let

h,hl

h2,h3

of

ll Hom

u

Fu

TI

(V,). TI

cC

A

If

u

FlJ

n

II TA, U

:= { ( h , ) / U ( l i m

U

as well.

is an i d e a l (dim Im(h,))

I t i s e a s y t o show

F,

A

:=

Tf

(V,)

Hom

u

U E

We r e m a r k t h a t t h i s i n c l u s i o n

S.

I

I t c a n b e shown t h a t f o r e a c h i d e a l

t h e r e is a c e r t a i n o r d i n a l

(V,)

Then t h e r e e x i s t s

Then c l e a r l y

is an o r d i n a l a n d

h,

Im(h).

f o r some

h2 o h o h 3 o h l = h l .

i s a n i d e a l of

TAo w h e r e

5 dim

dim I m ( h l )

such t h a t

can be s t r i c t ( [ 3 ] ) .

II Horn

with

I = Ti

i f and o n l y i f

b e a s e q u e n c e of o r d i n a l s .


HomF(V)

HomF(V)

E

(A,),ES

Let

E

HomF(V)

and f o r

F

<*A}

A

such t h a t

Ti

of

c 1 5 IIU

Ti.

More p r e c i s e l y :

2.4

Theorem:

ordinal

A

[3]

Let

such t h a t

{OI

f

I d

I = ll TA

U

-u

n

or

Horn

Fu

(V,).

I = Tf

OK

Then t h e r e e x i s t s a n Tf

I

n

U

TA.

P. Fuchs

84 If

is a successor ordinal then

A

are many i d e a l s s t r i c t l y b e t w e e e n

TI =

n

TA.

U and

TI

n

I n g e n e r a l , however. t h e r e TA.

The n e x t r e s u l t shows

U how many p r i n c i p a l i d e a l s a r e between

2.5

Theorem:

1)

If

U

Let

Let &lsl/U

U

be a n i n f i n i t e s e t a n d

S

Lxo}E

f ( I ( d i m (V,)

t h e n t h e r e is n o i d e a l s t r i c t l y b e t w e e n

is U-complete,

and

2)

[3]

Il To.

and

Ns/u.

d e n o t e t h e c a r d i n a l of

U

I01

To

u

If

is U-incomplete,

then there a r e exactly

;

(01

p r i n c i p a l i d e a l s between

and

To*

The p r o o f , which i s l e n g t h y , w i l l be o m i t t e d . s e c t i o n 1 w e now d e t e r m i n e when

U

(dcc) provided t h a t

2.6

Theorem:

1)

II Hom

u

Applying

Hog,(V,)

or

r e s u l t s from

OUK

Maff(Vu)

has a c c

is a - i n c o m p l e t e .

U

Let

(V,)

NIsI/U many

h e U-incomplete.

has the dcc (acc) f o r i d e a l s

Fu

<+n

Horn

(V,)

is a

Fu

simple ring. 2)

Il Maff(V,)

If

has dcc ( a c c ) on i d e a l s then

Maff(Vo)

has e x a c t l y

U one p r o p e r i d e a l

I = II Mc(Vo).

and

I

U

-Proof:

1 ) By Theorem 1.6 we c a n c h o o s e

fulfills (I

6

C

1

(x') 1

dim f m (x;)

dim I m (x:+~)

If

U.

E

Suppose t h a t

t h e r e are elements

F

I dim I m

I

n, k

x1,x2

E

Horn

=

F :=

...'xk+ 1

x

n,k

E

{U

E

p4

s u c h t h a t {,(Horn

Idim(VU) > ( n + l )

Hom(VU)

E

u.

If

1

= n+l

= (n+l)k

(Va),

stops a f t e r

k+l

{(IIHom

fulfills C

(V,)

1

(V,)

with

-

x2 = p(xl)

f o r some

p E Pn

then

n dim I m ( x ). Now o n e c a n e a s i l y show by i n d u c t i o n t h a t 1

F,

k

steps for a l l

n, k

1

E

u.

U E

F.

dim I m (x,)

'

x1 .**

i

(I

k+ 1

T h i s is a c o n t r a d i c t i o n s i n c e

Consequently

{U(dim(VU)

i (n+l)kl

E

u.

The ideal structure in ultraproducts of affine nearrings S i m i l a r a r g u m e n t s a s i n Example 1.1 show t h a t 2)

II HornF (V,)

U

i s simple.

o

Follows by 1, and P r o p o s i t i o n 2.1.

ACKNOWLEDGMENT T h i s p a p e r forms a p a r t of t h e a u t h o r s d i s s e r t a t i o n . h e w a n t s t o t h a n k h i s t e a c h e r P r o f . G.

A t t h i s place

P i l z f o r h i s c o n s t a n t encouragement

and advice.

REFERENCES

[I] [2]

[3] 141 IS]

[6] 171

R. Baer, L i n e a r a l g e b r a a n d p r o j e c t i v e g e o m e t r y , New York, Academic P r e s s (1965). C. C. Chang-H. J. K e i s l e r , Model T h e o r y , N o r t h - H o l l a n d , Amsterdam ( 1973). P. F u c h s , UltKapKOduCtS of n - g r o u p s , D i s s e r t a t i o n , Univ. L i n z (1985). P. Fuchs-G. P i l z , U l t r a p r o d u c t s and u l t r a l i m i t s o f n e a r - r i n g s , Mh. Math 100, 105-112, (1985). H. Lausch-W. NEbauer. A l g e b r a of p o l y n o m i a l s , N o r t h H o l l a n d , Ams t e r d a m ( 1 9 7 3 ) . G. P i l z , N e a r - r i n g s , R e v i s e d e d i t i o n , N o r t h - H o l l a n d , Amsterdam (1983). K. Wolfson. Two s i d e d i d e a l s o f t h e a f f i n e n e a r - r i n g , Americ. Math. Monthly 6 5 ( 1 9 5 8 ) 29-30.

D e p a r t m e n t of Mat hema t i c s T e x a s A6M U n i v e r s i t y C o l l e g e S t a t i o n , TX 77843 U.S.A.

85

This Page Intentionally Left Blank

Near-rings and Near-fields, G . Betsch (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1987

87

RADICALS OF $2-GROUPS DEFINED BY MEANS G.K.

OF ELEMENTS

Gerber*

Mathematics Department, U n i v e r s i t y o f P o r t E l i z a b e t h , P 0 Box 1600, P o r t E l i z a b e t h 6000, South A f r i c a . ABSTRACT. I n t h i s paper r a d i c a l s o f Si-groups a r e d e f i n e d by means o f elements. The f o l l o w i n g g e n e r a l i z a t i o n s o f r a d i c a l s o f r i n g s a r e obtained: The n i l r a d i c a l , t h e E 6 - r a d i c a l , t h e A-regular r a d i c a l and the f-regular radical. 1980 Mathematics subject c Z a s s i f i c a t i o n (Amer. Math. Soc.): primary 20 N 99; secondary 16 A 12, 16 A 22, 08 A 99.

1.

INTRODUCTION Wiegandt ( [ l l ] ) p u b l i s h e d an a r t i c l e where he d e f i n e d r a d i c a l s o f r i n g s by

means o f elements.

The d e f i n i t i o n s , theorems and p r o o f s o f Wiegandt can be We a p p l y these theorems t o o b t a i n t h e n i l r a d i c a l

g e n e r a l i s e d t o R-groups.

(Buys and Gerber, [ 2 ] ) and g e n e r a l i z a t i o n s o f t h e E 6 - r a d i c a l f o r r i n g s , t h e r a d i c a l c l a s s o f A-regular r i n g s as w e l l as the r a d i c a l c l a s s o f f - r e g u l a r r i n g s f o r v a r i e t i e s o f &groups. 2.

NOTATION AND DEFINITIONS Throughout t h i s paper we s h a l l use t h e d e f i n i t i o n s and n o t a t i o n o f H i g g i n s

([?I).

Whenever we r e f e r t o G i t i s meant t o be an R-group.

P

Q

G denotes an

In particular i d e a l P o f G. AG w i l l denote t h e i d e a l generated by A 5 G i n G. G G Higgins ( [ 7 ] ) c a l l e d words which i n v o l v e o n l y o p e r a t i o n s we w r i t e a f o r { a }

.

w i

R monomials.

We s h a l l c a l l monomials 0-words.

2. 1 D e f i n i t i o n L e t $2 be a f i x e d s e t o f o p e r a t i o n s . a v a r i e t y V o f n-groups i f

w E 12 i s c a l l e d a trivia% operation i n

= 0 for all

operationif i t i s not t r i v i a l .

w E

g E G E V.

R i s a non-trivial

An $]-word i n v o l v i n g o n l y n o n - t r i v i a l o p e r a t i o n s

w i 11 be c a l l e d a non-trivial R-word. I f f(x1,x2 f(x).

A A

1 2

,...,x n )

2

f ( x ) i s any word, then f ( x , x

,...,x)

w i l l be denoted by

Furthermore

...Anw

=

{a a 1

2

...anu I ai

E A . , i = 1,2 1

,...n),

Ai

5 G,

i = 1,2

,...,n .

The a u t h o r g r a t e f u l l y acknowledges f i n a n c i a l a s s i s t a n c e from t h e U n i v e r s i t y o f Port Elizabeth.

C. K. Gerber

88

2.2

Definition

t G

An R-group G i s c a l l e d zero-symnetric i f f o r each w E R and a ,a ,...,an w = 0,

i t i s t r u e t h a t al...ai-lOai+l...a

2.3

...,n.

i= l,Z,

n Theorem (Buys and Gerber, [ 3 ] )

1

2

G i s a z e r o - s y m n e t r i c R-group i f and o n l y i f f o r each w i (2 and H1,H

2,...

,Hn a G i s i s t r u e t h a t

H1H2.. .Hnw

2.4

5

n

0 Hi. i=1

Definition

,...,x n ) = x 1 ( t h e s o - c a l l e d i d e n t i t y - w o r d o f H i g g i n s ,...,Hn a G. D e f i n e t h e i d e a l f'(H1,H2 ,...,Hn) as f o l l o w s : f'(H1,H2 ,...,Hn) = H . F o r any [?-word f(xl,xz ,...,xn) and H ,H ,...,Hn a G, t h e i d e a l f ' ( H

( [ 7 ] ) and

L e t f(xl,x

H1,H2

1

i s defined inductively. R-word.

L

If

=

f i n e f ' ( H ,H

L

Let

1 t h e n t h e r e e x i s t s an G Hn) = H H Hnu . 1

2

K.

t R such t h a t f ( 5 )

ii)

e

=

2.5

,.. .,Hn)f:

= f ' (H ,H 1

,...,Hn)

Then we de-

= ~'LI.

ktl.

Then f ( x ) = f l ( s ) f 2 ( 5 )

where some f . ( x ) a r e i d e n t i t y words and t h e r e m a i n d e r a r e +words J Then we d e f i n e

,. ..,Hn)

2

has been d e f i n e d f o r any R-word f ( 5 ) w i t h

L e t f ( 5 ) be any i2-word w i t h

f ' (H1 .H2

,H 1

...

Suppose t h a t f'(Hl,H 2,...,Hn)

L 5

2

be t h e number o f o p e r a t i o n s w h i c h o c c u r s i n an

1

.

( H1 ,H2,.

, ,Hn).

...f r ( 5 ) w

with

e

5 k.

. .f I( H1 ,H ,. . . .Hn)w G .

2

D e f i n i t i o n (Buys and Gerber, [ 4 ] )

P a G i s c a l l e d an i1-prime (%semi-prime) i d e a l i f f o r e v e r y n o n - t r i v i a l w t

R and A ,A ,...,An 1

f o l l o w s t h a t Ai

2

5P

4

(A

G ( A a G) such t h a t A A

5 P)

1

2

...A n u 5 P

(AA. ..Aw

5 P)

it

f o r an i.

The c o n c e p t s a c c e s s i b l e R-subgroup, s u b d i r e c t l y i r r e d u c i b l e R-group, h e a r t o f a s u b d i r e c t l y i r r e d u c i b l e R-group, r e g u l a r c l a s s , r a d i c a l c l a s s and upper r a d i c a l o f a r e g u l a r c l a s s a r e d e f i n e d as i n t h e v a r i e t y of r i n g s .

3.

RADICALS DEFINED BY MEANS

OF ELEMENTS

A n a l o g o u s l y t o Wiegandt ( [ I l l ) we l e t P d e n o t e a p r o p e r t y an e l e m e n t o f an
P r o p e r t y P w i l l be an a b s t r a c t p r o p e r t y and we assume

t h a t 0 always possesses p r o p e r t y P f o r any P.

3.1

Definition

(a) (b)

An element a o f a n R-group i s c a l l e d a P-element i f i t has p r o p e r t y P. An Q-group G (+subgroup

A, H

0

G) w i l l be c a l l e d a P-R-group

subgroup, P - i d e a l ) i f e v e r y e l e m e n t o f G (A,H)

The f o l l o w i n g c o n d i t i o n s w i l l be used f r e q u e n t l y . (A)

(P-R-

i s a P-element o f G (A,H). L e t H a G.

I f a i s a P-element o f G, t h e n a+H i s a P-element o f G/H.

a9

Radicals of Sl-groups b.v means of elemenr.,s

(B) (C)

Ifa i s a P-element o f G, t h e n a i s a P-element o f H. Ifa t H # H i s a P-element o f G/H,

t h e n a i s a P-element o f H.

(D) I f a i s a P-element o f H, t h e n a i s a P-element o f G. (E)

I f a+H i s a P-element o f G / H and H i s a P - i d e a l , t h e n a i s a P-element o f o f G.

(F)

I f a i s a P-element o f G and H has no non-zero P-elements, t h e n a+H i s a P-element o f G/H. Let

-

3.2

denote

l o g i c a l negation.

Lemma (Wiegandt, [ l l ] )

P s a t i s f i e s (A),

(B), (C),

(D),

(E),

( F ) i f and o n l y i f -P s a t i s f i e s ( C ) ,

(01, ( A ) , ( B ) , ( F ) , ( E ) r e s p e c t i v e l y . S i m i l a r t o Wiegandt ( [ l l ] ) we can p r o v e t h e f o l l o w i n g theorems f o r (?-groups. 3.3

Theorem (Wiegandt, [ 1 1 1 , Theorem 1 )

Let property P s a t i s f y conditions (A), R1(P) = t G

IG

( D ) and ( E ) .

i s a P-$2-group} i s a r a d i c a l c l a s s .

Then

I f P also satisfies condition

( B ) , t h e n R (P) i s a h e r e d i t a r y r a d i c a l c l a s s .

Furthermore,

I Each

SR (P) = { G 1

non-zero i d e a l o f G has a n o n - z e r o -P-element)

i s t h e semisimple c l a s s o f R (P). ( C ) and ( F ) R*(T)

=

I f T = - P and T s a t i s f i e s c o n d i t i o n s ( B ) ,

, then {G

1

G has no non-zero T-elements)

i s a radical classwhich i s hereditary i f T satisfies condition (0).

3.4

Theorem (Wiegandt,

[ I l l , Theorem 2 )

I f p r o p e r t y P s a t i s f i e s c o n d i t i o n s ( A ) and ( B ) , t h e n R (P) = I G

morphic image o f G has no non-zero P-elements} i s a r a d i c a l c l a s s .

I

E v e r y homoThe semi-

s i m p l e c l a s s o f Rp(P) i s g i v e n by

I

SR ( P ) = { G

E v e r y non-zero i d e a l o f G has a homomorphic image c o n t a i n i n g non-zero P-elements}.

I f T = -P and T s a t i s f i e s c o n d i t i o n s ( C ) and ( D ) t h e n R*(T) = { G

I Every

homomorphic image o f G i s a T-f?-group]

i s a radical class. 3.5

Theorem (Wiegandt, [ T I ] ,

Theorem 3)

I f p r o p e r t y P s a t i s f i e s c o n d i t i o n ( D ) , then

R (P) = { G

I

E v e r y n o n - z e r o homomorphic image

F

o f G has a non-zero

a c c e s s i b l e 0-subgroup such t h a t each e l e m e n t i s a P-element

of GI i s a radical class.

I f T = -P s a t i s f i e s c o n d i t i o n ( B ) , t h e n

G.K. Gerber

90 R*(T) = I G

I

Every non-zero homomorphic image s i b l e n-subgroup

F

o f G has a non-zero acces-

c o n t a i n i n g no non-zero T-elements o f

T; 1

i s a radical class.

3.6

Theorem (Wiegandt, [ll],P r o p o s i t i o n )

(a)

Ri(P)

= R;(-P),

(b)

R:(P)

_c R1(P), R2(P) c_ R:(P)

(c)

R1(P) 11 R2(P) = R1(P) I > R:(P)

(d)

Ifp r o p e r t y P s a t i s f i e s c o n d i t i o n (A), t h e n R1(P) = R:(P)

(el

If p r o p e r t y P s a t i s f i e s c o n d i t i o n ( B ) , then RL(P)

i = 1,2,3.

c_ UR1(P). = {O}.

c_ R3(P).

5 SR2(P).

Note t h a t t h e f u l l analogue o f Wiegandt i s n o t proved ( p a r t ( i i i ) o f t h e p r o p o s i t i o n is mi t t e d )

4.

.

APPLICATIONS Wiegandt (1111) had many examples t h a t he c o u l d f i t i n t o t h i s mould. a-groups,

on t h e o t h e r hand, have few known examples o f r a d i c a l s .

4.1

D e f i n i t i o n (Buys and Gerber, [ 2 ] )

a E G i s c a l l e d a niZpotent eZement o f G i f t h e r e e x i s t s a n o n - t r i v i a l Rword f ( 5 ) such t h a t f ( a ) = 0. L e t P1 be t h e p r o p e r t y n i l p o t e n t . 4.2

Theorem

( D ) , (E) and ( F ) .

P1 s a t i s f i e s c o n d i t i o n s ( A ) , ( B ) ,

Proof Since f(a+H) = f ( a ) + H f o r any R-word f ( x ) , i t f o l l o w s t h a t P1 s a t i s f i e s cond i t i o n ( A ) . I t i s easy t o see t h a t ( B ) ,

(D), and ( F ) a r e a l s o s a t i s f i e d .

Consider t h e n i l p o t e n t element a+H o f G/H w i t h H being a P - i d e a l ( i . e . ideal).

Hence f ( a ) t H.

H.

a nil

Thus we can f i n d a n o n - t r i v i a l Q-word f ( 5 ) such t h a t f(a+H) = f ( a ) + H = Since H i s a P - i d e a l t h e r e i s a n o n - t r i v i a l n-word g ( y )

such t h a t g ( f ( a ) ) = 0.

Thus a i s a n i l p o t e n t element o f G s i n c e g ( f ( 5 ) ) i s a

n o n - t r i v i a1 R-word. I t f o l l o w s t h a t R ( P ) , R (P ) , R (P ) and R*(P ) are r a d i c a l c l a s s e s and 1

t h a t R (PI) 4.3

1

2

1

3

1

3

1

i s hereditary.

Definition

a E G i s c a l l e d a centraZ eZement o f G i f [a,G]

= 0.

From H i g g i n s ( [ 7 ] , Lemna 4.3) i t f o l l o w s t h a t a i s a c e n t r a l element i f and G only i f [ a ,GI = 0. L e t P be t h e p r o p e r t y being c e n t r a l .

4.4 Theorem P s a t i s f i e s conditions (A),

(B) and

(F).

Proof L e t H 4 G, a a c e n t r a l element o f G and 0 : G ism.

Then

+

G/H be t h e n a t u r a l homomorph-

91

Radicals of Sl-groups b y means of elements 0 = [a,G]e= [ae,Gel [a+H

=

Lemma 3.1)

(Higgins, [7],

, G/H]. Thus P2 s a t i s f i e s c o n d i t i o n ( A ) and a l s o

Hence a+H i s a c e n t r a l element o f G/H. S i n c e [a,H]

(F).

5

[a,G]

=

0 we have t h a t P2 s a t i s f i e s c o n d i t i o n (B).

By 3.4 i t f o l l o w s t h a t R 2 ( P 2 ) i s a r a d i c a l c l a s s . We now i n t e r p r e t t h i s p r o p e r t y f o r t h e v a r i e t i e s o f a s s o c i a t i v e r i n g s ,

r-

r i n g s (Barnes, [ l ] ) and c u b i c r i n g s (Nobusawa, [ 8 ] ) u s i n g r e s u l t s o f H i g g i n s Theorem 4B and Lemma 4 . 3 ) .

([7],

I t i s easy t o see t h a t a i s a c e n t r a l element o f an a s s o c i a t i v e r i n g R i f and

o n l y i f a C Ann R (Ann R b e i n g t h e a n n i h i l a t o r o f R ) .

Hence, by Wiegandt ( [ l l ] ,

p.123) R ( P ) c o i n c i d e s w i t h t h e E 6 - r a d i c a l o f Szasz ( [ l a ] ) . 2

2

I f we d e f i n e t h e a n n i h i l a t o r o f a i ' - r i n g M by Ann M Mra

=

{a E

M 1

al'M = 0 and

01 t h e n i t f o l l o w s t h a t a i s a c e n t r a l e l e m e n t o f M i f and o n l y i f

=

a C Ann M. I n t h e v a r i e t y o f c u b i c r i n g s i t i s easy t o see t h a t a i s a c e n t r a l element o f a c u b i c r i n g i f and o n l y i f a = 0 . rings

Hence R ( P 2 ) i s t h e c l a s s o f a l l c u b i c

.

4.5

Definition

a C G possess p r o p e r t y P 3 i f f o r each n o n - t r i v i a l

a t aG

4.6 P

ili

E I? i t i s t r u e t h a t

...Gw. Theorem

s a t i s f i e s c o n d i t i o n s ( A ) , ( 0 ) and ( F ) .

The p r o o f f o l l o w s t r i v i a l l y . By 3.5 i t f o l l o w s t h a t R 3 ( P 3 ) i s a r a d i c a l c l a s s .

I n t h e case o f a s s o c i a t i v e

r i n g s t h i s i s t h e c l a s s o f A - r e g u l a r r i n g s (Wiegandt, [ I t ] ) .

4.7

Definition

a E G i s c a l l e d iderrpotent i f aa...aw

4.8

= a f o r a l l non-trivial w

il 52.

Theorem

a i s i d e m p o t e n t i f and o n l y i f f ( a )

=

a f o r each n o n - t r i v i a l R-word f ( 5 ) .

Proof T r i v i a l by i n d u c t i o n on t h e number o f o p e r a t i o n s p e r f o r m e d i n an Q-word. Let P

4.9 P

be t h e p r o p e r t y i d e m p o t e n t . Theorem

s a t i s f i e s conditions (A),

(B),

(D) and

(F).

Proof L e t a be a P -ele!nent o f G and f ( 5 ) any n o n - t r i v i a l @word. f(a)+H

=

a+H.

Then f(a+H)

Hence c o n d i t i o n ( A ) i s s a t i s f i e d and t h u s a l s o ( F ) .

(B) and

f o l lows t r i v i a l l y . By 3 . 4 and 3.5 R2(PLI), R 3 ( P L I ) and R:(Ps)

4.10 (a)

a r e r a d i c a l classes.

Definition G i s c a l l e d a non-trivia2 i2-group i f t h e r e e x i s t s a n o n - t r i v i a l

2

(D)

G .K. Gerber

92 operation w (b)

E R.

e E G i s c a l l e d an identity-element o f a n o n - t r i v i a l R-group G i f f o r each n o n - t r i v i a l w

E R and g E

G i t i s t r u e t h a t e...ege...ew

= g where

g can be i n any p o s i t i o n . Let P 4.11

be t h e p r o p e r t y i d e n t i ty-element. Theorem ( B ) and ( F ) .

P5 s a t i s f i e s conditions ( A ) , Proof The p r o o f i s t r i v i a l .

By 3.4 and 3.5 i t f o l l o w s t h a t R (P ) and R*(P ) a r e r a d i c a l c l a s s e s . 2

4.12

5

3

5

Theorem

L e t G be a n o n - t r i v i a l s2-group and e an i d e n t i t y - e l e m e n t o f G. (a)

I f f o r a r b i t r a r y n o n - t r i v i a l w E R and H a G i t i s t r u e t h a t

5H

G...GHG...Gw

(b)

( H i n any p o s i t i o n ) , then eG

=

G.

I f G # 0 i s a zero-symmetric R-group then e # 0 and eG = G.

Proof (a)

E R we have g = e...ege...euix L e t g t G. For any n o n - t r i v i a l G G G Hence G = e G...e GG...Gw 5 e

.

.

(b)

.Gw c - H.

By 2.3 we have G.. .GHG..

L e t 0 # g i G.

Hence eG = G by ( a ) .

= 0 f o r any n o n - t r i v i a l

O...OgO...Ow

(ii

Since G i s zero-symmetric, C h?.

Therefore, e # 0.

qs

Zero-symmetric i s a necessary c o n d i t i o n i n 4.12(b). mu1t i p l ic a t ion by

7

T

Then G i s an R-group w i t h

u. a as

-

i d e n t i t y - e l e m e n t and

Let G = 2

aG= a #

2

and d e f i n e

G.

I t i s a l s o i n t e r e s t i n g t o i n q u i r e i n t o t h e uniqueness o f i d e n t i t y - e l e m e n t s .

The f o l l o w i n g example show t h a t t h i s i s n o t s o even i f t h e fi-group i s d i s t r i b utive.

Let G

=

i d e n t i ty-elements

4.13 (a)

a62

Z3. D e f i n e a t e r n a r y o p e r a t i o n as f o l l o w s :

G i s a d i s t r i b u t i v e R-group (hence a l s o zero-symmetric)

.

and both

= abc mod 3.

T and 7 a r e

Definition

E G i s c a l l e d f-reguZar i f f o r each n o n - t r i v i a l

a

w

E R

i t holds that

a E aGaG...aGwG. (b) (c) 4.14

G i s c a l l e d idempotent i f f o r each n o n - t r i v i a l o E R i t h o l d s t h a t

H

0

HH

...HwG

= H.

G i s c a l l e d h e r e d i t a r i l y idempotent i f each i d e a l o f G i s idempotent.

Theorem

H a G i s idempotent i f and o n l y i f f ' ( H ) = H f o r each n o n - t r i v i a l R-word

f(x)

93

Radicals of Ggroups by means of elements Proof The s u f f i c i e n c y i s t r i v i a l .

The n e c e s s i t y i s proved by i n d u c t i o n on t h e num-

b e r o f o p e r a t i o n s performed i n an R-word. Let P

be t h e p r o p e r t y f - r e g u l a r .

4.15

Theorem

P, s a t i s f i e s c o n d i t i o n s (A), ( D ) ,

( E ) and ( F ) .

Proof L e t a be an f - r e g u l a r element o f G and H

G.

Q

By d e f i n i t i o n o f a homomorph-

ism and Lemma 3.1 ( H i g g i n s , [ 7 ] ) i t f o l l o w s t h a t f o r each n o n - t r i v i a l w E R

.(a+H)G/HwG/H =

(a+H)ti/H(a+H)G/H..

(aGaG.. .aGuG+ H).

Since a E aGaG...aGwG i t f o l l o w s t h a t a t H E (a+H)G/H(a+H)G/H...(a+H)G/HwG/H f o r each n o n - t r i v i a l w E 12.

Thus P

s a t i s f i e s c o n d i t i o n ( A ) and hence a l s o ( F ) .

C o n d i t i o n (D) i s s a t i s f i e d t r i v i a l l y . L e t H be an f - r e g u l a r i d e a l o f G. G/H and

(0

E 52 a n o n - t r i v i a l o p e r a t i o n .

L e t a+H E G/H be an f - r e g u l a r element of Hence

a+H E (a+H)G/H(a+H)G/H...(a+H)G/HwG/H

(aGaG...aGk?+ H)/H

=

G G - G Hence t h e r e e x i s t s c E aGaG.. .a w c a G G - Since Thus bG 5 a Since c E aG, we have -c+a E a

I t f o l l o w s t h a t a t aGaG.. .aGwGt H. such t h a t -c+a

=

b

E

H.

.

.

H i s an f - r e g u l a r i d e a l , i t f o l l o w s t h a t f o r any n o n - t r i v i a l

b C bHbH.. .bHoH

5 bGbG.. .bGwG G G c -a a

...aGwG .

Since b,c C aGaG...aGwG i t f o l l o w s t h a t a Hence P

E 12

=

c t b E aGaG...aGwG.

also s a t i s f i e s condition ( E ) .

By 3.3 and 3.5 we have t h a t R ( P 6 ) and R 3 ( P 6 ) a r e r a d i c a l classes. more, by 3.6,

Further-

R1(P6) = RZ(P6).

4.16 Theorem L e t G be a s u b d i r e c t l y i r r e d u c i b l e n i l p o t e n t o r idempotent.

-group.

The h e a r t , H, o f G i s e i t h e r

(An i d e a l H i s c a l l e d n i Zp o t en t i f t h e r e e x i s t s a non-

t r i v i a l R-word f ( 5 ) such t h a t f ’ ( H ) = 0.) Proof

i f H i s n i l p o t e n t t h e n we a r e f i n i s h e d . f ’ ( H ) # 0 f o r each n o n - t r i v i a l R-word f ( 5 ) .

Suppose H i s n o t n i l p o t e n t . Since f ’ ( H ) -

we must have H = f ’ ( H ) f o r any n o n - t r i v i a l R-word f ( 5 ) .

I=

Hence

H and 0 # f ’ ( H )

a G

By 4.14 i t f o l l o w s t h a t

H i s idempotent.

The n e x t theorem g e n e r a l i z e s two theorems i n r i n g t h e o r y (Szasz,

[lo],

P r o p o s i t i o n 13.4 and Courter, [ 5 ] ) .

4.17 Theorem I f G is a zero-symmetric R-group then t h e f o l l o w i n g statements a r e e q u i v a l e n t

G . K . Gerber G i s h e r e d i t a r i l y idempotent. G i s f - r e g u l a r t h a t i s , every element o f G i s f - r e g u l a r . F o r each n o n - t r i v i a l w E 2 and Bl,B2

,...,Bn

d

G we have B B 1

2

...Bnw G = i 11=n 1Bi.

G has no s u b d i r e c t l y i r r e d u c i b l e homomorphic image w i t h n i l p o t e n t h e a r t . Every homomorphic image o f G i s a s u b d i r e c t sum o f s u b d i r e c t l y i r r e d u c i b l e a-groups, each w i t h an idempotent h e a r t . Every I

Q G i s the i n t e r s e c t i o n o f a l l 2-prime i d e a l s P o f G such t h a t I ? P and G/P has a minimal i d e a l . Every i d e a l o f G i s an R-semi-prime i d e a l .

Proof L e t a E G and w be a n o n - t r i v i a l o p e r a t i o n .

( a ) =z ( b ) ( b ) => ( c ) BIB 2 . . . B n ~

b E bGbG...b (c)

G

=

Thus G i s f - r e g u l a r .

aGaG...aGuG.

2.3,

Then a E a

w

R be a n o n - t rni v i a l o p e r a t i o n and B 1 ,B 2 ,...,Bn a G. Since G i s f - r e g u l a r , we have L e t b E ,(I Bi.

Let w t G n

5

B 1. i=i [ I

c -B B 1

2

...Ew, G .

By

'='

Hence e q u a l i t y f o l l o w s .

( d ) I f ( c ) holds f o r G, then i t a l s o holds f o r any homomorphic image L e t w 6 R be a n o n - t r i v i a l L e t 0 : G + G ' be a homomorphism onto G '

=:,

.

o f G.

o p e r a t i o n and B ' , B i

B . a G, i = 1 ,2 1

.

,...,B,',

,... ,n

Q

G'.

By Theorem 38 o f H i g g i n s ( [ 7 ] ) t h e r e e x i s t

such t h a t Ker O

5 Bi

and BiO

= B;,

i = 1 ,2

,...,n.

Hence

B ' B ' , .B;IwG = (B1B2.. .Bnw G ) O 1

2

n

= ( I ) Bi)8

i=i n c - I I B ~ .

i= I

Let b' E

n 11 B i and bi E Bi such t h a t bi8

=

b ' , i = 1,2

i =1

,...,n.

Hence biO = b '

b.8, i # j . Therefore, t h e r e e x i s t s k t Ker 8 such t h a t bi = k+b.. But J J Ker O B . , j = 1 ,2,. ,n and hence bi = k+b. E B . f o r j # i. Therefore, J J J n G , i = 1 ,2 ,n. bi E ,I1 Bi = B B Bw , 1 2 1 =1 n BnwG)O = Bi,Bi,.. .,Bl;oG'and hence n B! = B I B ' BJ,) Thus b ' = bi8 E (B B 1 1 1 2 iz1 1 1 2

=

..

c

...

,...

...

...

G'

.

L e t GO, w i t h n i l p o t e n t h e a r t H, be a s u b d i r e c t l y i r r e d u c i b l e homomorphic image o f G.

By what we have j u s t proved, i t f o l l o w s t h a t H i s idempotent.

T h i s c o n t r a d i c t s 4.16.

Therefore, ( d ) f o l l o w s .

( d ) => ( e ) By B i r k h o f f ' s Theorem (Gratzer [ 6 ] , 120 Theorem 3 ) every homomorphic image GO o f G, i s a s u b d i r e c t sum o f s u b d i r e c t l y i r r e d u c i b l e R-groups. Since each s u b d i r e c t sumnand i s a homomorphic image o f GO, and hence a l s o o f G, i t f o l l o w s t h a t the h e a r t o f each s u b d i r e c t summand i s n o t n i l p o t e n t and thus

idempotent by 4.16.

95

Radicals of .%groups by means of elements L e t I a G.

( e ) => ( f )

G / I i s a homomorphic image o f G and hence a s u b d i r e c t

sum o f s u b d i r e c t l y i r r e d u c i b l e R-groups and i I ( I a / I )

h e a r t s Ha/Ia

=

(G/I)/(Ia/I)

ilia =

Hence

0.

= G/Ia

I.

L e t w t R b e a n o n - t r i v i a l o p e r a t i o n and A / I a , A 2 / I a

0 # Ha/Ia

,... ,n. Then Ha/Ia G/ I a (Ha/Ia) ...(Ha/Ia)w G/Ia . (Aa/Ia). . . (An/Ia)w

= c

(Aa/Ia)

By Buys and Gerber ( [ 4 ] ,

Ia

g r o u p and hence

Theorem 2.3)

, ... ,An/Ia

.. ,n.

i = 1 ,2,.

i t follows t h a t G/Ia

being a minimal i d e a l o f G / I a .

G

Hence

i s an $]-prime R-

i s an 52-prime i d e a l o f G f o r each a (Buys and Gerber, [ 4 ] , Now ( f ) f o l l o w s e a s i l y .

( f ) => ( a ) L e t A a G and w E 12 be any n o n - t r i v i a l o p e r a t i o n . G AA.. .Am a G i t f o l l o w s f r o m t h e assumption t h a t

A A . . .Ao

a G/Ia

T h i s I = o I a , Ia a n R-prime i d e a l o f G such t h a t I F I a and

C o r o l l a r y 2.10). Ha/Ia

c_ Ai/Ia,

# 0, i = 1 ,2

such t h a t Ai/Ia

w i t h idempotent

= [){Pa

Q

G

Pa an R-prime i d e a l o f G such t h a t AA

Since

...AmG 5 P

and

G/Pa has a m i n i m a l i d e a l } .

...AwG 5 P A F I I P ~ = AA ...AmG: S i n c e AA

and Pa i s R-prime,

i t follows that A

E q u a l i t y f o l l o w s s i n c e A a G.

5 Pa

f o r each a.

Hence

Hence G i s h e r e d i t a r i l y

idenpotent. L e t P a G, w E R a n o n - t r i v i a l o p e r a t i o n and a t G such t h a t G G S i n c e G i s f - r e g u l a r . a C aGaG. ..a w and t h u s a E P. There-

( b ) ==> ( 9 )

- P. aGaG. ..aGwG c

f o r e , P i s an i2-semi-prime i d e a l . L e t a C G and w E R a n o n - t r i v i a l o p e r a t i o n .

( 4 ) => ( b )

aGaG...aGwG a G

and must be an R-semi-prime i d e a l by assumption. By d e f i n i t i o n o f Q-semi-prime G G G G and aGaG.. .a w 5 aGaG.. . a w we have aG 5 aGaG. ..aGwG. Hence a t aG 5 aGaG...aGb?.

Thus t h e r e q u i r e d r e s u l t f o l l o w s .

We a l s o g i v e t h e f o l l o w i n g r e l a t i n g r e s u l t . 4.18 Theorem The c l a s s o f i d e m p o t e n t R-groups i s a r a d i c a l c l a s s . Proof We f i r s t n o t e t h a t f ' ( G 0 )

f ' ( G ) B e a s i l y f o l l o w s by i n d u c t i o n on t h e number

=

o f o p e r a t i o n s p e r f o r m e d i n any R-word f ( x ) . L e t G be an i d e m p o t e n t S1-group and I word.

Q

G.

L e t f ( x ) be any n o n - t r i v i a l R-

Then

f ' ( G / I ) = ( f ' ( G ) + I ) / I = G/I Again, by 4.14,

u s i n g 4.14.

G/I i s i d e m p o t e n t .

P r o p e r t y R3 ( R j a b u h i n , [ 9 ] )

are satisfied.

L e t Ha be an i d e m p o t e n t i d e a l o f G and f ( x ) any n o n - t r i v i a l R-word. f'(ZHa)

5 ZHa

CHa = f ' ( Z H a ) .

i s satisfied.

by d e f i n i t i o n .

B u t Ha = f'(H,)

Then

c_ f ' ( C H a ) f o r any a and hence

Thus ZHa i s an i d e m p o t e n t i d e a l .

P r o p e r t y R5 ( R j a b u h i n , [ 9 ] )

G. K. Gerber

96

L e t H = ZHa be t h e sum o f a l l t h e i d e m p o t e n t i d e a l s . idempotent i d e a l .

f ' ( J ) / H = J/H s i n c e H = f ' ( H )

so J

5 H.

L e t J/H a G/H be an

F o r any n o n - t r i v i a l 9-word f ( x ) , f ' ( J / H ) = J/H, t h a t i s ,

c_ f ' ( J ) because H 5 J .

P r o p e r t y R6 ( R j a b u h i n , [ 9 ] ) i s s a t i s f i e d .

T h e r e f o r e , f ' ( J ) = J and

By Theorem 1.2 o f

R j a b u h i n ( [ 9 ] ) t h e c l a s s o f i d e m p o t e n t $2-groups i s a r a d i c a l c l a s s . REFERENCES Barnes, W.E., On t h e r - r i n g s o f Nobusawa, P a c i f i c . J. Math., 1 8 ( 1 9 6 6 ) , N0.3, 411-422. Buys, A. and Gerber, G.K., N i l and s-prime R-groups, J . A u s t r a l . Math. SOC ( S e r i es A), 3 8 ( 1985) , 222-229 : Buys, A. and Gerber, G.K., Prime and k-prime i d e a l s i n R-groups, Q u a e s t i o n e s Mathematicae, 8 ( 1 9 8 5 ) , No.1, 15-32. Buys, A. and Gerber, G.K., The p r i m e r a d i c a l f o r 52-groups, Communications i n A l g e b r a , 10(1982), No.10, 1089-1099. C o u r t e r , R.C., R i n g s a l l whose f a c t o r r i n g s a r e semi-prime, Canad. Math. B u l l . , 12(1969), N0.4, 417-426. G r a t z e r , George, U n i v e r s a l A l g e b r a (2nd Ed,), ( S p r i n g e r - V e r l a g , New York I n c . , 1979). H i g g i n s , P.J., Groups w i t h m u l t i p l e o p e r a t o r s , London Math. SOC. Proc., ( 3 ) , 6 ( 1 9 5 6 ) , 366-416. Nobusawa, Nobuo, On a g e n e r a l i z a t i o n o f t h e r i n g t h e o r y , Osaka J . Math 1 ( 1 9 6 4 ) , 81-89. R j a b u h i n , J u . M., R a d i c a l s i n R-groups I , General Theory, Mat. I s s l e d , 3(1968), 123-160 ( i n R u s s i a n ) . Szasz, F.A., R a d i c a l s o f Rings, (John W i l e y & Sons, 1981). Wiegandt, R i c h a r d , R a d i c a l s o f r i n g s d e f i n e d b y means o f e l e m e n t s , U s t e r r e i c h Akad. Wiss. Math. N a t u r w i s s . SB 11, 184(1975), No.1-4, 117- 25.

.

Near-rings and Near-fields,C. Betsch (editor) 0 Elsevier Science Publishen B.V. (North-Holland), 1987

NOTE

ON

THE

COMPLETELY

PRIME

91

RADICAL

IN

NEAR-RINGS

Groenewald

N.J.

U n i v e r s i t y o f P o r t E l i z a b e t h , P 0 Box 1600, 6000 P o r t E l i z a b e t h ABSTRACT: I n t h i s note a completely prime r a d i c a l i s d e f i n e d f o r nearr i n g s . I t i s then proved t h a t f o r t h e c l a s s o f a l l d i s t r i b u t i v e l y generated n e a r - r i n g s t h e completely prime r a d i c a l c o i n c i d e s w i t h t h e g e n e r a l i z e d n i l r a d i c a l . We a l s o show t h a t i n general t h e completely prime r a d i c a l and a l s o t h e prime r a d i c a l i s n o t h e r e d i t a r y . 1.

INTRODUCTION L e t N be a n e a r - r i n g .

An i d e a l I o f N i s completely prime i f a,b C N, ab C I

i m p l i e s a E I o r b E I. I f ( 0 ) i s a completely prime i d e a l then N i s c a l l e d a completely prime n e a r - r i n g . We d e f i n e t h e completely prime r a d i c a l o f t h e nearr i n g N as t h e i n t e r s e c t i o n o f a l l t h e completely prime i d e a l s and denote i t by C(N).

For t h e n o t a t i o n s about general r a d i c a l t h e o r y i n n e a r - r i n g s we r e f e r t o [ 7 ] L e t M be a c l a s s o f completely prime near-rings.

if A E M and I M.

0

A then

IE

M.

Clearly M i s hereditary, i.e.

L e t UM be t h e upper r a d i c a l c l a s s determined by

We have UM = {A : A has no nonzero homomorphic image i n M} = {A : every nonzero homomorphic image o f A has a nonzero d i v i s o r o f zero j

As i n t h e case o f r i n g s

.

[Z] and [ 3 ] we c a l l

UM t h e g e n e r a l i z e d n i l r a d i c a l and

denote i t by N

9' We have t h e f o l l o w i n g theorem:

Theorem 1. Let

UM = N ' = {A : A a n e a r - r i n g such t h a t A = '.?(A)].

D denote t h e c l a s s o f a l l d i s t r i b u t i v e l y generated n e a r - r i n g s .

If

M is

a c l a s s o f prime n e a r - r i n g s , K a a r l i [ 4 ] d e f i n e d t h e c l a s s M t o be D - s p e c i a l i f

M I I 2) i s h e r e d i t a r y w i t h r e s p e c t t o i d e a l s and i f U a S a N E D and S/U E N i m p l y U 4 N and N/(U:S) E N, where (U:S) = {n E N : n S i U j . I n what f o l l o w s , l e t M be t h e c l a s s o f a l l completely prime n e a r - r i n g s . Theorem 2. Proof.

The c l a s s M o f a l l completely prime n e a r - r i n g s i s D - s p e c i a l .

ki r I D i s c l e a r l y h e r e d i t a r y .

prime n e a r - r i n g s i s D - s p e c i a l .

Since a completely prime n e a r - r i n g i s a l s o a

prime n e a r - r i n g , we have U 4 S a N E t o show t h a t (U:S) x2

E (U:S).

a l s o have bx

D and S/U E M i m p l i e s

i s a completely prime i d e a l i n N.

Suppose x j?

From [ 4 ] we have t h a t t h e c l a s s o f a l l

? j'

(U:S),

U, f o r i f bx E

i.e.

U

4

N.

We o n l y have

L e t x E N such t h a t

t h e r e e x i s t s b E S such t h a t xb 5!

U then xbxb E U and s i n c e

S/U

U. We

E M, i.e. U

98 98

N.J. Groenewald Groenewald N.J.

completely prime i d e a l of S , i t follows t h a t xb E U. Let a E S be a r b i t r a r y . Since x 2 E (U:S) we have x 2 a E U and therefore bx2a € U. Since bx f U and U i s a completely prime i d e a l , we have xa E U and consequently x E ( U : S ) . Hence ( U : S ) i s a completely semi-prime i d eal . I t i s now easy t o show t h a t , since ( U : S ) i s a prime ideal ( c . f . [ 5 1 ) , ( U : S ) i s also completely prime. Hence N / ( U : S ) E M. I n the d e f i n i ti o n of P-spesial classes the following condition plays an important r o l e : ( F ) U q S 4 N and S/U E M imply U Q S where M i s a c la ss of near-rings. The following example shows t h a t in general condition ( F ) i s n o t s a t i s f i e d i f M i s e i t h e r the cl as s of prime near-rings or the c l a s s of completely prime near-rings. Example 1 . Consider the dihedral group N with a d d i t i o n and multiplication defined as in Pilz [ 6 1 , p.345, number l l . symmetric

it

N i s a near-ring which i s n o t zero

*

0

1

2

3

4

5

6

7

0

0

0

0

0

0

0

0

0

1

0

1

0

1

0

1

1

0

2

0

2

0

2

0

2

2

0

3

0

3

0

3

0

3

3

0

4

4

4

4

4

4

4

4

4

5

4

5

4

5

4

5

5

4

6

4

6

4

6

4

6

6

4

7

4

7

4

7

4

7

7

4

i s easy ro cnecK t n a t

F u r t h e r m o r e , K = {0,21

I = ~U,I,L,JI i s t n e only n o n t r i v i a l ideal o t N. I b u t 10,21 i s n o t an i d e a l i n N. L e t M d e n o t e t h e

Q

c l a s s of a l l p r i m e n e a r - r i n g s , i . e .

B of N , we have AB # 0. I and I =
C l e a r l y t h e n e a r - r i n g i n t h e example i s p r i m e , i . e .

Hence N i M.

i s a prime ideal. =

I.

N E M i f f o r e v e r y two nonzero i d e a l s A and

Hence <&

i s not s a t i s f i e d since K

0

Furthermore, i f a,b f K, we have I

K.

= ‘a>N = T h e r e f o r e I / K E M and c o n d i t i o n ( F )

fi N.

I f M i s t h e c l a s s o f c o m p l e t e l y p r i m e n e a r - r i n g s , we have N f M s i n c e 2.2

=

0.

I i s a c o m p l e t e l y p r i m e i d e a l i n N f o r i f x,y f I,t h e n xy f I . I f x,y a r e elements o f I such t h a t x,y f K, t h e n xy f K. Hence K i s a c o m p l e t e l y p r i m e i d e a l i n I and, t h e r e f o r e , I / K E M . C o n d i t i o n ( F ) i s n o t s a t i s f i e d f o r t h e c l a s s o f a l l c o m p l e t e l y p r i m e n e a r r i n g s s i n c e K fi N. The f o l l o w i n g theorem i s p r o v e d i n [ l ] . Theorem 3 ( [ l ] , Theorem 1 ) . L e t M be a r e g i l a r and e s s e n t i a l l y c l o s e d c l a s s o f n e a r - r i n g s s a t i s f y i n g c o n d i t i o n ( F ) . Then fi = {A : A = ( A a : Aa E M ) } i s a semisimple c l a s s , i n f a c t

fi

subdi r e c t

= SUM, moreover, t h e upper r a d i c a l c l a s s

99

Note on the completely prime radical

y = UM i s hereditary, i . e . I 4 A E y implies I E y. Theorem 4. I f M i s the cl as s of a l l completely prime near-rings, then f o r every PC E

D and I

__ Proof.

4 N we have N ( N ) fl I = N g ( I ) and N ( N ) = C ( N ) . 4 9 We show t h a t the conditions of Theorem 3 i s s a t i s f i e d .

Since M i s D-

s p e c i a l , we only have t o show t h a t M i s e s s e n t i a l l y closed. Let N be a nearring and H an e ss en t i al ideal of N . We show tha t i f A E M , then N E M. I t i s easy t o show t h at [ ( A ) = {n E N : nA = 01 i s a two-sided ideal of N. But ([(A) n A)'t_ C ( A ) . A = 0. Since L ( A ) IIA a A and A F M , we have [ ( A ) il A = 0. From the f a c t t h a t A i s an es s en t i al ideal i t follows t h a t [ ( A ) = 0. Let x,yCN such t h a t x,y # 0. Since [ ( A ) = 0 , we can find a ,b E A such t h a t xa and yb # 0. Furthermore, we have ax # 0 f o r i f ax = 0 , then xaxa = 0 and since A E M , i t follows t h a t xa = 0. Now axsyb # 0 and hence a lso xy # 0. T h u s N E M. From Theorem 3 we have N i s hereditary and from [ I ] , Proposition 3, we have SUM 4 hereditary. Hence, f o r every N 6 U and every I 4 N we have N ( I ) = N i i N ( N ) . 9 9 For N E D we have t h at C ( N ) = N ( N ) . To show t h i s , l e t {TJ be the s e t o f a l l 9 ideals T of N f or which NITa F M. We have N (N): B = (ITa. Let us suppose 9 In this case w e have N ( B ) = B i ) N ( N ) = N ( N ) ,f B. Hence there N 9 ( N ) # B. 9 9 9 e x i s t s an ideal C 0 B with 0 # B/C E M. From Theorem 2 we have N/(C:B) E M. Consequently ( C : B ) E lTa}. ( C : B ) i l B = C f o r i f x 6 (C:B) rl B , then x E B a n d x 2 C C. Now, since B/C i M , we have x E C. Hence ( C : B ) 1 1 B f C. Inclusion i n the other d i r e c ti o n i s cl ear . T h u s B = IiTa = B I I ( C : B ) = C # B a contradiction. Hence B = N ( N ) = C ( N ) . 9 Remark. From [ 4 1 i t follows t h a t i f N E U , then P(N) 11 I = P ( 1 ) where I o N and P(N) denotes the prime radical of N. In general, the prime and completely prime r a d i c a l s a r e not hereditary. To show this we use the example above. Since N i s not a completely prime near-ring, we have C ( N ) = I since I i s the only completely prime ideal of N. I , considered as a near-ring, i s n o t completely prime since 2.2 = 0. Since K i s the only completely prime ideal of I , we have I Jl C(N) = I. C(1) Since N i s a Kurosh-Amitsur radical ( as the upper radical of a hereditary c l a s s ) 9 and N = 1N : N = C ( N ) I b u t N ( N ) # C ( N ) in general, the completely prime radical 9 4 i s n o t a Kurosh-Amitsur radical f o r near-rings.

KF

We note t h a t N i s a prime near-ring, hence P(N) = 0. The near-ring I = {0,1,2,3} i s n o t prime since 0 # K = {0,2} 4 I and K2 = 0 . Since K i s the only prime ideal of the near-ring I , i t follows t h a t P ( 1 ) = K ? P(N) i I I = 0. This shows t h a t , i n general, Theorem 5.62 o f Pilz [ 6 ] i s not s a t i s f i e d if I i s not a dire c t summand of N. REFERENCES

[ I ] Anderson, T . , Kaarli, K. and Wiegandt, R., Radicals and subdirect decompositions, Comm. in Algebra, 13(1985), 479-494.

100

N.J. Groenewald

( 2 1 Andrunakievic, V.A. and Rhabuhin, Ju. M., Rings w i t h o u t n i l p o t e n t elements and completely simple i d e a l s , Dokl. Akad. Nauk. SSSR, 180(1968, 9-1 ( i n Russian - E n g e l i s h t r a n s l a t i o n i n S o v i e t . Math. Dokl. 9(1968), 565-568). [3]

D i v i n s k y , N . , Rings and r a d i c a l s (Math. E x p o s i t i o n s No.14, Toronto Press, 1965).

[4]

K a a r l i , K., Special r a d i c a l s i n n e a r - r i n g s , T a r t u R i i k l . UL Toimetesed 610( 1982), 53-68.

[5]

McCoy, N.H., Completely prime and completely semi-prime i d e a l s , Rings, Modules and Radicals, C o l l . Math. SOC. S. B o l y a i , 6, North Holland (1973), 147-1 52.

[6] [7]

P i l z , G . , Near-rings (North-Holland Mathematics Studies, 1977).

University o f

Wiegandt, R., Near-rings and Radical Theory, Conf. on n e a r - r i n g s and nearf i e l d s , San Benedetto del T r o n t o ( I t a l y ) (1981), 49-58.

Near-rings and Near-fields,G . Betsch (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1987

101

ON p-ADIC NEARFIELDS The0 GRUNDHOFER Mathematisches Institut, Universitat Ttibingen Auf d e r Morgenstelle 10 D-7400 Tiibingen 1 , BRD This is an announcement of a result concerning t h e classification of locally c o m p s c t nearfields, namely

t h e following: all p-adic

nearfields with continuous

multiplication a r e Dickson nearfields. W e indicate a proof for some e a s i e r special

cases, and give a rough sketch of a proof for t h e general case. A topological nearfield is a nearfield with a topology such that addition, multiplication and inversion a r e continuous. The locally c o m p a c t connected nearfields have been classified by Kalscheuer [ 3 ] : these a r e t h e fields of real and complex numbers and Dickson nearfields derived from t h e skew field o,€ Hamilton's quaternions ( cp. also T i t s [ 7 ] , Salzmann [ 6 17.26 ). The following is an extension of Kalscheuer's result. Theorem

1

Every

non-discrete

locally c o m p a c t

nearfield

zero is a Dickson nearfield derived from a non-discrete

of

characteristic

locally c o m p a c t skew

field. The nearfields covered by this theorem a r e those of t h e reals

W or t h e p-adics Q

finite dimension over

with continuous multiplication. Theorem 1 is a

P consequence of t h e following group-theoretic assertion.

Let G be a closed subgroup of GL(n, Q ) or GL(n, R ) which a c t s P transitively on t h e nonzero vectors. Assume t h a t t h e stabilizer Gv is d i s c r e t e Theorem 2

for every nonzero vector v. Then G is a subgroup of I'L(I,D) for s o m e skew field D of dimension n over Q resp. IR P

.

Such a group G is a p-adic resp. real linear Lie group, and, a s a consequence

of our assumption on t h e stabilizers G

its linear Lie algebra L has t h e special v ' property t h a t all nonzero e l e m e n t s of L a r e invertible ( Kalscheuer [ 3 1 p.414 1.

T h e Lie algebra L c a n be used t o construct t h e skew field D, as t h e proof of t h e following proposition illustrates.

102

Th. Grundhiifer

Proposition

Let F be a non-discrete locally c o m p a c t nearfield of c h a r a c t e r i s t i c

zero. Then t h e following a r e equivalent: a ) The multiplicative group F* of F has an open abelian subgroup. b) The Lie algebra L of F* is abelian.

c) F' is a Dickson nearfield derived from a finite field extension of R o r

%.

Proof. a ) and b) a r e equivalent by Lie theory, and c) implies a ) , a s every finite extension of t h e reals o r t h e p-adics has only finitely many continuous field automorphisms. Suppose b) holds. Then L is contained in its centralizer C , which is a skew field by Schur's Lemma. Comparing dimensions gives

is a field.

L = C , hence L

Furthermore, F* a c t s on L via t h e adjoint representation, i.e. L is

normalized by F*, which gives c). The non-abelian Lie algebra of dimension 2 has no faithful linear representation with t h e special property mentioned above. For every topological nearfield F of

or IR , t h e Lie algebra of F* is therefore abelian, and t h e P Proposition applies, thus proving Theorem 1 for this special case. dimension 2 over Q

Finally we give a very sketchy description of a proof for Theorem 2 in t h e p-adic case. The Lie algebra L of F* is reductive, i.e. t h e direct sum of i t s c e n t e r and iLs semisimple c o m m u t a t o r algebra L'. The simple p-adic Lie algebras have been classified by Veisfeiler, Kneser 1 4 1, Bruhat-Tits [ 1 ] , T i t s [ 9

1.

This classifi-

cation yields t h a t L' is a direct sum of c o m m u t a t o r algebras D! of skew fields Di

. The

Proposition ( o r r a t h e r i t s group-theoretic analogue ) allows u s t o assume

that L' is not trivial. Then o n e of t h e D! a c t s irreducibly. Analysis of t h e representation of such an irreducible ideal D! shows that

D := D.

satisfies t h e con-

clusion of Theorem 2. Several s t e p s of t h e proof make use of t h e representation theory developed in Tits[ 81. It s e e m s t o be difficult t o remove t h e c h a r a c t e r i s t i c zero assumption in Theorem

1. Note t h a t Rink constructed non-discrete locally compact nearfields with

finite kernel ( see Hah1[2

1 p.279

). T h e Dickson nearfields derived from unramified

extensions of local fields have been classified in Rink [ 51. REFERENCES F. Bruhat and J. Tits, Groupes reductifs sur un corps local 1,Il , Publ. Math. I.H.E.S. 41 (1972) 5-251, 6 0 (1984) 5-184. [ 2 1 H. Hahl, Kriterien fur lokalkompakte topologische Quasiktirper, Arch. Math. (Basel) 38 (1982) 273-279. [ 3 ] F. Kalscheuer, Die Bestimmung aller stetigen Fastkdrper Uber d e m Korper d e r reellen Zahlen als Grundkbrper, Abh. Math. Sem. Hamburg 1 3 (1940) 413-435. [ 41 M. Kneser, Galois-Kohomologie halbeinfacher algebraischer Gruppen uber y - a d i s c h e n Kdrpern 1,11 , Math. 2. 88 (1965) 40-47, 8 9 (1965) 250-272. [I]

On padic near-fields

[5]

103

R. Rink, Zur Konstruktion lokal kompakter Dicksonscher FastkCirper, Geom. Dedicata 20 (1986). [ 6 ] H. Salzmann, Topological planes, Advances Math. 2 (1967) 1-60. [ 7 ] J. Tits, Sur les groupes doublement transitifs continus, Comment. Math. Helv. 26 (1952) 203-224. Correction et complkments, ibid. 30 (1956) 234-240. [ 81 J. Tits, ReprBsentations lingaires irrkductibles d'un groupe rBductif sur un corps quelconque, J. Reine Angew. Math. 247 ( 1 971 1 196-220. [ 9 ] J. Tits, Reductive groups over local fields, in: A. Bore1 and W. Casselman (eds.), Proc. Sympos. P u r e Math. 33 (19791, p a r t 1, pp. 29-69, Amer. Math. Soc., Providence [R.I.), 1979.

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Near-ringsand Near-fElds, G . Betsch (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1987

EUCLIDEAN SEMINEARRINGS AND NEARRINGS Udo HEBISCH and Hanns Joachim WEINERT 1.

INTRODUCTION

This paper is mainly based on the doctoral dissertation [ 6 1 of the first author, where the concept of a left Euclidean ring (cf. Def. 3.1) has been generalized to arbitrary (2,2)-algebras ( S , + , * ) (cf. S 2 and Def. 3 . 2 ) . Several statements on left Euclidean rings or on commutative ones, e. g. a well known characterization of Euclidean integral domains due to Motzkin [ l l ] , could be generalized in [ 6 1 to those left Euclidean (2,2)-algebras without any further assumptions. Other results of this kind have been proved using and the associativity merely the right distributive law for (S,+;) i. e. for (right distributive) seminearrings in the more of ( S , . ) , general meaning of this term (cf. 5 2). So 161 contains a first version of a theory of left Euclidean seminearrings and nearrings, which will be presented here in a self-contained and improved way. The only paper in this context known to the authors is the unwith left published manuscript [ 5 1 . It deals - in our notation Euclidean nearrings S which are embeddable into a nearfield of right quotients of S. We also mention that there are some recent papers on Euclidean semirings. They consider rather specialized (cf. § 2) for which both Euclidean functions on semirings ( S , + ; ) operations are associative and commutative. For an extensive discussion of these publications we refer to [8]. In the latter paper we have dealt with semirings for which both operations are only assumed to be associative, and which are left Euclidean algebras according to Def. 3.2 given here.

-

Coming back to this paper, there are some results on left Euclidean seminearrings which are essential for our subject, but in fact true for arbitrary (2,2)-algebras (S,+, * ) or those with associative multiplication. In order to see which assumptions are really needed and since the proofs cannot be simplified for seminearrings, we present those statements in their general version.

So, after some preliminaries in $ 2 , we define our basic concepts in § 3 for arbitrary (2,2)-algebras and show in particular, that a ring S is a left Euclidean (2,2)-algebra according to our

(I. Hebisch

106

Def. 3.2

and H.J. Weinert

i f f S i s a l e f t E u c l i d e a n r i n g . The n e x t s e c t i o n d e a l s w i t h

@ - d e r i v a t i o n s o f s e m i n e a r r i n g s i n o r d e r t o show t h a t l e f t E u c l i d e a n s e m i n e a r r i n g s can b e o b t a i n e d a s @ - d e r i v a t i o n s of o t h e r o n e s . I n t h i s way w e g e t examples of l e f t E u c l i d e a n n e a r r i n g s a s O-derivat i o n s of l e f t E u c l i d e a n r i n g s . I n 5 5 w e show t h a t each homomorphic image of a l e f t E u c l i d e a n ( 2 , 2 ) - a l g e b r a i s a g a i n l e f t E u c l i d e a n , and w e g i v e s u f f i c i e n t cond i t i o n s such t h a t t h e d i r e c t p r o d u c t of a f i n i t e number of l e f t E u c l i d e a n ( 2 , 2 ) - a l g e b r a s i s a l s o l e f t E u c l i d e a n . The l a t t e r y i e l d s t h a t such a d i r e c t p r o d u c t of l e f t E u c l i d e a n n e a r r i n g s a l l of which have l e f t i d e n t i t i e s i s a l e f t E u c l i d e a n n e a r r i n g . S u f f i c i e n t cond i t i o n s f o r t h e e x i s t e n c e of

( r i g h t ) i d e n t i t i e s a r e given f o r l e f t

E u c l i d e a n ( 2 , 2 ) - a l g e b r a s S such t h a t (S,.) i s a s s o c i a t i v e . A main p o i n t f o r e a c h l e f t E u c l i d e a n r i n g S ( s a y w i t h a l e f t

i d e n t i t y ) i s t h a t t h e l e f t Euclidean a l g o r i t h m y i e l d s t h e e x i s t e n c e of a g r e a t e s t common r i g h t d i v i s o r f o r any two e l e m e n t s a , b E S, and t h a t e a c h c h a i n of p r o p e r r i g h t d i v i s o r s i n S i s f i n i t e . According t o our c o n s i d e r a t i o n s i n

6 , t h e s e r e s u l t s a r e a l s o t r u e f o r each

l e f t E u c l i d e a n s e m i n e a r r i n g s a t i s f y i n g one supplementary a s s u m p t i o n which i s a l w a y s s a t i s f i e d f o r n e a r r i n g s . F u r t h e r , e a c h l e f t E u c l i dean r i n g w i t h a l e f t i d e n t i t y i s a l e f t p r i n c i p a l i d e a l r i n g . Also t h i s r e s u l t t r a n s f e r s t o l e f t E u c l i d e a n s e m i n e a r r i n g s and n e a r r i n g s , b u t w e emphasize t h a t one n e e d s i n t h i s c o n t e x t a c o n c e p t of “ l e f t i d e a l ” f o r s e m i n e a r r i n g s and n e a r r i n g s which i s i n t h e l a t t e r c a s e n o t t h e u s u a l one. In

the last section, we consider t h e constant subnearring S

and t h e zero-symmetric

1

one S2 of a n e a r r i n g S and t h e g e n e r a l i z a -

t i o n s of t h e s e c o n c e p t s f o r a s e m i n e a r r i n g S w i t h a l e f t a b s o r b i n g zero a s introduced i n S

1

5

4 of

[ 2 0 1 . Whereas a c o n s t a n t s e m i n e a r r i n g

i s always l e f t E u c l i d e a n , w e i n v e s t i g a t e c o n d i t i o n s s u c h t h a t

t h i s p r o p e r t y t r a n s f e r s from S t o S2 o r c o n v e r s e l y . 2 . PRELIMINARIES

(2,2)-ALGEBRAS AND SEMINEARRINGS

ON

W e consider (2,2)-algebras

S = (S,+,-)

with operations c a l l e d

a d d i t i o n and m u l t i p l i c a t i o n . The c a r d i n a l i t y of S , a l s o c a l l e d t h e o r d e r o f t h e a l g e b r a , i s d e n o t e d by ( S I . F o r subsets A,BES introduce

A+B= la+b

I

a E A , b r B} and

AB = { a b

I

aEA, bEB1,

u s e e . g . aB f o r t a l B . I f t h e r e e x i s t s a n e u t r a l e l e m e n t o of [e of

we and (S,+)

( S , - ) J ,it i s u n i q u e and c a l l e d t h e z e r o [ t h e i d e n t i t y ] of

t h e a l g e b r a S. Accordingly w e speak a b o u t Z e f t

( o r r i g h t ) identities

107

Euclidean seminear-rings and near-rings

of S. An e l e m e n t a € S i s c a l l e d left a b s o r b i n g i f f

a S = {a} h o l d s .

Corresponding l e f t - r i g h t d u a l c o n c e p t s and t h e i r two-sided v e r s i o n s a r e always t a k e n f o r g r a n t e d . W e f u r t h e r i n t r o d u c e t h e n o t a t i o n S *

s*

by

and by

0,

IS/ 2 2

otherwise.

S* = S

( 2 , 2 ) - a l g e b r a S i s c a l l e d multiplicatively

A

iff

i f S has a zero

=S\{o}

left c a n c e l l a t i v e

i s l e f t c a n c e l l a b l e i n (S,.).

h o l d s and each a E S *

If

i s a s s o c i a t i v e , w e have t h e f o l l o w i n g s t a t e m e n t ( c f . [181,

(S;)

Lemma 1 ) : S i s m u l t i p l i c a t i v e l y l e f t c a n c e l l a t i v e i f f e i t h e r a l l e l e m e n t s o f S , i n c l u d i n g t h e z e r o i f t h e r e i s o n e , a r e l e f t can( t w o - s i d e d ) a b s o r b i n g z e r o o and (S*,-)

cellable i n ( S , . ) o r S has a

i s a l e f t c a n c e l l a t i v e subsemigroup of

(S,.).

Our main i n t e r e s t i s w i t h ( 2 , 2 ) - a l g e b r a s (S,+;)

such t h a t ( S , . )

i s a semigroup and

( a + b ) c= a c + bc

d i n g t o Def. 1 . 1

[ 2 0 1 , w e c a l l such a n a l g e b r a a right seminear-

of

holds f o r a l l

r i n g , b r i e f l y a seminearring i n t h i s p a p e r .

a,b,c

E

S. Accor-

(The more r e s t r i c t e d

v e r s i o n of t h i s c o n c e p t i n c l u d e s a l s o t h e a s s o c i a t i v i t y of

(S,+).)

T o a v o i d c o n f u s i o n , t h e t e r m ( r i g h t ) nearring i s used f o r a s e m i n e a r r i n g such t h a t (S,+)

i s a group. A s e m i n e a r r i n g which s a t i s f i e s

c ( a + b )= c a + c b for a l l

also

a,b,ccS

i s c a l l e d a distributive

S 1, means sometimes such a d i s t r i b u t i v e s e m i n e a r r i n g , whereas i n i t s more f r e q u e n t v e r s i o n t h e a s s o c i a t i v i t y of b o t h o p e r a t i o n s i s i n (The term s c m i r i n g , i n t h i s p a p e r mainly used i n

seminearring.

cluded. ) F i n a l l y , l e t (S,+,')

be a ( 2 , 2 ) - a l g e b r a which may have a z e r o

o r n o t . Then a n a b s o r b i n g z e r o d i n g t h e o p e r a t i o n s by

o + x = x + o = x and xo = o x = o f o r a l l XESUCO}.

Clearly, the resulting (2,2)-algebra nearring i f f

(S,+,-)

can b e a d j o i n e d t o S e x t e n -

o#S

( S U { o } , + , - )i s a g a i n a s e m i -

i s one.

3. LEFT EUCLIDEAN (2,2)-ALGEBRAS AND SEMINEARRINGS The c o n c e p t of

a

E u c l i d e a n r i n g , s a y S i n o u r c o n t e x t , was o r i -

g i n a l l y d e f i n e d f o r commutative r i n g s w i t h o u t z e r o d i v i s o r s , and w i t h E u c l i d e a n f u n c t i o n s y from

S* = S x { o }

i n t o t h e s e t No of non-

n e g a t i v e i n t e g e r s . Various g e n e r a l i z a t i o n s ( c f . [Ill

,

[I51 , [ I 4 1

,

[ l o ], [ I 6 1

,

[171 , [12J , [ 9 J ,

[ I ] ) have i n c l u d e d non-commutative

r i n g s , o r t h o s e w i t h z e r o d i v i s o r s , o r r e p l a c e d No by any w e l l o r d e r e d set Wy. The most g e n e r a l o n e , g i v e n i n [ 2 1 f o r a r b i t r a r y ( a s s o c i a t i v e ) r i n g s , reads a s follows: DEFINITION 3.1.

A left E u c l i d e a n f u n c t i o n ( b r i e f l y LEF) on a r i n g S

U. Hebisch and H.J. Weinert

108

is defined as a mapping y : S* + W of S * into a well ordered set Y W with the following property: Y (3.1) For all a € S, b e S * there are q,r E S satisfying a =qb+r and r = o or y(r) < y(b). If such a LEF on

S

exists, S is called a left Euclidean ring.

Clearly, the zero o of S is used in (3.1) as an additive neutral to include a = q b by r =o. Moreover, since o is left absorbing, a = o b + a holds for all a,bES. Hence (3.1) is equivalent to (3.2) For all a , b e S * such that y(a) 1 y ( b ) there are q,r E satisfying a = q b + r and r = o or y(r) < y(b).

S

It is near by hand to apply Def. 3.1 also to any (2,2)-algebra (S,+,-) with a left absorbing zero, hence in particular to each nearring, and to look for a suitable generalization of the resulting concept for arbitrary (2,2)-algebras. The following definition presents such a generalization as we shall prove in Prop. 3.3. DEFINITION 3.2. A l e f t Euclidean function on a (2,2)-algebra S is defined as a mapping 6: S + W6 of S into a well ordered set W6 with the following property: (3.3) For all a,bE S such that 6(a) t 6(b) there are q,rc satisfying a = q b or a = qb+r and G(r) < 6(b).

S

If such a LEF on S exists, we call S a left Euclidean (2,2)-algebra, and e. g. a l e f t Euclidean seminearrJingif S is a seminearring. PROPOSITION 3.3. Let S be a (2,2)-algebra with a left absorbing zero 0 . If 6: S -t W6 is a LEF on S according to Def. 3.2, its restriction to S* is a mapping y : S* + W C W6 which satisfies (3.1 ) Y and, equivalently, (3.2). Conversely, each mapping y : S * + W of Y this kind can by extended to a LEF 6' on S satisfying (3.3) by adding a value 6' ( 0 ) 4 W y to W as a greatest element. Y In particular, a left Euclidean ring according to Def. 3.1 is just a left Euclidean (2,2)-algebra which is also a ring. Proof. If 6: S + W6 satisfies (3.3), its restriction y on S * satisfies (3.21, which is clearly equivalent to (3.1) for each (2,2)W algebra S with a left absorbing zero. Conversely, let y : S* Y satisfy (3.2). Then we extend W by one element, say - 4 W y y to the Y well ordered set W6' = W y U {-I such that w < - for all W E W YY and define 6': S W g I by 6 " s ) = y ( s ) f o r all S E S* and 6 ' ( 0 ) =-. Since 6 ' ( 0 ) > 6 ' ( s ) holds f o r all S E S*, (3.1) for y

-.

+

109

Euclidean seminear-rings and near-rings

implies ( 3 . 3 ) for 6 ' and all a = b = o one has a = o b .

a € S, b e S*.

In the remaining case

In the following, we use " l e f t E u c l i d e a n f u n c t i o n " (LEF) only with respect to Def. 3 . 2 . For a (2,2)-algebra S with a left absorbing zero, a mapping y : S* -t W according to Prop. 3 . 3 will be Y called a r e s t r i c t e d LEF henceforth. In this context we note that a similar proof as the above one provides PROPOSITION 3 . 4 . Let S be a (2,2)-algebra which may have a zero or not and let T = SU{o} be the (2,2)-algebra obtained from S by adjoining an absorbing zero o + S. Then S is left Euclidean iff the same holds for T. Various examples of distributive left Euclidean seminearrings (in fact those with associative addition) have been given in [ 8 1 . It is also clear that each (zero-symmetric) nearfield and each seminearfield S such that I S * I 2 2 (cf. [ 1 9 1 ) is left Euclidean. S o we present here in particular proper nearrings which are left Euclidean (Expls. 4 . 4 and 4 . 5 ) in the more general context of § 4 . Moreover, the first four statements of 5 5 can be used to obtain further examples of lzft Euclidean seminearrings and nearrings. So we close this section considering two special properties of left Euclidean functions needed later: DEFINITION 3 . 5 . Let S be a left Euclidean (2,2)-algebra and 6: S + W6 a LEF on S. Then 6 is called p s e u d o - i s o t o n e iff (3.4)

6(b) 6 6(qb)

holds for a l l

and a l t e r n a t i v e iff for all

q,be S,

a,b,q,,q2,rc

S

a = q1b excludes a = q2b+r for r E S* and 6(r) < 6(b). is associative, it is possible (and often convenient) If (S,.) to use only pseudo-isotone left Euclidean functions according to

(3.5)

PROPOSITION 3 . 6 . Let S be a multiplicatively associative left Euclidean (2,2)-algebra and 6 : S -+ W6 any LEF on S. Then g: S + W 6 defined by (3.6)

x(b) = min {6(x)

is also a LEF on

S

1

X E

SbU {bll

for all

beS

which is pseudo-isotone.

Proof. Note that 8(b) 2 6(b) holds for all b E S and that 'i; is pseudo-isotone since Sqb U {qb] G Sb U Cbl implies x(b) S x(qb). So it remains to show that 8 satisfies ( 3 . 3 ) and we assume %(a) b %(b). Then we have 6(a) 28(a) 2%(b) and either %(b) =6(sb) for some

L! Hebisch and H.J. Weinert

110

I n t h e f i r s t c a s e , w e apply ( 3 . 3 ) f o r 6 t o a

o r % ( b )= 6 ( b ) .

S E S

and s b and o b t a i n

a = q s b or

a = q s b + r and 6 ( r ) < 6 ( s b ) . Because of

% ( r )5 6 ( r ) < 6 ( s b ) = 8 ( b ) , t h i s i s ( 3 . 3 ) f o r a and b w i t h r e s p e c t

tog.

The second c a s e f o l l o w s i n t h e same way. REMARK 3 . 7 .

L e t S be a m u l t i p l i c a t i v e l y a s s o c i a t i v e l e f t E u c l i d e a n

( 2 , 2 ) - a l g e b r a w i t h a l e f t a b s o r b i n g z e r o . Then a r e s t r i c t e d LEF y on S i s c a l l e d p s e u d o - i s o t o n e (3.4’)

If

y:

y ( b ) 5 y(qb) S* + W

holds for a l l

q , b e S*

such t h a t

qb

E

S*.

i s any r e s t r i c t e d LEF on S , one o b t a i n s a pseudo-

Y

y:

i s o t o n e r e s t r i c t e d LEF (3.6’)

iff

y ( b ) = min{y(x)

I

S*

-*

W

on S s i m i l a r l y a s above by

Y

(SbU { b ) ) n S * }

X E

for a l l

b e S*.

Note t h a t ( 3 . 4 ’ ) i s o f t e n i n c l u d e d i n t h e d e f i n i t i o n of a ( l e f t ) Euclidean r i n g , i n p a r t i c u l a r f o r t h o s e without z e r o d i v i s o r s f o r which

i s s u p e r f l u o u s . By t h e above c o n s i d e r a t i o n s , t h i s

q b c S*

supplementary a s s u m p t i o n d o e s n o t r e s t r i c t t h e c o n c e p t of a ( l e f t ) Euclidean r i n g i t s e l f . F i n a l l y , e a c h p s e u d o - i s o t o n e LEF 6 on a n e a r r i n q S i s a l s o a l t e r native since

4.

6(r) < 6(b).

6 ( b ) 2 6 ( ( q -q ) b ) = 6 ( r ) c o n t r a d i c t s 1

2

@-DERIVATIONS OF (LEFT EUCLIDEAN) S E M I N E A R R I N G S

W e g e n e r a l i z e a w e l l known p r o c e d u r e f o r n e a r r i n g s a s f o l l o w s :

DEFINITION 4 . 1 .

be a s e m i n e a r r i n g . W e w r i t e

L e t (S,+,‘)

each endomorphism

n,I e E n d ( S , + ; )

endomorphism g i v e n by

a

-*

a

n1*‘2

of

and

(S,+;)

n1 = (a )“2.

(4.2)

@ ( S )G E n d ( S , , * )

aob = a Q P ( b ) . b f o r a 1

iff

a,bES.

p r o v i d e t h a t one d e f i n e s by a,bc S

called

again a seminearring (S,+,o)

L

(S,+,-)

Q ( a ) * @ ( b=) O ( a @ ( b ) . b ) h o l d s f o r a l l

The l a t t e r and

for

for the

04*n,, I

‘i

A mapping

0 : S + E n d ( S , + , - ) i s c a l l e d a c o u p l i n g map of

(4.1)

a+a

the

@ - d e r i v a t i o n of

(S,+,

.)

.

i s t h e 0-derivaOne a l s o checks t h a t e a c h s e m i n e a r r i n g ( S , + , o ) ) with t h e t r i v i a l m u l t i p l i c a t i o n

t i o n of a s e m i n e a r r i n g (S,+, a.b = a

for a l l

a,b

E

S,

g i v e n by

= aob.

a c o u p l i n g map of a s e m i n e a r r i n g (S,+;)

F u r t h e r , l e t 0 be

with a zero

0.

Then o @ ( ~ ) = o

h o l d s i f @ ( a )i s s u r j e c t i v e , and it h o l d s f o r a l l Q ( a ) i f l e f t o r r i g h t c a n c e l l a t i v e . Moreover, o @ ( ~ =o ) t h e equivalence

0-S =

{ol

w

o0S=

{oj.

for a l l

(S,+)

a

E

is

S* y i e l d s

111

Euclidean seminear-rings and near-rings

Now l e t ( S , + , o )

be a Q - d e r i v a t i o n of any s e m i n e a r r i n g ( S , + , * ) .

I f @ ( a )i s i n j e c t i v e f o r a l l a E S *

is multiplicatively

and (S,+,.)

r i g h t c a n c e l l a t i v e , t h e same h o l d s f o r ( S , + , o ) .

The i n v e r s e i m p l i -

c a t i o n i s t r u e i f @ ( a )i s s u r j e c t i v e f o r a l l @ ( a )be b i j e c t i v e f o r a l l

a E S*.

a E

S*.

Finally, let

is l e f t cancellative

Then ( S , + , O )

(The c o n v e r s e i m -

o r a s e m i n e a r f i e l d i f t h e same h o l d s f o r (S,+,.).

p l i c a t i o n s f a i l t o be t r u e by t h e f i r s t remark b e h i n d Def. 4 . 1 . ) The p r o o f s f o r t h e s e s t a t e m e n t s a r e more o r less s t r a i g h t f o r w a r d , b u t w e need Thm. 4 . 1

( f ) of

t o show t h a t

[191

(S,+,O)

is a s e m i -

n e a r f i e l d . W e o m i t t h e s e p r o o f s h e r e s i n c e o u r main i n t e r e s t i s with t h e following THEOREM 4 . 2 .

he a l e f t E u c l i d e a n s e m i n e a r r i n g .

L e t (S,+,-)

a ) R e g a r d l e s s whether t h e r e i s a z e r o , f o r e a c h c o u p l i n g map @ of

( S , + , * ) such t h a t @ ( b ) i s s u r j e c t i v e f o r a l l b e S,

the @-derivation

i s a l s o l e f t E u c l i d e a n , and each LEF 6 on ( S , + , - )

(S,+,o)

LEF on

is also a

( S , + , O ) .

b) I f there is a zero satisfying

t h e same s t a t e m e n t a s

S * o = {ol,

above h o l d s f o r e a c h c o u p l i n g map Q such t h a t @(b) i s s u r j e c t i v e f o r all

b

E

S*.

P r o o f . For a l l q,r

E

a,bE S

satisfying

S

& ( a ) 2 6 ( b ) , by ( 3 . 3 ) t h e r e a r e

such t h a t

a = q-b

or

6(r) < 6(b).

a = q - b + r and

I f Q(b)

i s s u r j e c t i v e , q = q ' Q ( b ) h o l d s f o r some q ' E S , and q . b = q t Q ( b ) - b q'ob = by ( 4 . 2 )

shows t h a t 6 i s a l s o a LEF on

. For

)

(S,+,O

b ) w e o n l y have

t o e x c l u d e b = o i n t h e s e c o n s i d e r a t i o n s , which i s t r i v i a l s i n c e q-o= o

h o l d s by a s s u m p t i o n and i m p l i e s

qoo =

0.

W e w i l l u s e b ) t o g i v e examples of l e f t E u c l i d e a n n e a r r i n g s a s

@ - d e r i v a t i o n s of l e f t E u c l i d e a n r i n g s ( S , + , - ) ,

s i n c e it i s conven-

i e n t t o u s e t h e z e r o endomorphism f o r @ ( o ) . W e f u r t h e r r e s t r i c t o u r s e l v e s t o one s i m p l e p r o c e d u r e a c c o r d i n g t o LEMMA 4 . 3 .

phism of fying

L e t (S,+,-) (S,+,')

be a r i n g w i t h o u t z e r o d i v i s o r s , ri an automor-

and n a homomorphism of

n(bn) = n ( b )

for all

bES*.

(S*,.)

into (INoy+) satis-

Then a c o u p l i n g map of

(S,+,-)

i s d e f i n e d by (4.3)

a'(b) = a'

n (b) if

b

E

S*

and

= o

for a l l

a

E

S.

The proof i s s t r a i g h t f o r w a r d and w e s h a l l see t h a t t h e r e a r e very natural realizations for those Example 4 . 4 .

Let

S

=K[xl

n

and n .

be a polynomial r i n g o v e r a skew f i e l d K .

Then K[x] i s l e f t (and r i g h t ) E u c l i d e a n ( c f . [ 1 5 ] ,

8 6 ) w i t h re-

s p e c t t o t h e r e s t r i c t e d LEF y g i v e n by t h e d e g r e e homomorphism

112

U. Hebisch and H.J. Weinert

n: ( S * , - ) -+ (may+). Each linear polynomial cx+d with coefficients by k n = k in the centre of K provides an automorphism I- of ( S , + , - ) i for all k E K and f (x)n = (Ekixi)' = Eki (cx+d) , and we clearly have n(bT7)=n(b) for all b E S * . So we can apply Lemma 4.3 and Thm. 4.2 b) and get for each automorphism q of this kind a coupling by (4.3) and a zero-symmetric left Euclidean nearmap Q of (S,+;) One easily checks that ( S , + , O ) is not left distriburing ( S , + , " ) . tive (provided that cx+d$x), but multiplicatively cancellative by the above remarks. We note that in a similar way S =K[xl over a commutative field K was used iri [51 to obtain examples for "Euclidean near domains" as considered in that paper (cf. $1). Example 4.5. Let (S,+,.) be a commutative Euclidean ring without zero divisors and with an automorphism I- which is not the identical mapping, e. g. the ring of Gaussian integers %+Xi. Each ring (S,+;) of this kind has an identity (cf. Thm. 5.7) and is hence a Gaussian ring in the meaning that each b e S * is either a unit or has an essentially unique factorization into prime elements. Denote by n(b) the number of the latter in this factorization, including n(b) = 0 if b is a unit. This clearly defines a homomorphism n of (S*,') into ( I N o , + ) such that n(bn) =n(b) holds f o r each automorphism TI of (S,+;). So q yields a coupling map 0 of ( S , + ; ) by (4.3). Hence each Euclidean ring as above provides for each (non-trivial) as a 0-derivation automorphism r- a left Euclidean nearring ( S , + , o ) is zero-symmetric and multiplicatively by Thm. 4.2 b). Again (S,+,O) cancellative, but in general not left distributive. For instance, if is the identical mapping as for S = X+%i, the left distributive law yields by p = po ( (p+l)-p) = po(p+I) + po (-p) the contradiction ' = p for each prime element p. p

TI^

5. SOME RESULTS ON LEFT EUCLIDEAN (2,2)-ALGEBRAS We first consider the question, whether the property "left Euclidean" transfers from given (2,2)-algebras to other ones. Clearly, subalgebras of a left Euclidean (2,2)-algebra S are in general not left Euclidean (otherwise each subring of a field would be Euclidean). As a contrast we have the following transfer without any further assumptions: THEOREM 5.1. Each homomorphic image T of a left Euclidean (2,2)algebra S is also left Euclidean. Proof. Let f: S - T be a homomorphism of ( S , + , . ) onto (T,+,*)and 6: S + W 6 a LEF on S . Then T : T + W 6 defined by

113

Euclidean seminear-rings and near-rings

(5.1)

- r ( f ( a ) ) = min{d(a')

I

f ( a ' ) = f ( a ) l for a l l

i s a LEF on T . To show t h i s we c o n s i d e r - r ( f ( a ) ) 2 ~ ( f ( b ) )Then . f o r s u i t a b l e elements and

f ( a ) , f ( b ) E T satisfying

. r ( f ( a ) )= 6 ( a ' ) and a',b'ES

a,a'ES

- r ( f ( b ) )= 6 ( b ' ) hold f ( a ' = f (a)

by ( 5 . 1 ) such t h a t

it f o l l o w s

f ( b ' ) = f ( b ) . From 6 ( a ' ) 2 6 ( b ' )

or

a' =qb

E S. T h i s y i e l d s i n T f ( a ) = f (9)f ( b ) + f (r), which t o g e t h e r w i t h

6 ( r ) < 6 ( b ' ) f o r come q , r

a ' = q b ' + r and f ( a ) = f ( 9 )f ( b )

or

- r ( f ( r )6) 6 ( r ) < 6 ( b ' ) = ~ ( f ( b ) proves ) that Next w e c o n s i d e r t h e d i r e c t product number of l e f t Euclidean

i s a LEF on

T

T = Sx

(2,2)-algebras

... x S '

S,...,S',

'1

.

of a f i n i t e also called

t h e i r d i r e c t sum. W e s h a l l s e e t h a t T i s a g a i n l e f t Euclidean i f one of i t s components, say S , s a t i s f i e s t h a t (5.2)

for a l l

q,bES

t h e r e i s some s

E

S

such t h a t

qb = s b + b ,

a n o t h e r one, say S ' , has a l e f t a b s o r b i n g z e r o , and t h e remaining components s a t i s f y both c o n d i t i o n s . Note t h a t ( 5 . 2 ) h o l d s e . g. f o r a nearring S with a l e f t i d e n t i t y e cf.

satisfies (6.11,

a l s o Thm. 5 . 7 ) .

by

s = q-eL

( o r one which

Since both,

(5.2)

and t h e

e x i s t e n c e of a l e f t a b s o r b i n g z e r o , t r a n s f e r from any two ( 2 , 2 ) a l g e b r a s t o t h e i r d i r e c t p r o d u c t , t h e above s t a t e m e n t i s a consequence of t h e f o l l o w i n g THEOREM 5 . 2 . S satisfies

L e t S and S ' be l e f t Euclidean

d i r e c t product Proof. L e t V=WxW'

i. e .

( 2 , 2 ) - a l g e b r a s such t h a t

( 5 . 2 ) and S ' h a s a l e f t a b s o r b i n g z e r o T = SXS'

6: S - t W

0 ' .

Then t h e i r

i s a l e f t Euclidean ( 2 , 2 ) - a l g e b r a .

be a LEF on S ,

6':

S'+W'

a LEF on S '

and

t h e w e l l o r d e r e d s e t o b t a i n e d by t h e l e x i c o g r a p h i c o r d e r , (u,u') < (w,w')

iff

u<w

i n W o r u = w and u ' < w '

in W'.

W e define

(5.3)

T:

SxS'

-+

V

by

T(s,s')= ( 6 ( s ) , 6 ' ( s ' ) ) .

To show t h a t T i s a LEF on

T = SxS',

assume

T (a,a') 2 T(b,b')

for

e l e m e n t s ( a , a ' ) , ( b , b ' ) E T . Then we have 6 ( a ) 2 6 ( b ) by ( 5 . 3 ) , hence a = q b o r a = q b + r and 6 ( r ) < 6 ( b ) f o r some q , r E S . I f 6 ' ( a ' ) > 6 ' ( b ' ) h o l d s , t h e r e are q ' , r ' E S ' s a t i s f y i n g a ' = q ' b ' + r ' and 6 ( r ' ) < 6 ( b ' ) i f r ' + o ' . But t h e same h o l d s f o r 6 ' ( a ' ) < 6 ' ( b ' ) a c c o r d i n g t o a ' = o ' b ' + a ' w i t h r ' = a ' . Now assume a = q b . Then we e i t h e r have

( a , a ' ) = (q,q') ( b , b ' ) o r , u s i n g

a = qb = s b + b by (5.2) ,

( a , a ' ) = (s,q')( b , b ' ) + ( b , r ' ) , where T ( b , r ' ) < T ( b , b ' ) f o l l o w s from 6 ' ( r ' ) < 6 ' ( b ' ) . For a = q b + r w e g e t ( a , a ' ) = ( q , q ' ) ( b , b ' ) + ( r , r ' ) , and 6 ( r ) < 6 ( b ) i m p l i e s T ( r , r ' ) < - t ( b , b ' ) .

114

(1.

Hebisch and H.J. Weinert

COROLLARY 5.3 Let S,...,S' be a finite number of left Euclidean nearrings such that all of them except at most one have a left identity. Then their direct product T = S X . . . x S ' is again a left Euclidean nearring.

This follows from Thm. 5 . 2 and the preceding remarks. We note that these statements for commutative Euclidean rings with identities (cf. Thm. 5.7) are due to [ 1 6 1 , but with a construction of a LEF on T = S x S ' which is more complicated than (5.3). To obtain other examples of left Euclidean algebras in this consatisfying text, note at first that each (2,2)-algebra (S,+,') (5.4)

S E Sb

for all

be S

is left Euclidean with any mapping 6 : S + W 6 as a LEF. There are plenty of those algebras, e. g . each constant seminearring (cf. § 7) or each seminearfield without a zero (cf. [ 1 9 ] ) , and each left sim(i. e. ( 5 . 4 ) ) provides such a seminearring if ple semigroup (S;) one defines an addition e. q. by a + b = a for all a,bE S. Now we state: PROPOSITION 5.4. Let S be any left Euclidean (2,2)-alqebra and S ' one which satisfies ( 5 . 4 ) and S ' G S ' + S ' . Then their direct product T = S X S ' is a left Euclidean (2,2)-algebra. Proof. Let 6 be a LEF on S. We show that -i(a,a')=6(a) defines a LEF T on T and assume T (a,a')2 .r(b,b'). Then we have a = q b or a = q b + r and 6(r) <6(b) in S. The first case is clear by ( 5 . 4 1 , and for the second one we only need that a' = q'b'+y' holds for a l l a',b'E S' with suitable q' , y ' S~' The latter clearly follows from a' = x'ty' E S ' + S ' and x' =q'b' by ( 5 . 4 ) .

.

In the second part of this section we use Prop. 3.6 to give sufficient conditions for the existence of (right) identities in left Euclidean (2,2)-algebras. We prepare them by the following LEMMA 5 . 5 . Let S be a multiplicatively associative left Euclidean (2,2)-algebra. Then one has (5.5)

M = {c E S

1

6(c) =min 6 ( S ) E W63 = {c E S

I

S G Sc}

f (8

for each pseudo-isotone LEF 6 : S+W6 of S and ( M y * )is a subsemigroup of ( S , . ) . Thus (My.) contains idempotents if M is finite. Proof. From a = q c for all a E S it follows 6(c) 6 6(a) by ( 3 . 4 ) . Conversely, each c E S such that 6(c) is minimal in 6 ( S ) satisfies a = q c for all a e S by (3.3) since 6(r) <6(c) is impossible.

Euclidean seminearrings and near-rings F i n a l l y , assume qiE S

c,,c2€M.

Then

c 2 = q 2 c 2 and q 2 = q l c l

115

f o r some

i m p l i e s c2 =q1c1c2 and hence 6 ( c1c 2 ) < 6 ( c 2 ) .

The p r o o f s f o r t h e f o l l o w i n g lemma and theorem are t h e same a s t h o s e f o r c o r r e s p o n d i n g s t a t e m e n t s on a d d i t i v e l y a s s o c i a t i v e l e f t E u c l i d e a n s e m i r i n g s g i v e n i n [ 8 ] (Lemma 5.1 and Thm. 5 . 3 ) . LEMMA 5.6.

L e t S be a m u l t i p l i c a t i v e l y a s s o c i a t i v e l e f t Euclidean

( 2 , 2 ) - a l g e b r a and M a s i n ( 5 . 5 ) . Then t h e f o l l o w i n g s t a t e m e n t s a r e e q u i v a l e n t and s a t i s f i e d i f M i s f i n i t e : has a r i g h t identity,

(i)

S

(ii)

M c o n t a i n s a n idempotent of

(S,.)

M c o n t a i n s a r i g h t c a n c e l l a b l e element of ( S , , ) . 2 Moreover, C E S i s a r i g h t i d e n t i t y of S i f f c = c E M h o l d s , and (iii)

cE

S

i s a n i d e n t i t y of S i f f c i s a c e n t r a l i d e m p o t e n t of

(S;)

c o n t a i n e d i n M. N o t e i n t h i s c o n t e x t t h a t , by t h e remarks p r e c e d i n g Prop. 5.4,

t h e r e a r e a l g e b r a s S of t h i s k i n d such t h a t e a c h e l e m e n t of S i s a r i g h t i d e n t i t y (e. g . any c o n s t a n t n e a r r i n g ) a s w e l l a s t h o s e w i t h o u t a l e f t or r i g h t i d e n t i t y . The l a t t e r h o l d s s i n c e t h e r e a r e l e f t s i m p l e semigroups ( S , - ) THEOREM 5.7.

w i t h o u t i d e m p o t e n t s ( c f . [ 3 ] , Chap. 8 ) .

L e t S be a m u l t i p l i c a t i v e l y a s s o c i a t i v e l e f t E u c l i d e a n

( 2 , 2 ) - a l g e b r a which i s m u l t i p l i c a t i v e l y c a n c e l l a t i v e ( c f . § 2) o r m u l t i p l i c a t i v e l y commutative. Then S h a s an i d e n t i t y e , and t h e group of u n i t s of (S,.)

c o i n c i d e s with M i n ( 5 . 5 ) .

6 . DIVISIBILITY I N LEFT EUCLIDEAN SEMINEARRINGS L e t S be any s e m i n e a r r i n g and a , b e S. W e say t h a t b i s a r i g h t d i v i s o r of a i f f a E S b h o l d s . For s i m p l i c i t y , w e d e n o t e t h i s by b l a i n s t e a d of b , 1 a . T h i s r e l a t i o n on S i s r e f l e x i v e i f f

(6.1)

SESS

holds for a l l

SES.

W e w i l l u s e ( 6 . 1 ) a s an a s s u m p t i o n i n some s t a t e m e n t s of t h i s sect i o n . I t i s weaker t h a n t h a t t o demand a l e f t i d e n t i t y ( c f . Thm. 5.71, s i n c e v a r i o u s t y p e s of semigroups ( S , . ) , e. g . i d e m p o t e n t o r r e g u l a r o r i n v e r s e o n e s ( c f . [31), s a t i s f y ( 6 . 1 ) . W e f u r t h e r c a l l d e S a g r e a t e s t common r i g h t d i v i s o r , a b b r e v i a t e d by g . c . r . d . ,

of a

s u b s e t A S S i f f d l a h o l d s f o r a l l a € A and t l a f o r a l l a € A i m p l i e s t l d . A main p o i n t f o r a ( l e f t ) E u c l i d e a n r i n g S i s t h a t t h e

( l e f t ) Euclidean algorithm

U.Hebisch and H.J. Weinert

116

r v E S*)

((6 a LEF on S and a l l

y i e l d s some r n E S a s a g r e a t e s t common ( r i g h t ) d i v i s o r of Now l e t

6: S+W6

a,bES.

be a LEF on a l e f t E u c l i d e a n s e m i n e a r r i n g S

which s a t i s f i e s ( 6 . 1 ) . Then (6.2) i s a p p l i c a b l e t o e a c h p a i r of elements a , b chain

E

6 ( a ) 1 6 ( b ),

S a s w e may assume

6(b) > 6(rl) > 6(r2)

:>.

..

a f t e r n s t e p s we o b t a i n a l a s t r b is a g . c . r . d . rnIrn-lY

E S*.

by

is finite,

( I f a = q l b holds, c l e a r l y

n of a and b.) Using ( 6 . 1 )

rnlrn-2

and s i n c e e a c h

i n a w e l l o r d e r e d s e t W6

f o r rn, w e have

r n - 2 = (qnqn-l

+X)rn""

which l e a d s t o r n l b and r n l a . Now w e assume t l a and t l b f o r some

t c S . Then w e g e t t l r l y t l r 2 , ing right divisor property (6.3)

xt=yt+z

3

Z E

St

...,t l r n

for a l l

i f S s a t i s f i e s t h e follow-

t , x , y E S,

Z E

S*.

S i n c e ( 6 . 3 ) c l e a r l y h o l d s f o r e a c h n e a r r i n g S, w e have shown the following THEOREM 6 . 1 .

L e t S be a l e f t E u c l i d e a n s e m i n e a r r i n g which s a t i s f i e s

( 6 . 1 ) and ( 6 . 3 ) , i n p a r t i c u l a r a l e f t E u c l i d e a n n e a r r i n g w i t h a l e f t i d e n t i t y . Then f o r each LEF 6 on S and a l l a , b E S such t h a t 6 ( a ) > 6 ( b ) , any performance of t h e l e f t E u c l i d e a n a l g o r i t h m ( 6 . 2 ) provides a n element

r n E

S which i s a 9 . c . r . d.

of a and b. Con-

s e q u e n t l y , e a c h p a i r of e l e m e n t s a , b E S and t h u s e a c h f i n i t e sub-

set A S S has a g . c . r . d .

in

s.

Note t h a t i n g e n e r a l t h e e l e m e n t s rv and q v i n ( 6 . 2 ) a s w e l l a s t h e number n of s t e p s a r e n o t u n i q u e l y d e t e r m i n e d , n o t e v e n i n t h e c a s e of r i n g s and f o r a s u i t a b l e c h o i c e of a LEF 6. W e f u r t h e r s t a t e with r e s p e c t t o t h e c o n d i t i o n ( 6 . 3 ) : LEMMA 6 . 2 .

L e t S be a n a d d i t i v e l y a s s o c i a t i v e l e f t E u c l i d e a n semi-

n e a r r i n g . Then S s a t i s f i e s t h e r i g h t d i v i s o r p r o p e r t y ( 6 . 3 ) i f f t h e r e i s a t l e a s t one a l t e r n a t i v e LEF 6 on S, and i n t h i s c a s e e a c h p s e u d o - i s o t o n e LEF on S is a l t e r n a t i v e . P r o o f . A s s u m e t h a t 6 i s an a l t e r n a t i v e LEF on S and x t = y t + z f o r some t , x , y E S and

ZE

S*.

Then 6 ( 2 ) < 6 ( t ) i s e x c l u d e d by (3.5).

Euclidean seminear-rings and nearrings

Hence

6 ( z ) 2 6 ( t ) implies

z = q t or

117

z = q t + r and

6(r)< 6 ( t ) for

some q e S and r e S*. Now w e u s e t h a t (S,+) i s a s s o c i a t i v e - f o r t o o b t a i n x t = y t + q t + r = ( y + q )t + r t h e only t i m e i n t h i s s e c t i o n

-

i n t h e second case. S i n c e t h i s i s a g a i n e x c l u d e d by (3.51, it remains

z = q t S~t and w e have proved ( 6 . 3 ) . For t h e c o n v e r s e w e show

t h a t e a c h p s e u d o - i s o t o n e LEF 6 on S i s a l t e r n a t i v e i f ( 6 . 3 ) h o l d s . BY way of c o n t r a d i c t i o n , assume a = q b 1

6(r)<6(b)

and a = q b + r , r e S* 2

and

f o r s u i t a b l e e l e m e n t s of S . Then ( 6 . 3 ) and ( 3 . 4 ) f o r 6

y i e l d r = xb E Sb and

6(b) I6(r).

be a s e m i n e a r r i n g and b E S . I t i s r e a s o n a b l e t o cons i d e r t h e s u b s e t A S S of a l l e l e m e n t s which have b a s r i g h t d i v i -

L e t (S,+,.)

sor, i . e . A = S b . (6.4)

A+A C A

Such a s u b s e t A o b v i o u s l y s a t i s f i e s

and

SA C- A .

I f S i s a r i n g , a subset A of S s a t i s f y i n g ( 6 . 4 )

is called a l e f t

i d e a l of S. The same n o t i o n i s common i f S i s a s e m i r i n g , and SAGA

i s a l s o t h e u s u a l d e f i n i t i o n of a l e f t i d e a l of a semigroup (S,.). This j u s t i f i e s the following A subset A o f a s e m i n e a r r i n g S is c a l l e d a l e f t i d e a l of S i f f A s a t i s f i e s ( 6 . 4 ) . I f such a l e f t i d e a l A i s g e n e r a -

D E F I N I T I O N 6.3.

t e d by one of i t s e l e m e n t s b e A ( i . e . i f A i s t h e s m a l l e s t s u b s e t o f S w i t h ( 6 . 4 ) which c o n t a i n s b ) , A i s c a l l e d a p r i n c i p a l l e f t

i d e a l of S . I n t h e f o l l o w i n g , w e u s e " l e f t i d e a l " always i n t h i s meaning, i n p a r t i c u l a r i f S happens t o be a n e a r r i n g . I n t h e l a t t e r case w e speak a b o u t a n e a r r i n g l e f t i d e a l A of S i f f A i s a normal subgroup of (S,+)

and s a t i s f i e s s ( t + a ) - s t E A

for a l l

s , t ~ Sand a e A.

Note

t h a t f o r a zero-symmetric

n e a r r i n g S each n e a r r i n g l e f t i d e a l i s a l so a l e f t i d e a l a c c o r d i n g t o D e f . 6 . 3 .

REMARK 6 . 4 .

I f S i s a s e m i n e a r r i n g which s a t i s f i e s ( 6 . 1 ) , t h e p r i n -

c i p a l l e f t i d e a l s of S are j u s t g i v e n by A = S a f o r a l l a E S , one h a s Sb ZSa i f f b l a f o r a l l a , b E S .

and

The n e a r by hand c o n c e p t of a l e f t p r i n c i p a l i d e a l s e m i n e a r r i n g would be a s e m i n e a r r i n g S f o r which a l l l e f t i d e a l s a r e p r i n c i p a l o n e s . But s u c h a c o n c e p t would be much t o s t r o n g and i n p r a c t i c a b l e . DEFINITION 6 . 5 . L e t S be a s e m i n e a r r i n g . A l e f t i d e a l A of S i s called l e f t k*-closed (6.5)

a, +x = a 2

iff

f o r some ai

E

A , x E S*

implies x

E

A,

and S i s c a l l e d a l e f t p r i n c i p a l i d e a l s e m i n e a r r i n g i f f e a c h l e f t

LI. Hebisch and H J.

118

Weinert

k * - c l o s e d l e f t i d e a l A of S i s a p r i n c i p a l l e f t i d e a l of S . REMARK 6 . 6 .

L e t S be a s e m i n e a r r i n g s a t i s f y i n g ( 6 . 1 ) .

Then, o b v i -

o u s l y , a l l p r i n c i p a l l e f t i d e a l s A = S a of S are l e f t k * - c l o s e d i f f S s a t i s f i e s the r i g h t divisor property (6.3). THEOREM 6 . 7 . L e t S be a l e f t E u c l i d e a n s e m i n e a r r i n g . Then S i s a

l e f t p r i n c i p a l i d e a l s e m i n e a r r i n g , and e a c h l e f t k * - c l o s e d l e f t i d e a l A o f S s a t i s f i e s A = S b f o r some b E A . Proof. L e t A be any l e f t i d e a l of S which i s l e f t k * - c l o s e d , s a y 6 ( b ) f o r some b E A . hence

a = q b or

Since r

and

a LEF on S . The s u b s e t 6 ( A ) of W6 h a s a minimal e l e m e n t ,

6: S+W6

E

Then one h a s

a = q b + r and

h o l d s by ( 6 . 5 )

A

,

6 ( a ) 2 6 ( b ) f o r each a E A ,

6 ( r ) < 6 ( b ) f o r some q E S , r E S * . t h e l a t t e r c a s e i s impossible. This

y i e l d s AE SbG A. THEOREM 6.8.

L e t S be a l e f t p r i n c i p a l i d e a l s e m i n e a r r i n g s a t i s -

f y i n g ( 6 . 1 ) and ( 6 . 3 ) . Then e a c h p r o p e r c h a i n

A1cA2cA

c

...

3 l e f t k * - c l o s e d l e f t i d e a l s of S i s f i n i t e . Hence S s a t i s f i e s t h e

c h a i n c o n d i t i o n f o r r i g h t d i v i s o r s , i. e . e a c h c h a i n i n S such t h a t

a v / a v - , and a v - , f a v

P r o o f . Due t o Remarks 6 . 4 and 6 . 6 , by At, = S a v and

avlav-,

*

hold

of

atYa2,a3,...

is finite.

both statements a r e e q u i v a l e n t

Sa D S ~ ~ -Now ~ .we assume t h a t t h e r e V-

a r e i n f i n i t e c h a i n s o f t h i s k i n d . Then t h e union

A = U{AV

I

v

E

IN)

i s c l e a r l y a g a i n a l e f t k*-closed l e f t i d e a l of S . Hence A = Sa h o l d s f o r some a E A , This yields tradiction

which i s c o n t a i n e d i n one of t h e A v ,

a = x a n and a n + l = y a f o r some

X,YE

S,

s a y An.

hence t h e con-

anla,+,.

A s a supplement t o o u r s t a t e m e n t on n e a r r i n g s i n Thm. 6 . 1

w e ob-

t a i n from t h e l a s t b o t h theorems: COROLLARY 6 . 9 .

(6.11,

L e t S be a l e f t E u c l i d e a n n e a r r i n g which s a t i s f i e s

e . g. one w i t h a l e f t i d e n t i t y . Then e a c h p r o p e r c h a i n of

r i g h t d i v i s o r s i n S a s d e s c r i b e d above i s f i n i t e .

7 . CONSTANT AND ZERO-SYMMETRIC SUBSEMINEARRINGS OF LEFT EUCLIDEAN SEMINEARRINGS AND NEARRINGS I n t h i s s e c t i o n we only consider seminearrings IS,+,*)

such t h a t

(S,+) i s a s s o c i a t i v e and h a s a z e r o . LEMMA 7.1.

L e t S be a n a d d i t i v e l y a s s o c i a t i v e l e f t E u c l i d e a n s e m i -

n e a r r i n g with a zero

0.

Then o2 = o h o l d s and f o r e a c h pseudo-

i s o t o n e LEF on S w e have t h e e q u i v a l e n c e

I19

Euclidean seminearrings and nearrings (7.1)

6(b) 6 6(0)

Proof. I f

ob=o

Q

6(b) 2 6(0)

for a l l bES.

h o l d s , by ( 3 . 3 ) t h e r e a r e q , r E S s a t i s f y i n g

o = q b + r , where r may be t h e z e r o o r n o t . T h i s y i e l d s o b = o and o 2 = 0 by 0 = ( o + q ) b+ r = ob + gb t r = ob + o = ob. The o t h e r i m -

thus

(7.1)

p l i c a t i o n of

i s immediate by ( 3 . 4 ) .

The f o l l o w i n g theorem a p p l i e s i n p a r t i c u l a r t o each l e f t E u c l i dean n e a r r i n g S w i t h a l e f t i d e n t i t y and i t s c o n s t a n t and zerosymmetric s u b n e a r r i n g s S1 and S 2 . W e u s e t h e g e n e r a l i z a t i o n of t h e l a t t e r c o n c e p t s f o r a s e m i n e a r r i n g S a s i n t r o d u c e d i n 5 4 of

[ZOI.

THEOREM 7.2.

L e t S be an a d d i t i v e l y a s s o c i a t i v e l e f t Euclidean seminearring w i t h a l e f t a b s o r b i n g z e r o 0 , S1 i t s c o n s t a n t and S2

i t s zero-symmetric subseminearring. A s s u m e t h a t S2 ( o r (6.1),

and t h a t

S = S 1 + S 2 h o l d s o r t h a t (S1,+)

S)

satisfies

does n o t c o n t a i n a

b i c y c l i c subsemigroup ( c f . [31 , 1.12). Then b o t h , S1 and S 2 , a r e l e f t Euclidean w i t h r e s p e c t t o t h e r e s t r i c t i o n s of any pseudoi s o t o n e LEF 6 on S . P r o o f . Each pseudo-isotone (7.2)

6(s)S 6(0) =6(t1)

LEF 6 on S s a t i s f i e s

for a l l

S E

S and

due t o ( 7 . 1 ) and t l o = t l y which y i e l d s each c o n s t a n t s e m i n e a r r i n g S., s t a t e d a s a consequence of

tl

E

Sly Moreover,

6(0)2 C ( t l ) .

i s always l e f t Euclidean a s a l r e a d y

(5.4).

By assumption, w e have

a 2 c S2a2

o r a 2 E S a 2 , say a 2 = x a 2 , f o r each a 2 E S 2 . Then 6 ( a 2 ) 2 6 ( s l + a 2 + t l ) h o l d s f o r a l l a 2 E S2 and

(7.3)

because of

sl,t,

( s l + x + t l ) a 2= s l a 2 + xa2 + t l a 2 = s , + a 2 + t l .

E

S1

To show

t h a t S 2 i s l e f t Euclidean, assume Due t o

6 ( a ) 2 6 ( b 2 ) f o r some a 2 , b 2 E S2. 2 S = S 1 + S 2 , by ( 3 . 3 ) f o r 6 and S t h e r e a r e q , + q 2 and r + r

1

which s a t i s f y

a2 = (q1+q2)b2 o r

6 ( r 2 ) 5 6 ( r l + r 2<) 6 ( b 2 ),

w e multiply

a 2 = ( q 1 + q 2 ) b 2+ r , + r 2

2

and

t h e l a t t e r by ( 7 . 3 ) . I n t h e f i r s t c a s e ,

a 2 = q 1 b 2+ q 2 b 2 from t h e r i g h t by o and g e t q1 = o ,

I n t h e second c a s e we have a 2 = q 1 b 2 + q 2 b 2 + r 1 + r 2 . q 2 + r 1 = s1+s2 f o r some si E Siy which W e apply t h i s t o y i e l d s m u l t i p l i e d by o from t h e r i g h t r l = s l . q 2 b 2 +r1 = q2 b 2 + r 1b 2 and o b t a i n q2b2 + r , = r l + s 2 b 2 . A s above, from a 2 = q b 2 + r , + s b + r 2 it f o l l o w s q b + r l = q l + r l = o . So 2 2 1 2 i t remains a 2 = s2 b 2 + r 2 a s we were t o show. hence

q1+q2 E S2.

Again by

S = S1+S2 we have

Without t h e assumption

S = S , + S 2 , we merely have

a2=qb2 or

= q b + r and 6 ( r ) < 6 ( b 2 ) f o r s u i t a b l e q , r E S . The f i r s t c a s e 2 2 y i e l d s o = a - o = q b o = q o and hence q E S2. I n t h e second one w e L 2

a

U,Hebisch and H.J. Weinert

120

obtain

q o + r o = o . T h i s y i e l d s e i t h e r r o + q o = o , or qo and r o

g e n e r a t e a b i c y c l i c subsemigroup (which c o n t a i n s an i n f i n i t e set of i d e m p o t e n t s ) of

(S,+)

a c c o r d i n g t o [ 3 1 , Lemma 1.31.

If t h e latter

i s e x c l u d e d by a s s u m p t i o n , w e go on w i t h a 2 = q b + r o + q o +r = ( q + r o ) b 2+ q o + r . 2

From ( q + r o ) o= o i t f o l l o w s q + r o E S2,

So i t

and l i k e w i s e q o + r E S2.

6 ( q o + r )< 6 ( b 2 ) , which f o l l o w s a g a i n by ( 7 . 3 ) from

r e m a i n s t o prove

6 ( q o + r ) 6 6 ( r o + q o + r )= 6 ( r ) and 6 ( r ) < 6 ( b 2 ) .

THEOREM 7.3. L e t S be a n e a r r i n g w i t h a n i d e n t i t y , S 1 i t s c o n s t a n t s u b n e a r r i n g . Then S i s l e f t E u c l i d e a n i f f

and S2 i t s zero-symmetric S2

i s and

(7.4)

S satisfies that

for a l l

b2,q2E S2

such t h a t

and bl E S1

t h e r e e x i s t s an x

E S

2

x (b +b2) = x b + q b 2 1 2 1 2 2 ’

(7.4) h o l d s i f S1 i s a n e a r r i n g i d e a l of

In particular,

2

S.

P r o o f . A s s u m e t h a t S i s l e f t E u c l i d e a n . S i n c e S h a s an i d e n t i t y e ,

w e c a n a p p l y Thm. 1 . 2 and o b t a i n t h a t

S2 i s l e f t E u c l i d e a n . F u r t h e r ,

6(a) = 6 ( s l + a + t l ) holds f o r a l l

(7. j )

a E S and

which f o l l o w s from a E S a a s ( 7 . 3 ) s i n c e ( S , + )

s l , t 1E S 1 ,

i s a group. I t i s

w e l l known t h a t e = e 2 b e l o n g s t o S 2 ’ and 6 ( e 2 ) i s minimal i n 6 ( S ) f o r e a c h p s e u d o - i s o t o n e LEF 6 on S due t o Lemma 5 . 6 . The same h o l d s f o r 6 ( b l + e 2 ) by ( 7 . 5 ) . So f o r each a l + a 2 E S = S1+S2 t h e r e i s some x +x E S s a t i s f y i n g a l + a 2 = ( x 1 + x 2 )( b l + e 2 ) = x1 + x ( b + e 2 ) . S i n c e 1 2 2 1 w e have x 2 ( b l + e 2 )= s 1 + s2 E S1+ S 2 w e o b t a i n s1 = x 2 b l and s 2 = a 2 , i n particular for a2 = q 2 x 2 ( b l + e 2 )= x 2 b 1 + q 2 , hence x 2 ( b l + b 2 ) = x ( b +e ) b = x b + q b 2 1 2 2 2 1 22’ T h i s shows (7.4). C o n v e r s e l y , l e t

6: S2+W6

be any LEF on S 2 . W e

T : S-r W6 by ? ( a , + a 2 ) = 6 ( a 2 ) f o r a l l a + a 2E S l + S 2 = S , 1 . r ( a + a ) 2 r ( b l + b 2 ) = 6 ( b 2 ) . Then t h e r e are q 2 , r 2 E S 1 2 2 s a t i s f y i n g a 2 = q 2 b 2 + r 2 and 6 ( r 2 ) < 6 ( b 2 ) i f r 2 + 0. W e choose X ~ SE

extend 6 t o and assume

a c c o r d i n g t o (7.4) and

x1 E S1 such t h a t a l = x1 + x2bl h o l d s . Then

( x , + x 2 )( b l + b 2 ) +r 2 ’ = x1 + x 2 b l + q 2 b 2 + r 2 = a, + a 2 and

we obtain

? ( r2 ) = 6 ( r 2 ) < 6 ( b 2 ) = ? ( b l + b 2 ) i f

r2$o.

This proves t h a t T i s a

LEF on S . Now l e t S1 be a n e a r r i n g ( l e f t ) i d e a l of S and b 2 , q 2 ’ b , q2 ( b l + b 2 )- q 2 b 2 E S1 y i e l d s q 2 ( b1 + b2 ) = s 1 + 9 2 b 2 M u l t i p l y i n g by o from t h e r i g h t w e g e t q 2 b l = s , . So w e even have shown q 2 ( b l + b 2 )= q 2 b 1 + q 2 b 2 , which c l e a r l y i s more

a s i n (7.4). Then f o r some

s1

t h a n (7.4).

E

S.

121

Euclidean seminearrings and near-rings

REFERENCES [ l ] Albis-Gonzdles, V. S . and R. K. Markanda, Rings of fractions

-

360. of euclidean rings, Corn. Algebra 6 (1978 , 353 H. H., Left Euclidean Rings, Paci ic J. Math. 45 27 - 33. Clifford, A. H. and G. B. Preston, The algebraic theory of semigroups, Amer. Math. SOC. Vol. I (1961), Vol. I1 (1967). Graves, J. A. and J. J. Malone, Embedding near domains, Bull. Austral. Math. SOC. 9 (1973), 33 - 42. Graves, J. A. and J. J. Malone, Euclidean near domains, Manuscript 1975. Hebisch,U., (2,2)-Algebren mit euklidischen Algorithmen, Diss. TU Clausthal, Germany 1984. Hebisch,U., Left euclidean functions on (2,2)-algebras, to appear. Hebisch,U. and H. J. Weinert, On Euclidean semirings, to appear. Jacobson, N., A note on non-commutative polynomials, Ann. of Math. 35 (19341, 209 - 210. Jategaonkar, A. V., Left Principal Ideal Rings, Lecture Notes in Mathematics 123, Springer, Berlin-Heidelberg-New York 1970. Motzkin, Th., The Euclidean algorithm, Bull. Amer. Math. SOC. 1146. 55 (19491, 1142 Ore, O., Theory of non commutative polynomials, Ann. of Math. 34 (1933), 480 - 508. Pilz, G., Nearrings, North-Holland Publ. Comp. 1977. Poll6k, G., Mengentheoretische Betrachhng der euklidischen und Hauptidealringe, Magyar Tud. Akad. Mat. Kutat6 Int. K o z l . 7 (1962), 323 333. RBdei, L., Algebra I, Akademiai Kiadb, Budapest 1954, Leipzig 1959, London 1967. Samuel, P., About Euclidean Rings, J. Algebra 19 (1971), 282 - 301. Wedderburn, J. H. M., Non-commutative domains of integrity, J. Math. 167 (1932), 129 141. Weinert, H. J., Multiplicative cancellativity of semirings and semigroups, Acta Math. Acad. Sci. Hungar. 35 ( 1 9 8 0 ) , 335 338. Weinert, H. J., Seminearrings, seminearfields and their semigrouptheoretical background, Semiqroup Forum 24 (1982), 254. 231 [20] Weinert, H. J., Partially and fully ordered seminearrings and nearrings, this proceedings.

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Near-rings and Near-fields, G. Betsch (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1987

123

IDEALS AND REACHABILITY I N MACHINES Gerhard HOFER I n s t i t u t f u r Mathemati k, Johannes-Kepler-Universitat L i n z A-4040 L i n z , A u s t r i a ABSTRACT I t i s well-known t h a t semigroups a r e v e r y u s e f u l i n t h e study o f automata. These semigroups c o n s i s t o f mappings f r o m t h e s t a t e s e t Q o f t h e automaton i n t o i t s e l f . I f , as i t i s o f t e n t h e case, Q bears t h e s t r u c t u r e o f a module, we a r e s t u d y i n g mappings o f a module i n t o i t s e l f . The n a t u r a l a l g e b r a i c t o o l f o r t h a t i s t h e s t r u c t u r e o f a n e a r - r i n g , c a l l e d t h e s y n t a c t i c n e a r - r i n g o f t h e automaton. I n t h i s paper we t r y t o show how i d e a l s o f t h e zerosymmetric p a r t o f s y n t a c t i c n e a r - r i n g s can be used f o r d e t e r m i n i n g r e a c h a b i l i t y i n automata.

We s t a r t w i t h t h e d e f i n i t i o n o f (semi-) automata. See e.g.

[21, [71 o r I101

f o r t h e t h e o r y o f these c r e a t u r e s . D e f i n i t i o n 1. A 4emiaLLtomaton i s a t r i p l e A = (Q,A,F) where Q and A a r e s e t s ( c a l l e d t h e 4.tate and inpu* A & )

and F i s a f u n c t i o n f r o m Q

x

A into Q (called

t h e n&~Xe-,t~~~m,LCion 6unction). I f Q and A a r e R-modules ( w i t h t h e same R) and F i s an R-homomorphism we c a l l A a module-semiautomaton and a b b r e v i a t e t h i s by

MSA and A = (Q,A,FlR. For any semiautomaton A=(Q,A,F)

we g e t a c o l l e c t i o n o f mappings Fa:Q

-t

Q,

one f o r each a E A, which a r e g i v e n by

I f t h e i n p u t a,EA

F a ( q ) := F(q,a) i s f o l l o w e d by t h e i n p u t a2, t h e semiautomaton "moves" f r o m

t h e s t a t e q € Q f i r s t i n t o Fa ( 9 ) and then i n t o Fa (Fa ( q ) ) . I f we extend (as 1 2 1 u s u a l ) A t o t h e f r e e monoid A* o v e r A ( c o n s i s t i n g o f a l l f i n i t e sequences o f elements o f A, i n c l u d i n g t h e empty sequence A ) , we t h e r e f o r e g e t

moreover M(A) := {F;la€A*} i d e n t i t y FA = i d

Q

= F F Fala2 a2 al ( w i t h composition o f maps) i s a monoid w i t h

( c a l l e d t h e oyntaotic mofloid o f A ) . I n t h e case o f MSA's we

a r e a l s o a b l e t o study t h e s u p e r p o s i t i o n Fa

t Fa2 ( d e f i n e d p o i n t w i s e l y ) o f two 1 "simultaneous" i n p u t s a, ,a2 E A. Hence i t i s n a t u r a l t o c o n s i d e r { F a l a E A} u {FA}

and a l l i t s sums and products ( = composition o f maps). The obvious framework f o r t h a t i s t h e s t r u c t u r e o f a near-ring:

124

G. Hafer

D e f i n i t i o n 2. A ( r i g h t ) n u - h i n g i s a t r i p l e (N,+,.), group ( n o t n e c e s s a r i l y a b e l i a n ) , (N,.)

where (N,+)

is a

i s a semigroup and (a+b)c = a c + bc f o r

a l l a,b,cEN. Prominent examples o f n e a r - r i n g s a r e (w.r.t.

p o i n t w i s e a d d i t i o n and

composition o f mappings) t h e f o l l o w i n g Examples. L e t G be an a d d i t i v e group w i t h i d e n t i t y 0 and A an R-module. Near-rings a r e e.g. M(G) := (f:G

-t

GI

Mo(G) := I f EM(G) l f ( 0 ) = 0) Mc(G) := I f E M(G) I f c o n s t a n t } Maff(A)

:= {f:A

--t

A l f a f f i n e map)

Thus n e a r - r i n g s a r e g e n e r a l i z e d r i n g s and most s u i t a b l e t o s t u d y mappings on groups. See 1121 f o r t h e t h e o r y o f n e a r - r i n g s . Since n e a r - r i n g s a r e &groups, t h e r e i s no need f o r d e f i n i n g homomorphisms, i d e a l s , etc.. We need a few more concepts f o r n e a r - r i n g s . L e t N be a n e a r - r i n g . n E N i s c a l l e d z m o - o y m t n ~ c i f nO = 0, com.tant i f nO = n and diba?~ibLLtiue i f n ( a + b ) = n a + n b f o r a l l a,bEN.

L e t No, Nc, Nd denote t h e s e t s o f a l l zero-

symmetric (constant, d i s t r i b u t i v e , r e s p e c t i v e l y ) elements. Note t h a t NdzN0. I t i s well-known t h a t N o U Nc generates ( N , + ) . N i s c a l l e d abn,txm.t

I f (N,+)

addine nem-hing (abbr.:

i s a b e l i a n and No = Nd then

a.a.n.h.).

Maff(A) i s o f t h i s

type. The l i n e a r case. L e t A = (Q,A,FlR be a MSA. Because o f Fa(q) = F(q,a) = F((q,O) + (0,a)) = F(q,O) + F(O,a) = F o ( s ) + F a ( 0 ) we g e t F a = F o + F a , where Fo i s an R-homomorphism. w h i l e Fa i s t h e map w i t h constant v a l u e F,(O).

By i n d u c t i o n one sees t h a t n = F: + z F : - ~ ( F ~ (01) (see [51 o r 161) Fala2. ..an i=l i where t h e sum denotes a c o n s t a n t map. Each power F: i s an R-homomorphism, 0

hence M(A) i s a subset of Maff(Q) and t h e n e x t d e f i n i t i o n comes i n a n a t u r a l way (see [ 5 1 o r [ 6 1 ) . D e f i n i t i o n 3. L e t A = (Q,A,FIR be a MSA. The subnear-ring N(A) o f Maff(Q) generated by M(A) i s c a l l e d t h e oyntactic nem-hing o f A. Thus N(A) i s an a.a.n.r.

with identity. I n the following,
group generated by X. The zerosymmetric p a r t , which we need i n t h e f o l l o w i n g ,

125

Ideals and reachability in machines i s given by No(A) := (“A)),

1

= = { z o i dQ + z l Fo + . . . +

I

z n Fn o

~ ~ € i Z , n € N ~ l

hence No(A) i s a commutative r i n g w i t h i d e n t i t y i d , which i s generated by (see [51).

{id,Fol

We motivate t h e next d e f i n i t i o n .

2

Example. L e t G = (9. .) E (Z,) 1J

Z,),

where a t l e a s t one gij,

( = the collection o f a l l 2

x

2-matrices over

( i # j ) i s i n v e r t i b l e . By Cayley-Hamilton and our

assumption on G i t f o l l o w s t h a t <E,G> = {zoE+ zlG/zo,zl E Z n l has c a r d i n a l i t y 2 n (E i s the i d e n t i t y m a t r i x ) . L e t A=(Q,A,FIR be a MSA where R i s commutative and Q a f i n i t e dimensional u n i t a r y f r e e R-module, then Fo corresponds t o a (see [l],191 o r [131), thus No(A) =<E,A

>. Since t h e r e seems t o be FO no hope t o g e t e x p l i c i t e r e s u l t s about r e a c h a b i l i t y i n general, we r e s t r i c t o u r

matrix A

FO

considerations t o t h e f o l l o w i n g special s i t u a t i o n s . D e f i n i t i o n 4. L e t R be a r i n g w i t h i d e n t i t y 1 , char R = n E where rL= a1 + Qrf o r some a,aE

hyntn&-hing n..&.)

(a,B;n)

(abbr.:

n.h.1

IN. I f R = <1 ,r> Z,o r d r = n and < l > n = {O} then R i s c a l l e d a

w i th genehatoh r and d y n t u c t i c h..ipLc (abbr.:

€ Z X Z x IN. We abbreviate t h i s s i t u a t i o n by R(a4;n).

I n the f o l l o w i n g , we w r i t e z instead o f z l ( f o r z E

Z 1.

Remark 1. L e t R be a s y n t a c t i c r i n g w i t h generator r. I zo,zl E H ) has c a r d i n a l i t y n2 and R i s commutative. The map

R = { z o + zlr zo+zlr

-P

i s a group isomorphism between (R,+) and (Z,xZ,,+).

(zo,zl)

I n the following, we study t h e s t r u c t u r e o f s y n t a c t i c r i n g s and t r y t o apply t h e r e s u l t s i n o r d e r t o describe “ r e a c h a b i l i t y “ i n machines. Theorem 1. L e t R(a,B;p)

be a s y n t a c t i c r i n g w i t h generator r and p prime. 2

Consider x 2 + a x - a = O , y = 4 u + ~as equations over then the f o l l o w i n g hold:

Z

P

(abbr. by (*),(+)),

a ) i s never an i d e a l o f R; i s an i d e a l i f f z i s a s o l u t i o n o f ( * ) . b ) (R,+) has p t l n o n t r i v i a l subgroups, namely <1> and f o r z €

a,...,p-

11.

c ) ( * ) has 2 no s o l u t i o n i f f R i s isomorphic t o GF(p ) one s o l u t i o n z i f f i s t h e unique n o n t r i v i a l i d e a l i n R two s o l u t i o n s z1,z2

i f f ,

a r e t h e o n l y n o n t r i v i a l i d e a l s i n

R.

G. Hofer

126

d ) I f p 2 3 then we get: ( t ) has no s o l u t i o n i f f R i s a f i e l d , isomorphic t o GF(p 2 ) one s o l u t i o n , namely 0, i f f <(?

8 ) t r > i s t h e unique n o n t r i v i a l i d e a l i n R

( p i ) t) r>, <(?

two s o l u t i o n s ig i f f <(?

( a - i ) ) t r> are

the o n l y non-

t r i v i a l i d e a l s i n R.

-

e ) R has a t most two n o n t r i v i a l ideals. Ph006.

a ) i s an i d e a l i n R

va,b E Z :

(bztbf,ta)rE o Va,b E Z : 2 zatba = zb(z+a) + za o a = z ( z t B ) *) z t b z - a = 0 <1> i s n o t i n i d e a l i n R since r.1 = r f <1>. ( a + b r ) ( z t r ) = (zatba)

t

i s isomorphic t o (Z x Z P , t ) /Remark l ) , hence a 2-dimensional vector P This proves b ) . space over Z P' c ) and e ) are t r i v i a l consequences o f a )

b) (R,t)

d ) f o l l o w s by c ) and elementary transformations o f ( * ) (see [ 3 1 and [41). Remark 2 . L a t e r we w i l l see t h a t , w i t h t h e n o t a t i o n o f Theorem 1 , , zEZ,

are t h e " e s s e n t i a l " subgroups o f R f o r the a p p l i c a t i o n s . Hence t h e

f o l l o w i n g d e f i n i t i o n s a r e explicable. D e f i n i t i o n 5. L e t p prime. < ( z , l ) > f o r each i n t e g e r ~ E { l , . . . ~ p l , and t h e t r i v i a l subgroups o f

Z

P X ZP

a r e c a l l e d impohturd 6ubgtaup6.

For a g e n e r a l i z a t i o n we need t h e f o l l o w i n g Notation. l e t R(a,B;n)

be a s y n t a c t i c r i n g , where n = n pi i s squarefree. iEA

Then Tl(a,B;n)

:=

T2(a,B;n)

:=

t i E A I x2+bx-a = 0 has no s o l u t i o n over Z

I t i E A I x2+Dx-a - 0 Ii€ A

1,

Pi x2+gx-a = 0 has e x a c t l y one s o l u t i o n zi over

Z I,

Pi over Z 1 , Pi We abbreviate, if no confusions can a r i s e , Tl(a,@;n) by T 1 , etc.. h,:(zo+zlr) + (zo,zl) denotes t h e group isomorphism between R and (Z,) 2

T3(a,D;n)

:=

h2:(zo,z1)

0

icA

-.

((zoyzl~,...,(zo,zl))

the group isomorphism between ( Z n ) 2 and

(Zp. ) 2 (Chinese-Remainder-Theorem). 1

D e f i n i t i o n 6. Let R(a,b;n) f r e e and U 6 (R,+).

@ Hi iEA

has two s o l u t i o n s zli ,z2i

:= h2(h,(U))

be a s y n t a c t i c r i n g , where n =

U i s c a l l e d a we6u.t subghaup i f a l l Hi's ( 2 @ (Zpi iEA

12) a r e important subgroups.

Now we have t h e i n g r e d i e n t s f o r t h e next Theorem.

n

iEA of

pi i s square-

127

Ideals and reachability in machines

Theorem 2. L e t R(a,B;n)

be a s y n t a c t i c r i n g w i t h g e n e r a t o r r, where n =

i s squarefree and UI (R,+).

The t h e f o l l o w i n g h o l d :

n

iEA

pi

a ) U i s c y c l i c i f and o n l y i f I U [ d i v i d e s n. b ) A l l u s e f u l subgroups o f R a r e g i v e n by < n pi> i < n p . ( z t r ) > where iET jcs J n p13. ScTcA, ~ € 7 1 , lET\S

...

Phood. hl(U)

@ Hi,where i€A U and h,(U) a r e c y c l i c i f f each Hi i s c y c l i c . Hi

i s a subgroup o f ( Z n ) 2 and isomorphic (by h 2 ) t o

.

2 i s a subgroup o f ( Z ) Hi Pi i s c y c l i c iffIHiI 6 p i' hence a ) i s proved. With t h e a b b r e v i a t i o n s t := inET pi, s := n p. f o r T c A and SET, we g e t f o r z E Z :

jcs

J

hl(+ <s(z+r)>} = <(t,o)>+

<s(z,l)>

=:

H.

By t h e l a s t expression one can e a s i l y see t h a t t h e sum i s d i r e c t .

@ Hi t h e n

I f h2(H) =:

i€A Hk : = <(o,O)>

i <(O,O)>

for k€S,

H1 : = <(O,O)> :<s(z,l)> = < ( z , l ) > f o r 1 E T S - = <(t,O)> :< s ( z , l ) > = <(1,0)> i <(0,1)> = Hm * Hence < s ( z + r ) > i s a u s e f u l subgroup.

...,a1

Ifz runs through a l l i n t e g e r s o f {I,

Z XZ Pr

f o r m$T.

Pm

where a :=

n

p,

1ET\S

, then,

1 E T\S, t h e H1's r u n through a l l n o n t r i v i a l i m p o r t a n t subgroups o f ( Z reason i s t h e isomorphism ( z , l )

-,( ( 2 ,...,z),(l,...,l))

between ( Z a )

for )

2

. The

2p1and

@ Z ) 2 (Chinese-Remainder-Theorem). Hence we g e t a l l u s e f u l subgroups o f 1€T\S p1 R i f T runs through a l l subsets o f A , S through a l l subsets o f T and z through (

a l l i n t e g e r s o f 11,

...,

n

p,}.

1€T\S

C o r o l l a r i e s . L e t R(a,B;n)

n

be a s y n t a c t i c r i n g w i t h generator r, where

@ Hi : = h 2 ( h , ( U ) ) i€A iEA 2 Hence, by Theorem 1 and b ) o f Theorem 2, a l l i d e a l s o f i s an i d e a l i n ( Z ) Pi R a r e now e x p l i c i t e l y known ( f o r t h a t , t h e r e a r e o n l y t o s o l v e some q u a d r a t i c 2 equations x + p x - a = O over Z , i € A ) . Note t h a t A = T 1 O T 2 0 T 3 . I n p a r t i c u l a r Pi we get: n =

p

i s squarefree. U i s an i d e a l i n R i f f each Hi o f

.

a ) A l l c y c l i c subgroups o f R which a r e i d e a l s a r e g i v e n by < n pi(z+r)>, i€S z.(mod p . ) f o r j E T 2 S J where T 1 z S _ c T = A and z ,z: o r z2, (mod p k ) f o r k E T3 S

{

b ) I f T1 = A , then a l l i d e a l s o f R a r e g i v e n by < n pi> TcT,. iET

4 <( n p . ) r > where jET

G. Hofer

128

< n p.(z+r)> i € T pi> ics J -where S c T z T 2 and z i s an i n t e g e r such t h a t z = z k (mod pk) f o r k € T 2 .

c ) If T 2 = A , then a l l i d e a l s o f R a r e g i v e n by <

.

Now we g i v e some a p p l i c a t i o n s f o r both theorems. F i r s t o f a l l (compare [ l l l ) l e t A = (Q,A,FIR be a MSA. I f M c ( Q ) ~ N ( A )(which i s o f t e n t h e case) t h e n c l e a r l y f o r a l l q , q ' € Q t h e r e i s some F a E N ( A ) , a = a l a 2...anEA*,

=q'.

such t h a t F,(q)

But t h i s says v e r y l i t t l e about A, because t h i s Fa can be chosen t o be t h e constant map w i t h v a l u e q ' . I n some sense No(A) r e p r e s e n t s t h e "autonomous p a r t " o f A, which i s d e f i n e d and s t u d i e d f o r l i n e a r automata ( c f . [ l l ] ) . D e f i n i t i o n 7. L e t A = (Q,A,F)R be a MSA, U a subset o f Q. U i s c a l l e d machabte dhom q i f U i s a subset o f No(A)q. U i s neachabLe i f U i s a subset o f t h e i n t e r s e c t i o n o f a l l No(A)q, q E Q \ { O l . A i s c a l l e d ~ L t - i C t R yconnected i f Q i s reachable. N o t a t i o n . L e t ( G , + ) any ( a b e l i a n ) group, g E G , n E I N , then

..,(n-l

izng := {O,g,.

191. be a MSA where No := No(A) i s a s.r.

Theorem 3. L e t A=(Q,A,F)R generator Fo and s . t r .

with

(a,B;p,),

p 1 prime. Then t h e f o l l o w i n g h o l d : a ) I f TI = { l l then Noq= i f o r a l l q E Q . b ) I f T p = { l } t h e n Noq= f o r a l l q € K e r ( z l i d + F o ) ,

Noq=

i

otherwise. c ) I f T 3 = C 1 1 t h e n Noq= f o r a l l q E K e r ( z l l i d t F o ) u K e r ( z 2 1 i d t F o ) , Noq =

t

otherwise.

Pmod. Consider t h e map hq:No

+

( n o t e t h a t Q i s a l s o a N,-module). q E K e r no. C l e a r l y Noq = I f TI

t

Noq, no

+ noq. h i s an No-epimorphism q Furthermore n o € Ann q i f and o n l y i f

.

Now we use m a i n l y b ) o f Theorem 1.

= { 1 I then No i s a f i e l d and a ) i s proved.

If T7 = 11) then < z l i d + Fo> i s t h e unique proper i d e a l i n No, hence N,

is a

l o c a l r i n g . Thus e x a c t l y t h e elements o f < z l i d t F o > a r e n o t b i j e c t i v e ; moreover, t h e y have t h e same k e r n e l ( z ( z l i d t F o ) ( q ) = O where ~ € 7 2 , ~ $ 0 , i m p l i e s ( z l i d + F o ) ( q ) = O s i n c e 2-l e x i s t s i n i s trivial

1.

If q E Ker(zlid

t

(Z ,.), t h e converse d i r e c t i o n

PI Fo) then Fo(q) = -zlq, hence Noq_c. B u t

=Z

q and p1 i s prime, t h u s Noq=. i f q $ K e r ( z l i d + F o ) P1 {O} and b ) i s proved. I f T3 = 111 then < z l l i d t

Fos,
The isomorphic images hl(),

< ( ~ ~ ~ , l )Since > . z l l* z 2 1 y we g e t

then Ann q i s

Fo> a r e t h e unique p r o p e r i d e a l s i n No. h 1 ( < z Z 1 i d t F o > ) a r e g i v e n by <(zll,l)>,

Z X Z =<(zll,l)> p1

p1

i <(z21,1)>, hence

129

Ideals and reachability in machines

N =

. The i d e a l generated by an element

0

* 0 of

No i s

o r o r No i t s e l f . Since No i s a c o m u t a t i v e r i n g

e i t h e r

w i t h i d e n t i t y , exactly the elements o f U a r e n o t i n v e r t i b l e , t h a t means n o t b i j e c t i v e . The r e s t f o l l o w s s i m i l a r t o b ) . Remark 3. With t h e same assumption and n o t a t i o n as i n Theorem 3, t h e f o l l o w i n g i s a l s o v a l i d . I f T 3 = l l I , then No= < z Z 1 i d t F o > hence = 0. Thus Im(zl,id+Fo) is (zllid+Fo) 0 ( z 2 1 i d + F o ) = (zZlid+Fo) o(zllid+F0) a submodule o f Ker(z2,id+Fo) and Im(zZlid+Fo)

i s one o f Ker(zllid+Fo).

Since Ann q * N o f o r a l l q * O , t h e i n t e r s e c t i o n o f Ker(zllid+Fo) Ker(zZlid+Fo)

with

i s t r i v i a l . I f Q i s f i n i t e o r a finite-dimensional

vector space,

Fo) = Im(zl1id+F0),respectively. The l a s t a s s e r t i o n needs an explanation. I f IQ( = n F: M, then = Im(z21id+Fo), Ker(zZlid+

then Ker(zllid+Fo)

Ker(zZlid+ Fo) = Im(zllid+ Fo) f o l l o w s by n = IKer(zllid+Fo)[

IIm(zllid+Fo)[

5 [Ker(zllid+Fo)(

( K e r ( z 2 1 i d + F o ) (< n .

The e q u a l i t y holds because of t h e homomorphism theorem, the f i r s t i n e q u a l i t y because of Im(zllid + Fo) 5 Ker(z2.id + Fo), the second because the sum o f both kernels i s d i r e c t , together w i t h t h e homomorphism theorem. I f Q i s a f i n i t e dimensional vector space then t h e a s s e r t i o n f o l l o w s s i m i l a r l y by dim Q = d i m Ker(zllid+Fo)+dim +

Im(zllid+Fo)

< d i m Ker(zllid+Fo)

+

dim Ker(zZlid + Fo) < dim Q. Now we present two methods f o r determining r e a c h a b i l i t y i n A i f char No(A)

i s a s.r. w i t h

i s n o t prime. L e t A=(Q,A,F)R be a MSA where No := N,(A) generator F,

s.tr.

(a,B;n),

and l e t q € Q * :

F i r s t method. Determine a := o r d q and b : = minCm6 IN1 mFo(q) E ). Then the following i s easy t o see (cf. [41): Za i d + Zb Fo i s a system o f representatives and, by considering hq : No o f No,Ann "smallest" representation o f Noq. If n =

n

-+

Noq, Noq = Z a q

t

Zb Fo(q) i s t h e

pi i s squarefree, then we can use a l s o the f o l l o w i n g

i€A Second method. 1) Determine a l l i d e a l s i n No ( t h a t means determine T1, T2, T3) 2) Determine Ann q

3 ) There are some

TcA,

SsT, z € Z (because of Th. 2 and i t s Cor.) such t h a t

p.(z+r)>. T h e n Z a i d + Z b F o , w h e r e a : = n pi, jEs J iET i s a system o f representatives o f No,Ann (see [41). Hence

A n n q = < n pi>;<" b :=

n jcS

iET

pj,

Noq = Za q + Zb Fo(q) i s the "smallest" representation o f Noq.

130

G. Hofer

The f i r s t method works more g e n e r a l l y . B u t t h e second method works sometimes f a s t e r and o f f e r s t h e o p p o r t u n i t y t o determine Noq f o r a l o t o f s t a t e s q

s imul t a neous 1y : P r o p o s i t i o n 1. L e t A=(Q,A,FlR be a MSA where No : = No(A) i s a s.r. generator Fo and s . t r .

(a,B;n)

where n =

n

pi i s squarefree and T1 = A .

a := o r d

i€A then t h e f o l l o w i n g holds:

a ) Ann

i

4, ;EQ, i =

with If

b ) Noq = Za; + Z a F o ( { )

Mb de-

c ) I f B i s the s e t o f a l l proper d i v i s o r s o f a, Mb : = K e r ( b i d ) f o r b E B ,

notes the complement of Mb, then we get: Noq = Z a q

+

Za F o ( q ) f o r a l l q E n M, n itb.

bEB

Pmod. n q = n i d ( 4 )

= 0 f o r a l l q € Q because No i s a s.r.,

hence o r d q

d i v i d e s n f o r a l l q € Q . Note t h a t f o r t q E Z n q we get:

.

t q = 0 o q E Ker t i d

Q t i d E Annq The i d e a l generated by t i d i s g i v e n by and a l s o c o n t a i n e d i n Ann q. Since T 1 = A , Ann 6 must be o f t h e

did>

form d i d >

i (see b ) o f t h e Corol!ary). Now a = o r d

4 yields

the desired

minimal t. T h i s proves a ) , and b ) f o l l o w s . q E r l M a n M b = q € K e r a i d h A q 4 K e r b i d o o r d q = a . This proves c ) . bEB b€B Hence, f o r T, = A , a l l q E f l ManRb have t h e same o r d e r , consequently t h e bEB same a n n i h i l a t o r , which i n t u r n g i v e s i n s t a n t knowledge about Noq f o r a l l these q. The s i t u a t i o n i n t h e case

T2 = A i s s i m i l a r .

P r o p o s i t i o n 2. L e t A = (Q,A,F)R be a MSA where No : = No(A) i s a s.r. generator Fo and s . t r . Let z E Z with z=zi

i :=

(a,B;n) (mod pi)

where n =

n

with

pi i s s q u a r e f r e e and T2 = A .

iEA f o r i € A , q € Q and a := o r d q, b := o r d

4, where

( z i d + F o ) ( q ) , then t h e f o l l o w i n g hold:

a ) Ann q = b ) Noq Pk006.

Ha q

+

Z b Fo(q)

On t h e one hand we have Ann q =

by c ) o f t h e

C o r o l l a r y f o r some c,d 6 n. Hence cq = 0 and dq = 0, t h a t means a \ c and b l d . But then

;
i d + F o ) > z < a id>

and bq = 0, so

;
i d + F o ) > . On t h e o t h e r hand, a q = O

FAnn q, and we a r e done.

F o r "mixed" cases (i.e.

T,

A, T2 9 A) and f o r more i n f o r m a t i o n about

p r o p e r t i e s o f i d e a l s i n s y n t a c t i c r i n g s , t h e Jacobson-radical o f a s y n t a c t i c r i n g and so on. see [41.

Ideals and reachability in machines

131

In private communications, G. Betsch (Tubingen) suggested the following point of view. I f a s t a t e q E Q i s reachable from another s t a t e q ' E Q ( i . e . i f qENq' o r qENoq'), we can say t h a t q ' d i u i d a q ( q ' l q ) . The relation I i s then a preorder on Q.The corresponding equivalence classes consist of s t a t e s which can mutually be reached from another. These and related subjects will be studied in a subsequent paper.

REFERENCES

101 [ll] [121 [131

Blyth, T.S., Module theory, Clarendon Press, Oxford, 1977. Eilenberg, S., Automata, languages and machines, Academic Press, New York, 1974. Hlawka, E. and Schoibengeier, J., Zahlentheorie, Manzsche Verlags- und Universitatsbuchhandlung, Wien, 1979. Hofer, G., Near-rings and group-automata, Doctoral d i s s e r t a t i o n , Univ. Linz, Austria, 1986. Hofer, G. and P i l z , G., Group-automata and near-rings, Contrib. Gen. Algebra 2 , Klagenfurt, Austria, 1983. Holcombe, W.M.L., Algebraic automata theory, Cambridge University Press, Cambridge, 1982. Holcombe, W.M.L., The s y n t a c t i c near-ring o f a l i n e a r sequential machine, Proc. Edinba. Math. SOC. 26 (1983). 15-24. Kalman, R.E:, Algebraic theory o f . i i n e a r systems, i n : Topics i n math. systems theory, McGraw-Hill, New York, 1969. Lang, S., Algebra, Addison-Wesley Publishing Company, Reading, Mass., 1984. Lidl, R. and P i l z , G . , Applied a b s t r a c t algebra, Springer-Verlag, New YorkHeidelberg-Berlin, 1984. P i l z , G . , Near-rings, 2nd e d . , North-Holland/American Elsevier, AmsterdamNew York, 1983. P i l z , G., Algebra, Universitatsverlag R. Trauner, Linz, Austrla, 1984. Pilz, G . , S t r i c t l y connected group automata, submitted.

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Ne~-ringsand Near-fElds,G. Barh(editor) 0 Elsevier Science Publbhm B.V.(North-Holland), 1987

133

COUPLINGS AND DERIVED STRUCTURES Helmut KARZEL Mathematisches Institut, Technische Universitat Miinchen, Arcisstr. 21, D - 8 0 0 0 Miinchen 2 Near vector spaces which can be derived from usual vector spaces by the method of couplings will be discussed. The main results are stated in the theorems ( 8 ) and ('16). INTRODUCTION A structure group (P,c,*) (cf.[21, §2) is a group (P,t) where the set P is provided with a structure such that for any a EP the map at: P P; x 4 ax is an automorphism of the structure (P,C) If Aut(P,L) denotes the group of all automorphisms of the is structure (P,c) then a map rp : P + Aut(F,C, - ) ; a + a CP called a coupling if for all a,b EP the functional equation

'I.

.

(C)

a b = (a-a (b))rp cprp

c

+

cp

holds.

Then we have the statement: Let (P,C,.) be a structure group, let cp : P +Aut(P,C) be a coupling and let a o b : = aaa ( b ) , then ( P , c , o ) = : ( P , c , * ) e p is cp again a structure group called the rp-derivation of (P,c,*) For a class & of structure groups we denote by D(B) the class of all structure groups which are cp-derivations of structure groups of the class 6 . Since the constant mapping rp: x +id is a coupling for any structure group we have Q c a ( & ) . A l l members o f a(&) are called Dickson structure groups with respect to ('I)

.

c. Now we can formulate the following Dickson problems: PI' Let S I be a class of structure groups. Find couplings and determine XI(&) P2 Let E be a class of structure groups and 6 a subclass of Q characterized by additional properties. When do we have D(6)=B?

.

Remarks. .I' A n y (left) nearfield ( N , + , * ) can be a structure group if we set P :=N = N \ ( O ) , (P,.): = if c denotes the additive structure + on N = P ~ ( of the left distribution law for any a EP the map is an automorphism of (P,C).

*

considered as (N , * ) and 0 ) Because a&: x a.x

* .

+-

H. Kanel

134

2. If 'J1 denotes the class of all left nearfields and 8 the subclass of all fields, then the members of the class %(a) are called Dickson nearfields. For the classes 'J1f of finitenearfields and 2f of finite fields we know by the theorem of Zassenhaus that 91f\'S)(3f) consists of exactly seven nearfields. Till now all known examples of infinite nearfields belong to D ( 3 ) (cf. [ I l l ) 3 . Let I be the class of all affine incidence p r o u p s , i.e.the class of all structure groups (P,L!,*) where (P,i?) is either an or a translation plane. Then 2l conaffine space of dim(P,Q)23 tains the subclass Z of all translation geometries (€',Q,+) defined by the property, that for each a EP the map a+: P P;x a + x is a translation. (Here (P,+) is always a commutative group.) By [2] we have here the result U=S(Z),which tells us that any affine incidence group is Dicksonian withrespect to a translation geometry. Any translation geometry (P,Q,+)has the nice property that the set R of all lines through 0 obeys the followingconditions: (F'l) For each A € R , As(P,+) and A {O],P,

.

+

+

+

(F2) U R = P . (F3) For each A,B € I , A = B or A n B = (O}. Now let (G,+) be any group (not necessarily commutative) and R a collection of subgroups of G satisfying (F'l,2,3). Then (G,R) is called a fibered g r o u p and R a fibration (or partition). R is called a kinematic fibration if further

(FK) For each a E G , for each A E R , a + A - a € R is satisfied. For a fibered group (G,R) let EndG be the set of all endomorphisms of the group ( G , + ) and E(G,L) : = (f €End G I v X €R,f(X) cX] Then E(G,R) is a semigroup with identity. But if (G,+) is not abelian in general the sum of two endomorphisms is not an endomorphism and the sets d.g.(EndG) : = [CbifiI b i t ( l , - 7 } , fi EEndG) and

.

d.g.(E(G,R)) : = {c bifil 6iE ('l,-'l], fie E(G,R)) are near-rings distributively generated by EndG and E(G,R) resp. using pointwise addition and composition (cf.[8] Part 111, B6). F o r g E d.g.(E(G,R)) we have still g(X) c X if X E R At the Harrisonburg Conference on near-rings and near-fields in 7983 it was suggested to study near-rings of the type d.g.(E(G,R)). By [51,[71 we have the result: (2) If ( G , R ) is a kinematic fibration with IRI >I and E(G,R)$(o,id) then G is abelian, E(G,R) an integral domain and ( G , R ) can be embedded into

.

Couplings and derived structures

135

h

a vector space ( t , L ) where G = G4 and LZZ(E(G,R))~ with A =Z(E(G,K))\{O) are the quotient structures. Therefore if ( G , R ) is kinematic and (G,+) not commutative then E(G,R) = [o,id) and hence d.g.(E(G,R)) is isomorphic to one of the rings ( Z , + , * ) or (z,,+,.) with nEn\l. To avoid this trivial case and also the case where d.g.(E(G,R)) =E(G,R) , we have to consider non abeliangroups (G,+) with a fibration R which is not kinematic such that

.

E(G,R) 4 l0,idI Here we will discuss the problem how to obtain examples of fibeyed groups using the method of couplings. As the simplest c l a s s 8 of fibered groups we consider all examples (V,R) where (V,+) is over a skewfield the additive group of a left vector space ( V , K ) K and where R is a fibration consisting of vector subspaces of the same dimension. F o r instance the set s l : = [ K a l a E V \ [ o ) of all one dimensional vector subspaces is such a fibration. If ( V , R ) is any fibered group of the class 9) then L :-E(V,R) is a skew field containing K and any X E R is a L-vector subspace over L. Therefore we will always assume K=L=E(V,R). R contains a subset 8 such that V = c {BI B € B is the direct sum of the subspaces of B ; we call 181 thedimension of the fibered group ( V , R ) If we set !2 : = [a+Xl a € V , X E R ) then it is easy to prove that (V,D) is an affine space of dimension 181 where V is the set of points and Q the set of lines, and ( V , ! 2 , + ) is a translation geometry. If ) % I 2 3 , then ( V , D ) is a desarguesian affine space (cf.p.e.[6] §‘lo) and hence R=(Lxl xEV ). If 181 = 2 ,this result is true if and only if(V,Q) is desarguesian; in all other cases ( V , O ) is a proper translation plane. In C4lp.207 and [ J ] the so called near vector spaces (G,@,P,*) are studied, i.e. (G,@) is a group, (I?,+,’) is a near field and : F xG G is a operation such that (V’I) V X , y t F , V x t G : ( X + p ) * x = X * x e p * x (V2) t r X , b E F , V X E G : (X*p)*x=X*(p*x) (V3) V x E G : ’I *x = x T o any near vector space there belongs also a fibration R : = (F *a] a t G\[O)) such that each fiber F *a is isomorphic to the additive group (F,+) of the near field F . So we have the questions:

.

*

*

P3

+

h

If 3 denotes the class of all additive groups ( V , + ) belongA ings to a vector space ( V , K ) does then P ( 8 ) contain examples which can be made to near vector spaces?

H. Karzel

136

Do t h e examples of n e a r v e c t o r s p a c e s g i v e n i n [ 3 ] and [ 4 ] belong t o ~ ( 8? )

P4

2.

CONSTRUCTION OF NEAR VECTOR SPACES, SENILINEAR CASE To d e a l with t h e problems P3 and P4 l e t (V,K) be a ( l e f t ) v e c t o r space. By M ( V ) we denote t h e n e a r - r i n g o f a l l maps from V i n t o V. and $ : V M ( V ) ; a -+a Let cp : V M(V); a a rp 0 : = acp- id Then g f u l f i l l s t h e c o u p l i n g f u n c t i o n a l e q u a t i o n (C)

Va,btV:

i f and only i f va,btV:

(C')

+

+

+

a b = ( a + a (b)) cprp

T

c

-

p

fulfills

$

( a + b + a (b)) %

( 3 ) For t h e o p e r a t i o n

a @b

% :=

=a +b t a b

w

00'

UI

a + a (b) cp

=

a + b + a (b) Ib

we have

a)

xbo =x

0

~ ~ (= 0 O )

b)

O@x=x

0

0 -0

c)

(V,@)

d)

Let a E V , t h e n f o r each b E V t h e e q u a t i o n a Q x = b h a s a s o l u t i o n i f and only i f a = a + i d i s a permutation.

4-

(zero map)

i s associative

0

( C ' ) and

$ ( V ) c End ( V , + )

.

c p u

From now on we assume f o r t h e map 0

(C'),

4

=

group with (4)

Let

0 and 0

S ( V ) c End(V,t)

.

$ : V +M(V)

Then by (3)

the conditions (V,&) i s aserni-

a s n e u t r a l element. By i n d u c t i o n we have

n,mEbI,

a E V ,

...

n * a :='a@aQ @ a , then n-t ime s

c)

n * a @ m * a = ( n + m )* a

e)

I f there i s a respect t o Q

f)

a h a s a n e g a t i v e ea with r e s p e c t t o d l e t Then c ) and d ) a r e v a l i d for a l l (-n) * a : = n * ( e a ) and hence z * a i s a c y c l i c subgroup of ( V , e t ) If

nEhJ with n * a = O t h e n a h a s a n e g a t i v e w i t h and M*a i s a f i n i t e c y c l i c subgroup of ( V , a ) ,

.

.

m,nEz

137

Couplings and derived structures

g)

If

a$

(5) I f

q

i s n i l p o t e n t then

has t h e additional property t h a t for each a Q i s semilinear then we have e i t h e r

: V 4 M(V)

t h e map

a EV

e a e x i s t s and

i s l i n e a r , or

a)

for each

b)

t h e r e i s an aEV such t h a t a i s not l i n e a r , then u v = 0 B 9 Qf o r a l l u , v E V and t h e r e i s an automorphism : K K; h + A such t h a t b (Ax) = x b (x) f o r a l l b,xEV and X E K . yl B

a E V , a$

-+

Proof. For

a,bEV

let

a

b 40

0' V

and l e t

a,p

phisms corresponding t o t h e semilinear maps By ( C ' ) we have f o r

A EX,

xEV

a V ,b

:

(a@b)o(lix)= a ( > ) * a(x) + p(X).bJl(x)+ap(A)a b (x)

vul

$

i s semilinear we have (6) (V'I)

a=P

and i f

be the automorrespectively

. Since

a = p $ i d then

JlJl

*

( A * a > (cl * a > = (P * a I p ( h * a >

b)

( h * a ) 0 (P * d 9 (cl =*a)*(X*a)$.

(a@b))

a b =O.

If t h e r e i s an e x t e r i o r operation : KxV j i s v a l i d then f o r a l l h , E~K and a l l a E V :

a>

.

V such t h a t

ul

Proof. a) i s a consequence of (V'l).

By (C') we have

( a Ceb)$ =

= a + b + a b hence "a b = b a o a 8 b = b 8 a f ' . Now by (V'I) 0 'p UlJI V Q vul obtain X * a @ p * a = ( A + p ) * a = p * a @ A * a which gives us b ) .

we

Next we w i l l show t h a t t h e examples of near vector spacesgiven i n [ 4 ] belong t o 9 ( B ) L e t (V,K) be a 3-dimensional vector space over a commutative f i e l d K with char K S 2 , l e t : K -3 K be an automorphism of the f i e l d K and l e t e,l,e2,e3 be a b a s i s

.

of

(V,K)

. For

xEV

let

X , ~ , X ~ , X € K~

be defined by

, .

x=x,e + x e + x e and f o r aEV l e t a (x) : = a x e Then I 1 2 2 3 3 UI -a y E Eud(V,+) and since a (Ax) = : a3Ax,,c2 = A a3x,r e2 = X a (x) f o r 9 Ib A E K , t h e map a i s semilinear. I f a 4 0 then a ( V ) =Ke and JI 3 V 2 a -1 (0) =Ke2 +Ke and i f a3 = O then a i s t h e zero map. This yl 3' Yl shows u s a b = O and ( a ( b ) ) = O for a l l a , b E V . Since

--

ulyl

- Q-

Q

(a+b) (x) = (a3 + b3)x,le2 = a3x,,e2 +F3%,e2, (a+b)* = a yl

+b4

and s o ( C ' ) :

= a + b + ( a (b)) = a + b = a + b + a b u l u l y l yl B V S Y I J l V I y l ' (V,@) with a 8 b : = a + b + a(b) i s a group which i s not 0

( a + b + a (b)) yl

By ( 3 )

$

138

H. Karzel

el @ e 3 = e,l + e3 f e,, + e3 + e2

commutative because a (b) =a3E,le2,

0 t h e formula ( 4 ) a ) i s reduced t o

=

-

= n . a + -I' n ( n - l ) a 3 a l e 2 .

Since

2

t h e same o r d e r with r e s p e c t t o

(7)

: V

The map

e5 8 e,,

and s i n c e

i s t h e a d d i t i o n d e f i n e d i n ['I.], 8ec.V. S i n c e

@

v

=

n *a = n . a

+ (z)aI(a)

(Xe2)y = 0 ,

each element

+ and

So we have:

6.

=

a EV

has

Ehd(V,+) d e f i n e d above has t h e f o l l o w i n g

p r o p e r t i e s a ) , b ) , c ) and (C') i s a consequence of a ) , b ) , c ) . a EV

=O,

b) a b

vul

,

i s s e m i l i n e a r and i f 0 (a$(b)) = O

a ) F o r each

C K

a

,

t h e n (Aa)

=

ya

u I J 1

UI

c ) ( a + b ) $ = a $+ b

0

Now l e t u s assume t h a t t h e r e i s an

a E V such t h a t t h e semii s not l i n e a r . Then by ( 5 ) b ) t h e r e i s an automor4 of ( K , + , * ) d i f f e r e n t from t h e i d e n t i t y , such t h a t phism u (Av) = T u ( v ) and u v = 0 f o r a l l u , v f V I f x (x) = 0 f o r Q 0 0 ) v a l l xEV t h e n b y ( 4 ) a ) n * x = n . x for n t N and X a & p a = l i n e a r map

a -B

.

=

( h + p ) a + ( a a ) (pa) $

x(pa) i s an

(Xp)a

=

with

1

( h + p ) a + pX

that

SO

aCV

=

(V,@,K,.)

a (a) + O

I

then

Trying t o extend t h e o p e r a t i o n

(Aa)#(Aa) = ( A

+p)a

and

i s a n e a r v e c t o r space. I f t h e r e a , a ( a ) a r e l i n e a r l y independent.

*

uI

of ( 4 ) a ) onto K x V s u c h t h a t

(V,CH,K,*) becomes a n e a r v e c t o r space we make t h e f o l l o w i n g approach: Let f , g , h : K +K be f u n c t i o n s such t h a t for each xEV

we have

Then

0

h('l)

=

Since

=

0 *a

f('l) ='I

X*x : = h ( h ) - x + g ( X ) x b ( x ) and

.

and

a ='I + a

We s e t

implies

a

h(0)

=

.

(x*~)~=f(X)x$ g(0)

and r e c a l l t h a t

=

g('l)

=

and

X * a @ p * a = @ ( h ) + h ( p ) )a + ( g ( X ) + g ( p ) ) a ( a ) + f ( X ) a ( h ( p ) a +

+ g(l-l)a(a)) = (h(A) + h ( ~ l ) ) +a ( g ( h ) +g(l-l) + f ( A > h ( c ) ) a ( a )

,

( A +I)

+ a =h(A + v ) a + g ( X + p ) a ( a > (v") i s e q u i v a l e n t with

( a ) h(X +>I

=h(X) + h ( p )

( p ) g(X+Ci) = g ( X ) + g b ) +f(X).h(P) and s i n c e h + ( p * a ) =h(A).p + a + g ( A ) . ( p * a ) Q ( k * a ) = h ( A ) h ( p ) a + h(h)g(l-l)a(a) + g ( A ) e f ( p ) a ( h h ) a + g ( p ) a ( a ) ) = h ( h ) h ( p ) a + ( h ( h ) g ( p ) + g(X>* f ( u ) h ( p ) ) a ( a )

=

( b p ) * a = h ( X p ) a +g(A p > a ( a >

(V2) i s e q u i v a l e n t with

(y) (6)

0

a2 = O .

~ ( X * V=)h ( X ) * h ( F ) g(Xep) = h ( X ) * g ( p ) + g ( A ) * f ( p ) E ( P )

=

Couplings and derived structures

139

+K

Since h('l) ='I the equations (a) and ( y ) tell us that h : K is a monomorphism, and from ( p ) and f ( l ) ='I we obtain f(h) =g(h+'l) - g ( X ) -g('l) =h(X) so that ( 5 ) gets the form ( 5 ' ) g(X*p) =h(X).g(P) +g(X)E(P)+E(P) By g(l) = 0 , h('1) = f(l) ='I , (a) and ( p ) we have h(n) = nh(1) for n E N , g(2) ='I and ( 5 ' ) gives us g ( 2 . h ) =h(X) +g(X).4= =

-

2.g( a ) + H( X) -E(a )

h(1-p)

(Y) -

= h(p).h(i)

of

and

Vr

c(p>

=

(PI =

$ (m - h( A ) ) . g ( h +I)

n

Furthermore

-g(X) - g ( p ) =f(k)

x(X)

=

= h ( p rA ) shows that K has to be commutative.

A s in ]I'[

and

=f(X)

h(X).E(p)

=

thus g( X)

=

:=

and [2] we consider the s u b g r o u p s U := (x E VI x = 0 ) 9 (x EVI V (x) = 01 of (V,+) Then V r is also a subgroup

.

9

and if V V = O

(V,@),

ulu x,yE V ) c V r

I

(x,(y)

.

then also U

is a subgroup of ( V , @ )

(u + v ) = ~ u +v

In any case

o

or v E U u V r and "I, is a vector subspace of (V,K)

v

if u E U U v r

because VV (h( X)x) +

consists of semilinear maps. Therefore (A +x) = o and ( 6 ) implies (h( A h ) ,(x> + (g( A ) . x,(x)) ,(x> = + (g( X>x)(x)) 1-2 = h(A)X9(x) , hence CI x,(x> = (px)$(x) + ( ~ ( c r - ~)xv(x))~(x) if

.

Eh(K) On the other hand one shows by calculation that also the part 2. of c) of the following theorem is valid:

P

be a vector space, let c p : V +Aut(V,+) be a coupling such that f o r each x € V the map x : = x -id is u r p semilinear and there is an a E V where a is not linear, and let V - : K jK ; a + T denote the automorphism of a Further let 0' x & t y = x + x (y) = x + y + x (y). Then ( 8 ) Theorem. Let ( V , K )

V

cp

a) For all u,v E V means that

c ) Let

(V,e,K,*)

a EV

X EK , u

v ) = x u (v) and

V( A

I

is a stronq couplinq i.e.

cp

b) If x (x) = O Q space. f,g,h : K

,

f o r all

with

4K

with

x EV

a (a) + O .

u

then 'I.

u v

= 0 which O Q ( U + V ) ~ =up vv.

(V,@,K,,)

is a near vector

If there are functions

+ g ( X ) . x (x)) f(A).xg such that 0 yl= 1 *x : = h( X),x + g( A ) -x,(x) becomes a near vector

with

(h(X).x

space; then K is commutative, Char K + 2 , h is a monomorphism and of the field K , f(h) =h(h),g(h) = $(m'-h(X)),

(N) For each x E V

,

p

E h(K)

,

-p x,(x)

=

(px) ,(x) +(&i -p)xy(x)&(x).

H Karzel

140

2. If the field K is commutative with Char K 4 2 and if there is a monomorphism h such that (N) is valid then (V,@) becomes a 'I 2 - h(ok))x,(x), near vector space (V,@,K,*) for X *x : = h( k)x + ~ ( h 4. I f

Remark.

(px) = E x

o

and

v

(x (x)) = O for each x E V

v 1 u

,

EK then (N) is valid. In this case h =id determines exactly the exterior multiplication for the examples given in [4l p.206.

p

3.

CONSTRUCTION OF NEAR VECTOR SPACES, LINEAR CASE The near vector spaces considered in [ 3 ] one obtains in this way: (9) Let ( A , K ) be an associative nilpotent algebra and let m E N

such that x m = 0 for all x E A but ym-"f 0 f o r at least one y E A. If char K = O or m ( p :=char K then (A,O,K,*) with m-7

xely :=x+y+xty and A*x= ( I t x ) x -'I space.

=

For an associative nilpotent algebra with

a,(x)

(ay(b))v

:=

=a b

9u1

a €A

each

a.x has the properties hence

the map

( a + b + a (b)) 0

a$

$

.C,(h)x

i

1=I 1

is a near vector

(A,K) the map $ (a+b)$ = a + b and

e

C

$

M(A)

*

= a + b +a b

$

:A 3

$

and f o r

is K-linear and nilpotent. Therefore by

(3)~) and (4)g) (A,$) with a&b = a + b + a JI (b) = a+b + a - b is a Hence group and so the examples of [ 3 ] are also members o f a($ P4 is answered positively. We remark that (A,CB,K,*) is not a near vector space if in (9) m > p : = char K ; then there is an a € A with ap f 0 but aP+''= 0 in and hence p *a = C(Pi)ai = ap f 0. Further if x E A then (x,) = O and there is an y E A with (yq)m-2 f 0.

.

The result of (9) can be reversed. ('10) If : V + M(V) has the following properties is linear, a) for each a E V , a b)

c)

there is a m € k t , there is an y E V for all

then

(V,@>

a,b EV with

,

0 m>l

with (a+b)*

with (X,)~-"(X) = O f o r all (~$)~-~(y) 4 0 if m > 2 , =

a$ + b $ and

a@b : = a + b + a (b) 0

(ag(b))

x EV

and

c = a9b J I '

is a group and (V,+,r) with xmf"= 0 for all

is an associative nilpotent ring with

a - b= a (b) JI x E V and there are y , z E V with ym-"z$O. (V,+,*) is a K-algebra if and only if (ha) = Xa for all X € K and all a E V . P

O

141

Couplings and derived structures

Finally we djscuss the problem P3 for the case of a coupling cp : V +Aut(V,+) where the corresponding map 0 : a - % avi := aV - id has the properties a) and b) of ('lo), If m = 2 , i.e. x (x) = O for all x E V then ( V , @ , K , * ) is a

v

near vector space (cf.(7)b)). Therefore let m > 2 . Let a E V am-2(a)) such that (a0)m-2(a) f 0 , a : = a and A : = (a,aa Then dim A

g

: KxN

-+K

=

m-l

and

a(A) c A

such that for

.

,...,

0

be

.

Again we ask if there is a function m-2 h *x : = 1 = 0 g(l,i)(x$)i(x) , (V,w,K,*)

.c

m- 7

is a near vector space. Then by (6) (x*a) (a) =igog(h,i)ai+l(a). B If we further follow (4)b) and make the assumption that the linear i map (X*a)* is a linear combination of the a then

Now (V'l)

gives us g(a +p,i) al(a> =C(g(x,i) + g(p ,i>)ai(a> j+i+'l + CCg(h,j) g(p,i> a (a) and by comparing the coefficients we obtain for all i with al(a> + O :

The part 2. of b) of the following theorem one obtains bycalculation.

H. Karzel

142

('16) Theorem. Let rp : V --tAut(V,+) be a coupling such that for each x f V the map x v := xcp- id is linear, there is a m E N , m>'l with (x~)~-'' (x) = O at least one y E V

for all x E V

.

a)

(V,&,K,.)

If m = 2 then

and

40

(~,)~-'(y)

for

is a near vector space,

b) Let m > 2 . 'I. If there is a function g- : K x M + M with m- 2 iGog(l,i)(xk)i(x))v E (X ,xw,...,x 2 m-2 ) for all x E V such that

(

w

with

(V,@,K,*)

space then K

igo g(A,i)(x

i ) (x) becomes a near vector I is a commutative field with char K = O or 2 m , A *x : =

m-j

-

T=g(k,O) is a monomorphism of the field K , g(X,i> = (i:,,) for all i E {O,'l, m-2) and 0 fulfills (N'). 2. If K is a commutative field with char K = O or z m , h + y a monomorphism of K such that (N') holds then (V,@,K,*) with m-2 A *x : = .C ( (x >i(x> is a near vector space. 1=0 ' V I

-+

...,

i2,,)

Remark. 5. If the map 0 : V +End(V,+) is linear (i.e. for A €K) then (N') is valid (x+y), = x I + y o and (Ax), = Ax

I

(C'

hence

'

1

a + b +(a (b))yf=(a+b+a (b)) = a + b + a b P v v 0 0 1 i+l (av(b))v=a b in particular ((x,) (x))$= x)

because then

v v

.

v v

Remark. 6. By [J1,(4.'lO) the fibrations belonging to the nearvector spaces o f theorem (9) are kinematic so that by (2) we have either d.g.(E(V,R)) = E(V,R) or d.g.(E(V,R)) is trivial. It remains the question whether there are other couplings in the sense of theorem ('16) leading to near vector spaces whose fibrations are not kinematic. In the theorems (8)b) and ('16)a) we considered couplings with x (x) = O for all x € V . F o r these maps we have: 0

('17) Let u, : V +Aut(V,+) be a coupling such that for all x,y E V , xI :=xcg- id is semilinear, x (x) = 0 and x y = 0. Then V v v a) x,(Y) = -Y,(x) , x @ y O x = y + 2x (y) I b)

( X + Y ) , = X ~ + Y ~ (Ax), , =Ix

i.e.

$ :

(V,K) +Ehd(V,K)

is a

semilinear map. c)

vr

d)

The fibration R : = {K.x I xEV\{O] of the near vector space (V,fb,K,*) is kinematic if and only if char K = 2 o r x is I linear for each x € V

=

u

.

143

Couplings and derived structures

If U)V

e)

then E((V,@),R)

Proof. a) By =

x,(x)

=

0

(xJI+ y,>(x + y + xv(y))

= ( A CK

IT'= A )

, xhyY= 0 and (C'), =

x,(y)

0 = (x(i~)~(x&y) =

x ui (Y) = -ye(x)

+ yv(x> hence

,

-

(X+Y)~(Z) =-zY (x+y) = - z (x) -z,(y)

b) By a),

by (5) (XX>,(Y)

=

-yyl(XX)= -XYui(x)=kui(Y)

.

=x,(z) ty u (z) and

.

. .

c) Let u € U and x € V then 0 = u0 (x) = - xe ( u ) hence u E V r Now v E V r and x C V implies 0 = xIk (v) =-vUI (x) thus v € U d) is a consequence of a). e) Since U + V there are x,yEV\U with xy(y) 40. Then h(x&y)=Xx+Xy+Ax 0 (y) and by b) X x @ X y = X x + h y + X 2 x 0 (y) hence XEE((V,@),R) if A2=A. (Examples for (17). Let K be a commutative field and K 4K an automorphism of K then the map 0 : K3 4 End(K3,K) defined by (a,l,a2,a3)$(xl ,x2,x3) = (0,0,a,,x2 - a2x,,) has the prop('18)

-

:

- -

erties of ('17). If K = z (T) is the quadratic extension field of 5 z 5 with t 2 = 2 then E((V,e),ST)=(O,'l,2+2t,2+37J. Remark. 7. If (V,m,K,*) is a near vector space of the type of theorem (8)c) and if ('I,@) is not commutative then E((V,Q),R) = {a € K I A ) (cf.[41, p. 206).

=

x2=

REFERENCES ]I'[ Karzel, H., Unendliche Dicksonsche Fastkorper. Arch.Math. ('1965),

[21

[3] C41

C51 [6]

[7] [8]

[9] ['lo]

'16

247-256

-, Affine

incidence groups. Rend. Sem. Mat. Brescia Vol. 2 ('1984), 409-425 -, Fastvektorraume, unvollstandige Fastkarper und ihre Abgeleiteten Strukturen. Mitt. Math. Sem. Giessen, Coxeter-Festschrift , Teil IVI ' ( 984) , I' 27-7 79 -, and C.J. Maxson, Fibered groups with non-trivial centers. Res. Math. 2 ('l984), '192-208 -, and C.J. Maxson, Kinematic spaces with dilatations. J. Geometry 2 ('1984), '196-20'1 -, K. Sorensen and D. Windelberg, Einfiihrung in die Geometrie Gottingen I' 973 Marchi, M. and C. Perelli Cippo, Su m a erticolare classe di S-spazi. Rend. Sem. Mat. Brescia 4 (1979p, 3-42 $5.12, G,, Near-Rings. North-Holland Math. Studies 23, '1977.

'1983.

Pokropp, F., Gekoppelte Abbildungen auf Gruppen. Abh. Math. Sem. Univ. Hamburg 2 ('l968), '147-159 Waling, H., Bericht uber Fastkorper. Jber. Deutsch. Math. Verein. 76 ('1974). 'l-'l05 - _ . _ ,4

[ 11 1 Wahling

-H .

,

I

Theorie d e r Fastkiirper , Thales Verlag

E s s e n 1987.

This Page Intentionally Left Blank

Near-rings and Near-fields, C. Betsch (editor) 0 Elsevier Science Publishers B.V.(North-Holland), 1987

145

MAXIMAL IDEALS I N N E A R - R I N G S H e r m a n n KAUTSCHITSCH I n s t i t u t f u r Mathematik Un iv e r s it a t K 1 a g e n f u r t 9022 K1 a g e n f u r t The i n t e n t i o n o f t h i s p a p e r i s , t o r e l a t e t h e i d e a l s t r u c t u r e o f a s p e c i a l n e a r - r i n g N t o t h a t o f some h o m o m o r p h i c In particular image N ' o f N o r t o i t s c o n s t a n t p a r t N one g e t s i n f o r m a t i o n a b o u t maximal i d e s l s . Such a r e l a t i o n s h i p i s w e l l known i n p o l y n o m i a l - o r f o r m a l p o w e r s e r i e s r i n g s a n d i t t u r n s o u t , t h a t one i s s u c c e s s f u l i n f i n d i n a such a r e l a t i o n s h i p f o r those n e a r - r i n g s , which have a 1i t t l e b i t o f a "polynomial-" o r "power s e r i e s - s t r u c t u r e " .

.

1.

IDEALS AND H O M O M O R P H I S M S T h r o u g h o u t t h i s p a p e r we assume t h a t (N,+,,)

i s a r i g h t near-

r i n g w i t h i d e n t i t y 1 and R i s a c o m m u t a t i v e r i n o w i t h i d e n t i t y . R[[x]]

denotes t h e s e t o f a l l f o r m a l power s e r i e s o v e r R ( i n one

indeterminate x),

R o [ [ x ] ] t h e s e t o f a l l f o r m a l power s e r i e s o f

positive order. F i r s t we s h a l l t r y t o d e s c r i b e m a x i m a l i d e a l s b y c e r t a i n homomorphisms. We s t u d y t h e c a s e ,

i s a near r i n s w i t h i d e n t i t y 1,

t h a t (N,+,O)

w h i c h h a s a n e p i m o r p h i s m h : N-+N', o f i n v e r t i b l e elements i n (N,o).

such t h a t h - ' ( l ' )

consists only

h-'If')s 1").

i.e.,

ExampLes: 1 . Any f i n i t e d g n r N w i t h i d e n t i t y : By P i l z [ 4 1 , 6 . 3 1 we h a v e h ( I ( N ) ) = I ( h ( N ) ) .

... ) : = P o ;

we s e t h ( p .+ p 1 x + p 2 x Z +

2. F o r N = ( R [ [ x ] l , + , . , )

i s a n e p i m o r p h i s m f r o m N o n t o (R,.),

then h

such t h a t h-1(l)=(l+plx+p2xz...)

c o n s i s t s o n l y o f i n v e r t i b l e elements o f (N,.). 3. F o r N = ( R o [ [ x ] l + , ~ ) , where

series,we

set h(r,x+r2xz+

o n t o (R,.),

... ) = r l ;

such t h a t h - ' ( l ) + + r 2 x

t i b l e elements o f (N,o) THEOREM I :

d e n o t e s t h e c o m p o s i t i o n o f power t h e n h i s an e p i m o r p h i s m f r o m

'...

f

(see [ I ] ) .

I f ( M o i } i s t h e s e t o f a l l maxima

e p i m o r p h i c image N ' ,

N

consists only o f inver-

then fh-'(MIi)f

ideals of the

i s the set o f

aLL m a x i m a l

i d e a l s o f N.

P ~ u o ~1 .: h - ' ( M I i ) w i s e 1~ h - ' ( M U i )

otheri s a maximal i d e a l o f N h-'(MIi)+N, a n d t h e n l ' : = h ( l ) E M I i a n d M I i N'. I f 8,4N w i t h

H Kautschitsch

146

,

B 3 h-' (MIi

then h ( B ) 3 MIi,

hence h ( B ) = N ' and l ' e h ( B ) .

Since

' ) i s i n v e r t i b l e i n ( N , o ) , we g e t l = b o v e B a n d B=N. ( T h e 1 n o t a t i o n b = h - (1') means, t h a t we c h o o s e a n e l e m e n t b e N w i t h

b:=h-'(l

h ( b ) = l ' .) 2.

I f M i s any maximal i d e a l o f N,then h(M) i s a maximal i d e a l o t h e r w i s e 1 ' E h ( M ) and M c o n t a i n s t h e i n v e r t i b l e

i n N':h(M)+N',

h e n c e 1 = m o u s M a n d M=N ( a g a i n t h e n o t a t i o n

element m:=h-'(l'), m:=h-'(t')

means, t h a t we c h o o s e a n e l e m e n t m r N w i t h h ( m ) = l ' ) .

T h e r e f o r e t h e r e e x i s t s an element a e N ' \ h(M). e x i s t s an element b r N \ M w i t h h ( b ) = a .

Hence t h e r e

By t h e m a x i m a l i t y o f M we

g e t M+(b)=N. By [ 5 1 , p. 111, t h e r e a r e mEM and p € P o ( N ) := ( P ( N ) ) o w i t h l=m+p(b). Here P(N) i s t h e s e t o f polynomial f u n c t i o n s on N ( i . e .

the near-ring

generated by i d N and t h e c o n s t a n t maps on N). Since l ' = h ( l ) = h ( m ) + p ' ( h ( b ) ) = h(m)+p'(a) w i t h p ' € P o ( " ) ,

we g e t 1 ' E h(M)+(a) and t h e r e f o r e h(M)+(a)=N', so

h(M) i s maximal i n N.

COROLLARY I:

( i ) A l l maximal i d e a l s o f t h e f o r m a l power s e r i e s o v e r a commutative r i n ? R w i t h 1 a r e g i v e n by

r i n g (R[[xIl,+,.) the ideals

Mi+(x)

,

w h e r e Mi

i s any maximal i d e a l o f R.

( i i ) A l l maximal i d e a l s o f t h e z e r o symmetric n e a r - r i n g o f f o r m a l power s e r i e s ( R o [ [ x f l , + , o )

over a commutative r i n g R w i t h 1

a r e g i v e n b y t h e i d e a l s {mlx+m2xP+

... I ml

E Mi]

,

w h e r e Mi

i s any

maximal i d e a l o f R.

P t l u o , j : ( i ) By T h e o r e m 1 we d e s c r i b e a l l m a x i m a l i d e a l s M o f (R[[xll,+,.)

w i t h t h e e p i m o r p h i s m o f E x a m p l e 2,

...

x+m x 2 + 1 mi 6 M i 1 2 (R,+,.), SO M=M.+(x). \m,+m

] ,

where Mi

hence M=h-'(Mi

)=

i s any maximal i d e a l o f

1

( i i ) S i m i l a r l y we d e s c r i b e a l l maximal i d e a l s M of (R,"~ll,+,o) w i t h t h e e p i m o r p h i s m o f E x a m p l e 3, h e n c e M = h - ' ( M i ) = { m x+m2xz+ 1 (R,+, . ) .

... I m l c M i 1 ,

where Mi

i s any maximal i d e a l o f

COROLLARY 2: ( i ) A non-simple f i n i t e dgnr N w i t h 1 i s never a subdirect

product o f simple near-rings.

( i i ) The r i n g R [ [ x l l o f f o r m a l p o w e r s e r i e s o v e r a c o m m u t a t i v e r i n g R w i t h 1 i s never a subdirect

product of f i e l d s ,

even i f R

i s such a r i n g . ( i i i ) The z e r o s y m m e t r i c n e a r - r i n a R , [ [ x ] ]

o f f o r m a l power

s e r i e s over a commutative R w i t h 1 i s never a s u b d i r e c t o f simple near-rings,

even i f R i s a s u b d i r e c t

P t l u o d : L e t Rad N ( o r R a d N ' ,

product

product o f f i e l d s .

r e s p e c t i v e l y ) denote t h e i n t e r -

s e c t i o n o f a l l maximal i d e a l s o f N ( o r N ' ) and J ( R ) t h e Jacobson

Maximal ideals in nearrings

147

r a d i c a l o f R.

N ( o r R, r e s p e c t i v e l y ) i s a s u b d i r e c t product o f s i m p l e n e a r - r i n g s ( f i e l d s ) i f f Rad N=O ( J ( R ) = O ) . By T h e o r e m 1 we get: = A h - l ( M I ) = h - ' (AM ' ) = h - ' (RadN ) Rad N = n M M maximal M ' maximal in N in N' I

( i ) I f N i s n o t simple,

then there exists a non-injective

N and h - l ( R a d ( N / I ) ) + O , even i f Rad(N/I)=O. ( i i ) By C o r o l l a r y l ( i ) R a d ( R [ [ x ] ] , + , . ) = J ( R ) + ( x ) ~ O , even i f

e p i m o r p h i s m h:N-+N/I,

14

J (R)=O. ( i i i ) By C o r o l l a r y l ( i i ) R a d ( R o [ [ x l l , + , ~ ) = \mlx+m2x2+

.../ml

E J ( R ) ] PO,

even i f J(R)=O.

To g e t some i n f o r m a t i o n a b o u t m a x i m a l l e f t - i d e a l s o f N , we i n t r o d u c e an a d d i t i o n a l a s s u m p t i o n on N: I ( N ) i s c o m m u t a t i v e and e,~e2=mo(n+el)-m,n Vel,e2E

w i t h s u i t a b l e m,n E N ,

I(N).

Exampbeb: 1. Any n e a r - r i n g ,

i n which t h e i n v e r t i b l e elements

a r e z e r o symmetric and p e r m u t a b l e : elne2=e20(0+el)-e200. 2 . The z e r o s y m m e t r i c p a r t R o [ x I o f t h e p o l y n o m i a l n e a r - r i n g (R[x],+,o).

I n R o [ x 1 , + , ~ ) a l l i n v e r t i b l e elements a r e g i v e n by t h e

p o l y n o m i a l s ex,

e a u n i t i n (R,.),

which are permutable w i t h

respect t o o . THEOREM 2 :

I f { L a i } i s t h e s e t o f a l l maximal l e f t i d e a l s o f

t h e e p i m o r p h i c image N ' o f N,

then {h-'(Lti)]

i s the set o f a l l

maximal l e f t i d e a l s o f N. P4ua6: 1 . N:h-l(L'i)PN,

I f LIi

i s maximal i n N ' ,

otherwise 1 E h-'(LIi)

then h-'(Loi)

and l ' = h ( l ) € LIi.

i s maximal i n By t h e a b o v e

a s s u m p t i o n we g e t :

n ' = l ' ~ n ' = p ' ~ ( q ~ + l ' ) - p ' ~ qe 'L I i v n ' E N ' , I f B slN

and B = h - l ( L ' i ) y

so LIi=N'.

t h e n h ( B ) = N ' and

1 ' 6

h(B).

b & B , so B c o n t a i n s t h e i n v e r t i b l e e l e m e n t b : = h - ' ( l ' ) .

i.e.,

l'=h(b),

Then

l = b o u = m o ( n + b ) - m o n E B a n d B=N. 2.

I f L i s any maximal l e f t i d e a l o f N, t h e n h ( L ) i s a maximal

l e f t ideal o f N': h ( L ) + N ' , o t h e r w i s e l ' ~ h ( L ) ,h e n c e l ' : = h ( l ) l:=h-'(l') a:=h-l(a')c

i s i n v e r t i b l e i n L,

So l = l + p ( a ) w i t h some z e r o

Then

w i t h p ' c P,(N'), 1 ' e h ( L ) , hence ' h (~L ) + ( a ' ) a n d t h e n h ( L ) + ( a ' ) = N ' , s o h ( L ) i s m a x i m a l i n N ' .

l'=h(l)=h(l)+p'(h(a))=l'+p'(a') 1

s o L=N. Now l e t a ' € N ' \ h ( L ) , t h e n

N \ L. B u t then L+(a)=N.

s y m m e t r i c p o l y n o m i a l p~ P,(N).

f o r some l c L a n d

H. Kautschitsch

148

I f N i s a f i n i t e and d i s t r i b u t i v e l y g e n e r a t e d n e a r -

C U R O L L A R Y 3:

r i n g w i t h commutative I ( N ) , t h e n a l l maximal l e f t i d e a l s a r e g i v e n by t h e i d e a l s

{

,

h-'(LIi)}

where L I i

i s any maximal l e f t i d e a l o f

an e p i m o r p h i c image h ( N ) o f N .

2.

IDEALS A N D THE C O N S T A N T P A R T N c Now we c o n s i d e r t h e c a s e t h a t N d o e s n o t h a v e a n e p i m o r p h i s m

o f t h e type given i n s e c t i o n 1.

I n order t o describe the ideal

s t r u c t u r e o f N by t h a t o f i t s c o n s t a n t p a r t N c , N an a d d i t i o n a l m u l t i p l i c a t i o n - i s d e f i n e d , ( i ) (N,+,.,o)

we a s s u m e ,

that i n

such t h a t

i s a composition-ring

aN 3 s e N : n o ( m + i ) = n o m + i . s

(ii)vn,m,i T h i s means,

i n particular,

Exampeen: ( R [ X ] , + , . , ~ )

t h a t (N,+,.)

i s a ring.

and ( R N [ [ x l l + , . , o ) ,

where R " [ x l l

de-

n o t e s t h e s e t o f f o r m a l power s e r i e s w i t h n i l p o t e n t i n i t i a l c o e f f i c i e n t (see [2] ) . A nonempty s e t I i s an i d e a l o f a c o m p o s i t i o n - r i n a N w i t h property ( i i ) i f f ( i ) i - j€ 1 ( i i ) i,neI ( i i i ) i.n C I

and n . i C I

vi,j

€1, neN.

F i r s t we i n t r o d u c e t w o s p e c i a l i d e a l s ( s e e a l s o [ 3 ] ) : 1.

If I g(N,+,.,o),then

IC~(Nc,+,.,o,): V i , j e I c , c eNc. 2.

i - j E I n N c ,

If 1 ' s (Nc+,.),

we s e t I c : = I n N c 9 ( N c , + , . ) i.cEInNc,

a n d we g e t

c . i ~ I n C'N i o c r I n N c ,

then ( 1 ' ) denotes t h e i d e a l o f N generated

[>

pioj'ilpi e P O ( N ) , j l i t I ' > , where P o ( N ) finite denotes t h e zero symmetric polynomials over N (see i51). b y 1': ( I 1 ) =

R e m a r k : The r i n g - i d e a l s i d e a l s o f Nc,

3.

I' o f

Nc c o i n c i d e w i t h t h e c o m p o s i t i o n

because i c c = i f o r a l l i E I' and c € N C .

I f I'<(Nc,+,.),

t h e n [ I l l : =n{ ENlnoc 61' vctNc]QN:

F o r a l l i , j t [ I ' ] ,n t N we a e t : ( i - j ) o c = i o c - j o c E I , h e n c e i - j f [ I ' l , ( i ~ n ) o c = i o ( n o c ) = i o Ec ' I ' w i t h c ' E N ~ , h e n c e i o n € [ I 1 ] , (i.n)oc=(ioc).(noc)=(io ).c'E I ' ,

-

h e n c e i . n E [ I 'a n]d s i m i l a r l y

n . i E [ I l l . We l i s t some p r o p e r t i e s o f t h e s e t w o i d e a l s : L e t I ' , J ' d e n o t e i d e a l s o f Nc: 4. [ I ' I c = I ' I f i E. [ I l l c ,t h e n i t [ I 1 n N c y S O i = i o c e 1 ' . I f i 6 1 ' 3 Nc, t h e n i o c = € 1 ' f o r a l l c e N c , hence i e [ I ' ] .

Maximal ideals in near-rings

149

5 . I f I ' c J ' , t h e n [ I ' ] c [ J ' l and ( I ' ) C ( J ' ) a n d v i c e v e r s a . L e t i e [ I ' I , t h e n i o c t I ' C J ' Y c6NC, h e n c e i t ' [ J ' l . I f i e ( ~ l ) t, h e n i = x p i o j i w i t h jisO mod I ' c J ' , S O jizO mod J ' and p i o j ~ p i o O = O mod J ' and i e ( J ' ) . 6. [ I ' n J ' l = [ I ' l n [ J ' l and ( I ' n J ' ) = ( I ' ) n ( J ' ) . T H E O R E M 3 : For a l l i d e a l s I SN t h e r e e x i s t s a n u n i q u e l y d e t e r minded i d e a l I , $ ( N C , + , . ) , such t h a t ( I , ) S I C [ I c ] . I c i s c a l l e d t h e e n c e o b i n g i d e a L of I . PI LOO,^: We s e t I c = I nNc. If i € ( I c ) , then i = 7 p i o j i w i t h j i E I c and p i t . P o ( N ) . finite we g e t from jizO mod I a l s o p i o j i ~ p i o O = O mod I , Since Ic 5 1 hence i € 1 . I f i € 1 , t h e n i , C C I n N c = I c f o r a l l c eNC, h e n c e i & [ I c ] . I f K i s a n o t h e r s u c h an i d e a l w i t h ( K ) 5 I 5 [ K ] , t h e n (K) 2 [Ic: a n d eNc, h e n c e K G I , . f o r a l l k 6 K we g e t k = k o c c I C ,

v

From (I,)c[K]

we get I c 5 K in a similar way, hence K=Ic.

C O R U L L A R Y 4 : The i d e a l l a t t i c e of N i s a homomorphic image o f t h e ideal l a t t i c e of N . P/zo06: We s e t : h ( I ) = I c f o r I 5 N . a ) h ( 1 ) i s u n i q u e l y d e t e r m i n e d by Theorem 3 . b ) h ( I n J ) = (I r\ J ) c = (I r \ J ) ' \ N c = ( I oNc) A ( J o N c ) = I c A J c = h ( I ) n h ( J ) c ) h ( I t J ) = (I t J ) n N c = ( I n N c ) t ( Jr l N c ) = I c + J c = h ( I ) t h ( J ) .

To g e t a l l maximal i d e a l s , we assume f o r N : ( i ) ( N , . ) h a s an i d e n t i t y l c " N c . or ( i i ) N, g e n e r a t e s N , i . e . ( N c ) = N . T H E O R E M 4 : All maximal i d e a l s M of ( N , t , . , o ) a r e o i v e n by t h e i d e a l s M = [ M ' I , where M ' i s a maximal i d e a l o f ( N c , + , . ) . Phoo6: a ) I f M ' i s maximal i n N c , t h e n [ M ' ] i s maximal i n N : F i r s t [ M ' ] + N , otherwise [M'],=M'=N,. I f t h e r e e x i s t s a proper ideal B B N with N 3 B 7 [ M ' l , then B c 3 M ' , o t h e r w i s e B C [ B c l C [ M ' l , so Bc=N. By p r o p e r t y ( i ) : l c € B c & B and n = l c . n E B V n E N , s o B = N . By p r o p e r t y ( i i ) : I f B c = N c , t h e n N = ( B c ) C B , s o B = N . otherb ) I f M i s maximal i n N , t h e n M c i s maximal i n N c : M c 9 N , w i s e we g e t b y s i m i l a r a r g u m e n t s a s i n a ) : M = N . I f t h e r e e x i s t s a proper i d e a l A 4 N C with McC A CNc, then M E [McI c [ A I c [ N c I = N i n

H. Kautschitsch

150

c o n t r a d i c t i o n t o t h e m a x i m a l i t y o f M i n N.

C O R O L L A R Y 5: A l l m a x i m a l i d e a l s M o f t h e p o l y n o m i a l compos i t i o n - r i n g (R[X],+,.,~) o v e r a commutative r i n g w i t h i d e n t i t y a r e g i v e n by t h e i d e a l s M=(M':R), whereM' i s a maximal r i n g i d e a l o f R.

C O R O L L A R Y 6: to

I f N has o n l y t h e t r i v i a l a n n i h i l a t o r w i t h r e s p e c t

and g e n e r a t e s N o r possesses an i d e n t i t y w i t h r e s p e c t t o

m u l t i p l i c a t i o n , then t h e composition r i n g N i s a s u b d i r e c t product o f simple composition-rings

i f f N,

i s a subdirect

product o f

simple rings. Ph006:

Rad N = [ R a d ( N c ) ] b y Theorem 4 a n d P r o p e r t y 6, s o Rad N = O o t h e r w i s e t h e r e e x i s t s a n nlO w i t h noc=O v c eNC.

i f f Rad(Nc)=O,

C O R O L L A R Y 7: The p o l y n o m i a l c o m p o s i t i o n - r i n g ( R [ x ] , + , . , o ) an i n f i n i t e i n t e g r a l domain i s a s u b d i r e c t composition-rings

over

product o f simple

i f f R i s a subdirect product o f f i e l d s .

REFERENCES

[ I ] J e n n i n g s , A.S., [2] [31

[41 [51

S u b s t i t u t i o n g r o u p s o f f o r m a l power s e r i e s , 6(1954), 325-340. Canad.J.Math. K a u t s c h i t s c h , H. a n d M u l l e r , W.B., Ideale i n Kompositionsringen formaler Potenzreihen m i t nilpotenten Anfangskoeffi34(1980), 517-525. z i e n t e n , Arch.d.Math. L a u s c h , H. and N o b a u e r , W . , A l g e b r a o f P o l y n o m i a l s ( N o r t h 1973). H o l l a n d , Amsterdam, P i l z , G., N e a r - r i n g s ( N o r t h - H o l l a n d , Amsterdam, 1977). P i l z , G., N e a r - r i n g s , w h a t t h e y a r e a n d w h a t t h e y a r e g o o d f o r , C o n t e m p o r a r y M a t h . , 9(1982), 97-119.

Near-rings and Near-fields, G . Betsch (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1987

151

D. G. NEAR-RINGS ON THE INFINITE DIHEDRAL GROUP S.J. MAHMOOD and J.D.P. MELDRUM Department of Mathematics, University of Edinburgh, King’s Buildings, Mayfield Road, Edinburgh, EH9 3JZ. Scotland.

Department of Mathematics, University Studies Centre for Girls, King Saud University, P. 0. Box 22452, Riyadh, Saudi Arabia.

In this paper, we look at the behaviour of a l l the d. g. near-rings that can be defined on the infinite dihedral group. We are mainly interested in determining which d. g. near-rings are faithful and in characterizing the lower and upper faithful d. g. near-rings for those which are not faithful. The upper and lower defects are also found in all these cases and in three cases they turn out to be not equal. 1. INTRODUCTION.

This paper is concerned with the d. g. near-rings on the infinite dihedral group. These have been determined by J. J. Malone 121. Using the techniques of J. D. P. Meldrum

[Sl, we determine the upper and lower faithful d. g. near-rings for

[51 and

near-rings and two sets of generators for each d. g. near-ring.

a l l the d. g.

Following [Sl.we also

determine the upper and lower defects and defects in a l l these cases. Before giving more details, we give a summary of the definitions.

We write maps on the right and hence we use left near-rings. As a general source for definitions and results we refer to J. D. P. Meldrum [71.especially chapter 13. See also G. Pilz [8l for a more complete survey of near-rings.

(R,+,.) is a left zero-symmetric

near-ring if (R,+) is a group with neutral element 0, (R,.) for a l l x,y,z

E

R and Ox = x0 = 0 for a l l x

Mo(G) := {f: G If s

E

+

G; 0Gf =

E

i s a semigroup, x(y+z) = xy + xz

R. The standard example of such a near-ring i s

OG}, and all near-rings are embeddable in Mo(G) for a suitable G.

R satisfies (x+y)s = xs + ys for all x,y E R then s is called distributive.

R is

distributively generated (d. 9.) if there exists a semigroup (S,.) contained in R such that (R,+) = Gp<S>.

End G, the semigroup of all endomorphisms of G, forms a distributive

semigroup in Mo(G) and generates the d. g. near-ring (E(G),End G). We always write d. g. near-rings

in the form (R,S)

as which semigroup S

is taken t o be the distributive

generating set for R is very important. Let

8: (R,S)

+

(RS),

(T,U) be two

(T,U) such that

d.

g.

near-rings.

a

Then

near-ring

homomorphism

SO 5 U is called a d. g. (near-ring) homomorphism. We say that

(R,S) is faithful if there is a d. g. rnonomorphism 8: (R,S)

+

(E(G),End G) for some group

G. Not a l l d. g. near-rings are faithful, but to each d. g. near-ring (RS) are associated two canonical faithful d. g. near-rings. near-ring

(&S) and a

The lower faithful d. g. near-ring for (R,S) i s a d. g.

d. g. epimorphism 8: (RS)

+

(R.2) such

that S8 =

2,and

for any d. g.

152

S.J. Mahmood and J.D.P. Meldrum

4: (R,S)

homomorphism

+

(T,U)

where

(T,U)

is faithful, there exists a unique d. g

6: (R.3)+ (T,U) such that 41 = 0$. The upper faithful' d. g. near-ring for * * * * ( R S ) is a d. g. near-ring (R ,S ) and a d. g. epimorphism 0: (R ,S ) + (R,S) such that 01s is an isomorphism of semigroups, and for any d. g. homomorphism 4: (T,U) + (R,S) where (T,U) is faithful, there exists a unique d. 9. homomorphism 9: (T,U) + (R*,S*) such that homomorphism

4

=

$0. For these results see Meldrum [31 for the prehistory and Mahmood [11 for the

details. There is one minor change of notation from the earlier papers, involving the upper To end t h e definitions we define g(R,S) := Ker((R,S)+(R,S)) and

faithful d. g. near-ring. D*(R,S) := Ker((R

* * ,S )+(R,S)),

the

lower

and

upper

defects

for

(R,S)

respectively.

D(R,S) := Ker((R*,S*)+(R,S)) is the defect for (R,S).

We need the idea of presentations from group theory. For each set X, we can define Fr(X), the free group o n X. Let R b e a set of elements in Fr(X), N the normal subgroup of Fr(X) generated G = Gp<X;R>.

by

R. Then

G := Fr(X)/N

We write <X;R>

is

for Gp<X:R>.

said

to

be given

by the

presentation

Presentations are far from unique: for a

given group G , different generating sets X can be chosen, and for each X different sets R of defining relations can be used. If we start with a presentation R = c S ; Y > for tho d. g near-ring ( R S ) we wish to obtain presentations of R * = <S*;Y*>, and lower faithful d. g. near-rings. multiplication in S

*

and

2

group we start with is D,

Multiplication in R

*

and

5

=

<s;y> for the upper

i s defined completely by the

(12.7, [71),which in turn is given by that in S. In this paper the the infinite dihedral group. J. J. Malone [2] has determined all

the d. g. near-rings (f3.S) such that (R,+) is an infinite dihedral group.

For those that are

not faithful, w e determine the upper and lower faithful d. g. near-ring for t w o sets S of generators, and the lower faithful d. g. near-ring in a number of other cases. We cover the lower faithful d. g. near-ring first, in section 2, then the upper faithful d. g. near-ring, in section 3. Finally w e consider defects in section 4. Before w e leave this section we quote the results we need from Malone 121. We write (R.+) as the infinite dihedral group, with what is probably the standard presentation

From [2] we know that there are six d. g. near-rings defined on (R,+): (i) Ro, the zero d. g. near-ring on (R,+).

This i s a distributive near-ring,

i. e. all

elements are distributive. (ii) R1, with multiplication determined by the following table

-y b

b

b

This near-ring is also distributive. (iii) R2, with

multiplication determined

distributive near-ring.

by the following table,

again

giving a

153

D.g.near-rings on the infinite dihedral group

-

a

b

a

b

O

b

0

0

(iv) R3. with multiplication determined by the following table, another distributive near-ring.

(v) R4, with multiplication given by

xy = y if x t G p < a > ,

= O i f x Gp ~ The distributive elements are precisely the elements of finite order (vi) R5, with multiplication given by

xy = y if x tGp, = 0 if x E Gp

The distributive elements are again precisely the elements of finite order. Note that both R4 and R 5 have left identities, namely any element not in G p i a > for R4. any element not in Gp

for R5. So they are both faithful (13.12, 171). For the

rest of the paper, therefore, w e will only be interested in Ro, R1, R2 and R3.

2 . THE LOWER FAITHFUL D. G. NEAR-RING. We first have t o establish the method w e will use. Essentially we are using a form of the method used in section 3 of [51, based on theorems 2.5 and 2.6 of that paper, and on lemma 4.7 of [31. See also theorem 13.32 of [71. We adapt these results as follows. Let ( R S ) be a d. 9. near-ring without identity. Without loss of generality, assume that

(R,S)is not faithful, and has lower faithful d. g. near-ring

(R.2). Adjoin an identity t o (5,S) { S } U (1). Then (T,+) is

as described in theorem 13.32 of [71 t o obtain (T,U) say, where U = the free product of (+ ,!)

(T,+) = + (,!)

and a free group on the single element 1. say

< 1>

In fact the free product, denoted by *, can be taken t o be in any variety t o which (R,+) belongs. Which variety we choose does not affect the argument. From the construction of

(R,?),

it follows that (T,+) is the free (R,S)-module on the element 1 (lemma 13.20 and

theorem

13.21

of

[7]).

Let

N = Ker 8,

where

8 : (R,S)

+

(5,s).

is

the

canonical

S.J. Mahmood and J. D. P. Meldrurn

154

. . .,

homomorphism. Then w(s1, lw(s1, . . ., sn) = 0 i n (T,+). w(s1,. .

., sn)

such

Sn), a word in the elements of S, is in N precisely when

Using theorems 2.5 and 2.6 of 151, we find those words

that

wfsl, . . ., sn) E (T’,+)r,

where

r E X

for

a

presentation

(R,+) = <S;X>, and (T’,+) = (R,+)

* < 1>.

We add these words to the relations in the presentation of (R,+).

We then repeat the

process starting now with the new presentation. Eventually we reach a presentation for This is the method described in theorem 2.6 of [51. In practice, by judicious choice

(!,+).

of the relations t o add on in the first stage, the process takes only two steps, not a possibly infinite number. Familiarity with the details of the structure of the infinite dihedral group is assumed. For those who are not familiar with it, any reasonable text-book or tame group theorist colleague will supply the necessary information. We will start by listing some properties of the four d. 9. near-rings (Ri,S), 0

5 i 5 3,

which are easily deducible, and are some help in

subsequent work, though not all of them are completely necessary for what we do later. We use the presentation given in (1.1).

To show that Rj, for j

= 0.1.2.3, is not faithful, we use (R

* *

,S ) The definition of

R* i s

given by theorem 2.2 of [61 or theorem 13.26 of [71, namely for each s E S, define sd as an endomorphism of (T*,+) := (R,+) *

< 1 > by ls* =

s

E

R, rs* = rs, and extending, as we may

by the properties of free products. Then the d. g. near-ring generated by S

*

*

in Mo(T ) is

(R*,S*). Lemma 2.1.

* *

Using the presentation ( l . l ) , we have that a

+

b

+

a

+

b i s not a relation

in(Rj ,S ) f o r j = 0,1,2,3.

Proof.

We just need to find g E T* such that g(a+b+a+b) # 0 in (R,*,+)

For Ro, let g = 1 + 1. Then

* * * *

g(a +b +a +b ) = 2a + 2b + 2a + 2b = 4a.

For R1, let g = 1 + a

-

1. Then

g(a*+b*+a*+b*) = a

+

b - a

+

b

+

b

-b

+

a + b

-a+b

=a+b-a-b+a+b-a-b=4a.

For R2, let g = 1 + a. Then

g(a*+b*+a*+b*) = a

+

b

For R3, let g = 1 + b. Then

+

b + 0 + a + b + b + 0 = 2a.

+ b

- b

155

D.g.nearrings on the infinite dihedral group * * * *

g(a +b +a +b ) = a + 0 + b

+

b + a + 0 + b + b = 2a.

Corollary 2.2. (R,,S) is not faifhful forj = Proof. This follows directly from lemma

*

0.1.2.3. 2.1, since a + b

+

a + b is a relation in (Rj,+),

but not in (R, .+). The normal subgroups of D,

are well-known.

This enables us t o determine all the

ideals of Rj, for j = 0.1.2.3. First we list the normal subgroups of Dm. This list can be found in [21 or in [41.

Lemma 2.3. The normal subgroups of@,+) given by (1.1) are R, (0) and N1 := <2a,a+b>, N2 := i 2 a , b > ,

hA := , f o r h

2.4.

Corollary h

2

1.

The complere set of ideals of (R0,S) is given b y {Ro, N1, N2, hA;

2 0). Proof. This follows immediately from lemma 2.3 since in Ro, a l l products are 0. Theorem 2.5. (i) The complete ser of ideals of (R1.S) is given by (R1, N1, N2, hA; h an

even inregel). (ii) The complete set of ideals of (R2.S) is given by {Rg, N2, A, hA; h an even integelj. (iii) The complete set of ideals o f ( R 3 , S ) is given by(R3, N2, hA; h

2 0).

Proof. All three near-rings are distributive. So the ideals are normal subgroups H such that RjH

5 H.

HR,

5H

for each R,. The results are then found by a simple checking

process, using the l i s t i n lemma 2.3. Before w e start o n the determination of lower faithful near-rings, we remark that R/c2a> is an abelian group of order 4. Theorem 2.6. R,/

f o r j = 0,1,2,3, is always faithful, being a ring

Proof. This follows since a d. g. near-ring with an abelian group structure is a ring (Corollary 9.26 of [71). Another remark that needs t o be made is that since Rj, j = 0,1,2,3, is distributive, the semigroup S of distributive generators can be taken t o be any multiplicative semigroup that generates R. For each near-ring, we find the lower faithful d. g. near-ring for a number of different generating semigroups. Theorem 2.7. (i) lf S

5

Consider (R0.S). the zero d. g. near-ring on the infinite dihedral group

the set of elements of order 2, then (R0,S) is faithful

(ii) /f S = {a,b). then (I3o.S) is (Ro/<4a>,S),

the zero d. g. near-ring on the dihedral

group of order 8. (iii) l f S > {a,b) end S contains only one element of order 2, namely b, then (R0.S) is the same as in (ii). (iv) l f S

2

{a.b,a+b}, then

(50,s)is (R0/<2a >,S), the zero

ring on the group of order 4,

exponent 2. Proof. We use lemma 3.3 of [51. presentation

(R,+) = <S,X>

is

This states that (R,S) is faithful if and only if the

integrally

closed,

i.

e.

if

w(s1,

. . ., Sn)

E X

then

S.J. Mahmood and J.D.I! Meldrum

156

w(ms1,

. . ., ms,)

is a relation in R for a l l integers m. If S consists entirely of elements of

order 2, this is automatically satisfied, as w(ms1, w(s1,

. . ., sn)

. . .,

ms,)

= 0 if

m is even, and is

5 S.

The second example

if m is odd. This proves (i).

To proceed, w e note that (l+l)(a+b+a+b) = a + a + b + b + a + a + b + b = 4a So 4a must b e a relation in (@)

f o r j = 1,2,3, as long as {a.b}

on p. 55 of 151 gives us (ii). In (iii), S = (a,nja,b} for i E I. nja + b + nia + b, nia

- a -...- a

The relations are a

+

b

+

a

+

b.

o r nia + a + . . . + a, with an appropriate number of

summands, depending on whether n i is positive or negative. As for the second example on p. 55 of [51, we get again that the lower faithful d. g. near-ring is the same in case (iii)

as in case (ii). Finally we deal with case (iv). We are now in the position described in lemma 13.7 of [71. So in

(RO,~)w e

b = b + 5.

must have g +

This is sufficient t o prove the result.

We now turn t o R1. From theorem 2.5, we know that the ideals of R 1 are R1, N1, N2 and hA for all even integers h. The following result simplifies some calculations. Lemma 2.8. Ann(R1) = N1. -This result is immediate from the multiplication table.

Now a bit of notation which

will be used consistently in the rest of the section. Notation. An arbitrary word in (T',+) is of the form w := n i l + r1 + n21

+

, , ,

where n1 may be 0, rs may be 0, ni

+ n s l + rs E

2, ri E R1, for 1 5 i 5 s. Write

n : = n 1 + . . . + ns

Theorem 2.9. (R1.S) is faithful, where S = {b,a+b}. Proof. x := Q, y :=

A presentation for (R1,+) is given by R1 =
a+b.

b+b, a+b+a+b>.

Write

Consider w(x+x). Then

w(x+x) = n l b + r l b + . .

,

+nsb+rsb+nlb+rlb+.

. . +n,b+rsb

= 2(nl+m1+ , . , +ns+ms)b = 0

where rib = mib, m i E 2, f o r 1

i

5 s, from

the multiplication table. Now consider w(y+y).

Then

w(y+y) = nl(a+b)+O+ . . . +ns(a+b)+O+nl(a+b)+O+. . . +ns(a+b)+O = 2n(a+b) =

using lemma 2.8. Thus required.

0,

no new relations are introduced and so (Rl,{b,a+b}) is faithful, as

157

D.g.near-rings o n fhe infinite dihedral group Lemma 2.10.

Let (R1.S) be a d. g. near-ring. where 2ra + b

S. Then 4rg is a relation

6

in (Fl1.2).

Proof. Let

x := 2ra + b. Then x + x is a relation in (R1.f).

Consider (l+b)(x+x). We

obtain (l+b)(x+x) = 2ra +

& + b + 2ra + b + 5 = 4ra.

This suffices t o prove the result. We can use this in several cases. First we look a t the situation in which S consists of two elements of order 2.

Theorem 2.11.

Lets = {2ra+b.(Zrrl)a+b}.

Then (El,?) = Rl/<4ra>.

-

Proof. Lemma 2.10 shows that 4ra is a relation in (R1,+). Let x := 2ra

( 2 r i l ) a + b. Then wx = nl(2ra+5) 4r5 = 0,

it

m = mi +

follows

...

that

+ m,.

+

mlb

+

2ra commutes

.

,

. + ns(2ra+b)

with

b.

+

+ b, y :=

m&, where rix = mib. Since

Hence wx = n2ra + (n+m)b where

Also wy = nl((Zr*l)a+b) + 0 + . . . + ns((2ril)a+b) = n((Zr+l)a+b). We

now have w(x+x) = 2(nZrg+(n+m)b) = 4nra + 2(n+m)b = 0, w(y+y) = 2n((2rtl)g+b) = 0. Thus we have a new relation. namely 4r(x+y). Then

w(4r(x+y)) = 4r(wx+wy)

But in R1/<4ra>

all elements have order dividing 4r.

Hence w[4r(x+y)) = 0.

Thus

R1/<4ra> is faithful. Next we consider the case in which

S

= {b.a+b.Zra+b}.

Denote b by x, a+b by y and

Zra+b by z.

~Theorem 2.12. Let S = {b,a+b,2ra+b}.

Then (51.2) = Rl/.

Proof. Again lemma 2.10 shows that 4ra i s a relation in (El,+), and so 2ra commutes with

b.

By the above work, we know that the relations x + x, y + y and L + z do not

introduce new relations into (el,+). But we have also a relation in (R1,+) given by 2r(y+x)

+

x + z. Consider w(Zr(y+x)+x+z).

w(Zr(y+x)+x+z) = 2r(wy+wx) + wx + wz

= 2r(n(g+k)+(n+m)b) + (n+m)b + n2ra + (n+m)b where n = n1 + .

..+

ns, mb = ( r l +

. . . +rs)b

= (rl+

,

. . +rs)(2ra+b), since

We use a similar argument to that used in the proof of theorem 2.11.

(2.13) 2ra E Ann(R1). We divide the

consideration of this element into four cases, depending on whether n or m are even or Odd. (i) n even, m even. Then (2.13) is 0 as 4rg = 0. (ii)n even, m odd. Then (2.13) becomes 2rb +

+

b = 0.

S.J. Mahmood and J. D.P.Meldrum

158

(iii) n odd, m even. Then (2.13) becomes 2r(a+b+b) +

b+

2rg +

b=

2rg +

b+

2rg +

b=

0.

(iv) n odd, m odd. Then (2.13) becomes 2r(a+b) + 2rg = 2ra. Hence 2ra is a relation in 51. Having added this relation w e can repeat the process. The new relation is 2r(y+x). So w(2r(y+x)) = Zr(wy+wx). But in R1/<2ra> order dividing 2r. Hence w(2(y+x)) = 0 always. Thus R1/<2ra> Theorem 2.14.

L e t s = {b,a+b,(2r+l)a+b}.

all elements have

is faithful.

Then (51.2) = R1/<2ra>.

Proof. From the proof of theorem 2.9 we know that b + b and a + b + a + b do not

introduce new relations. As for a

+

b

does not introduce a new relation.

+

a

+

b, we can check that (2r+l)a + b

So we consider (2r+l)(y+x)

+

x

+

(2r+l)a + b

z, where x := b,

+

y := a + b and z := (2r+l)a + b. With n and m defined as in theorem 2.12, w e obtain, since

z

E

Ann(R1),

(2.15)

We consider separately four cases as in theorem 2.12. (i) n even,

m even. Then (2.15)

i s 0.

b=

(ii) n even, m odd. Then (2.15) becomes ( 2 r + l ) b +

0.

(iii) n odd, m even. Then (2.15) becomes (2r+l)(a+b+b)

= a

b=

+ (2r+l)g +

+

(iv) n odd, m odd. Then (2.15) becomes (2r+l)(g+b) + (2r+l)g

+

b=g

+

0.

Q + (2r+l)g + b

- (2r+l)g = -2ra. Hence 2ra is a relation in 51. Having added this relation w e can repeat the process.

The new relation is 2r(y+x). R1/<2ra>

As in theorem 2.12, this introduces no new relations.

Thus

is faithful.

There are a number of other results giving the lower faithful d. g. near-ring for (R1,S) where S consists entirely of elements o f order 2. They are somewhat more complicated than the previous cases. So w e list them without proof.

-Result 2.16. /f(r-r')/t

1. 1.

(i) Lets = {Zra+b,(2rfl)a+b,Zr'a+b}, with r # r', r,r'

is even, then(fl1,Z) = R1/<4ta>.

/f(r-r')/t

L e t t = H.C.F.(r,r').

is odd, then (FIl,S) = R1/<2ta>.

(ii) Let S = {2ra+b,(2r+l)a+b,(2r'+l)a+b}, with r # r', r,r'

Let t = H.C.F.(4r,2r-2r').

1.

Then (fi1,S) = R l / < t a > . (iii) Let S = {(2r-l)a+b,2ra+b,(2r'+l)a+b},

with r # r', r,r'

2

1. L e t t = H.C.F.(4r,2(r-r'-l)).

Th8n (51.5) = R l / < t a > . We now turn t o generating sets involving an element o f infinite order.

= R/<4a>.

Theorem 2.17. L s t S = (a,ra+b}, where r E. Z. Then (F11.3)

Proof. Write x :=a, y := ra + b. Then the relations in a presentation of R1 are y + y,

x + y + x + y. By considering (l+l)(x+y+x+y) = 2a + 2(ra+b) + 2a + 2(ra+b) = 4a, w e see that 4a is a relation i n !41. We f i r s t find wx and wy: wx = n l a ny, if y E N1, wy = n l y

+

mlb + . .

,

+ ny,

+

msb if y

Since 4g = 0, it follows that 25 commutes with

b.

4 N1.

+

mlb

+

-

. . . + nsg + msb, wy $ N1, then r is even.

Now if y

So wy = n y +

mb

if y

4

N1. Consider

159

D.g.nearrings on the infinite dihedral group

w(y+y). If r is odd, then y E N1, so w(y+y) = 2ny = 0 since y has order 2. If r is even, then w(2y) = 2(ny+mb). The possible values of ny + mb are ra. ra +

k, b,

0. Since r is even

Z(ny+mb) i s zero in a l l cases as 45 = 0.

look

Now

2(nla+m@+

at

w(x+y+x+y).

If

. , . +nsg+msb+n(ra+b)).

y

N1,

E

Suppose

r

then

n

is

is

odd,

even.

and

Then

w(x+y+x+y)

this

=

reduces

to

2(nla+mlb+ . . . +nsg+msb) and n = n1 + .

. . + ns. If m i + . . , + m, is odd, this is of the form Z(qZ+b) which i s 0. If m i + , . . + m s is even, this is of the form . , +Esnsg) = 2(n1+~2n2+., . +Esns)a where c i = f l for 2 i1 .s. But 2(nla+c2n2g+, n l + . , . + ns even forces n1 + c2n2 + . . . + cSnS even. So w(x+y+x+y) reduces to a multiple of

43

hence

is 0.

Now

suppose

. . . +nsa+msb+rg+b). Since n1 + 2(nla+mlb+ -

..

n is

that

odd.

Then

can repeat the above process to show that this is zero. Now l e t y and

w(x+y+x+y) =

- 2(nla+mlb+

.

2(nlg+rnlb+.

.

. +nsa+msb+ny+mb).

If

n

is

The

4 N1.

even, this

coefficient

m l + . . . + rns + 1 + m = 2m + 1. So this element i s of the form 2(qa + This covers a l l cases for the relation x + y The final relation is 4 3 i. e. 4x.

+

x

+

Then r is even becomes

If n i s odd, then this

. . +nsa+(ms+m)tj and can be treated as above.

2(nlg+m1Q+ . . . +nsa+msb+ra+b+mb).

becomes

w(x+y+x+y) =

. + ns + r is even (as n and r are odd), we

b

of

b)

is

which i s 0.

y.

But w(x+x+x+x) = 4wx = 0, since in R1/<4a>

all

elements have order divisible by 4. Thus no new relations are introduced and we have our result. Our final look at d. g . near-rings on R1 i s a t the case where Theorem 2.18.

Let

S = [a,b,ra), for r > 1.

(r-1)/2 is also odd, o r r is even andr/2 is odd

S

= {a,b,ra}, for r

m e n (Ft1.S) is R1/<2a>

>

1.

if r is odd and

Otherwise (El,?) isR1/c4a>.

Proof. Let x := a, y := b, z := ra. As at the beginning of the proof of theorem 2.17 we see that 45 is a relation in y + y, x + y + x + y, rx - z.

nlg+mlb+. nlz+m$+.

El.

The relations in a presentation for R 1 can be taken as

We first find what effect x, y and

. . +nsg+msE, wy = (n+m)k, as in . . +nS;+m& if L 4N1, i. e. r is odd,

z have on w: wx

=

the proof of theorem 2.17, and wz = and

WL

= nz if z E N1, i. 8. r is even. We

consider the innocuous relations first. We have w(y+y) = 2(n+m)F = 0 in a l l cases. Also w(x+y+x+y) = 2(nlg+mlh+. . . +nsa+msE+(n+m)b). This is of the form 2(qa+b) = 0 unless n i s even since m l

+

. ..+

ms = m. But then an argument similar t o that in the proof of

theorem 2.17 shows that we get 0.

Now we deal with rx

- z.

We have w(rx-z) = r(nig+rnib+

. . . +nsa+msb) -

wz. If r is

even then we have r(nla+mlb+ . . . +nsg+msb) - nre. If 4(r, then this i s 0 as R1/<4a>

has

exponent 4. If 2)r but r/2 is odd, then take n l = 1, rnl = 1, s = 1. We get r(a+b) - rg = -ra as r is even. But 4a = 0, -ra = 0 implies tg = 0, where t = H.C.F.(4,r) = 2, by hypothesis. So R1/<2a>

is (El,?) as it i s a ring (theorem 2.6).

r(nlg+rnlb+ . . . +nsg+mse)

-

Now let r be odd.

Then w(rx-z) =

. +nsra+msb). But nlg+mlh+ . . . +nsa+msb = n l + ~ 2 n p+ . . .. +ESnS for some choice of e i = fl,

(nirg+mib+. .

b, where m l + . . . + m, = m, q = 5 i 5 s. Then n1ratrnlb+ . . +nsrg+rnsb = qra + mb. So w(rx-z) = r(qi+h) - (qra+mb). If m is odd this is r(qe+b) + qra + b = 95 + b + qrg + b = q(1-r)g. By taking n1 = 1 = m l , qa +

2

,

160

S.J.Mahmood and J.D.P. Meldrum

s = 1, we get (1-r)a

0. If (l-r)/Z is odd, as before this leads t o R1/<2a>

=

= (I31.S).

If

41(r-l), then this is always zero. If m is even w(rx-z) becomes rqe - qra = 0 for all choices of q. The final check is whether 42 introduces new relations in the t w o cases 41r, 4\(r-l). has exponent 4, no new relations are introduced.

But as before, since R1/<4a>

This finishes our look at the lower faithful d. g. near-rings on R1. We turn t o R 2 next. Fortunately the work i s much easier here. The first result parallels lemma 2.8. Lemma 2.19. --

Ann(R2) = N2.

Again this is obvious from the multiplication table. The following result is useful.

-Lemma 2.20.

Ler (R2.S) be a d. g. near-ring

/f r is odd and ra + b E S, rhen 2rg = 0

in R2.

Proof. Put x b + rb + ra + b

.= r a + b. Then x + x is a relation.

So (l+a)(x+x) = x + ax + x

+

ax = r a +

rb = 2ra since b + rb = 0.

+

~Theorem 2.21. / f a + b

E

S , then(R2.S) = R2/<2a>.

Proof. Use lemma 2.20 and theorem 2.6. If a E S. rhen (t3.5) = R2/€2a

Theorem 2.22.

>.

Proof. Note that S must contain at least one element of order 2, say r a + b. Put x := a, y := r a

+

b. Consider (l+l)(x+y+x+y). As before this forces 42 = 0. If r is odd, then

lemma 2.20 forces 2 r a

0, which together with 42 = 0 gives 22 = 0. Another appeal t o

=

theorem 2.6 gives the result.

x_

+

12 + y + 5 + b + y.

So assume that r is even.

since (l+a)x = 5

2(1-r)2 = 0. Together with 4 :

+

5,

(l+a)y = y.

= 0 we again get

25

=

Consider (l+a)(x+y+x+y) =

So (l+a)(x+y+x+y) = Z(g+b+ra+Q) =

0 and the result follows from theorem

2.6.

Finally we look at the case when S = {2ra+b,(Zrfl)a+b} and r > 0. This gives the only non-ring lower faithful d. g. near-ring for R2. Theorem 2.23. Let S = {2ra+b,(Zrfl)a+b}.

Proof. Put

t = 2rfl.

Then (R2.S) = R2/c2(2r*l)a>.

That 2ta = 0 follows from lemma 2.20. We just need t o show

that no more relations come in. Let x := 2ra + b, y := ta + b. The relations w e do have are x + x, y

+

y, 2t(x+y). Also wx = nx, wy = nl(ta+lJ

0. ta commutes with

b.

+

rnlh + . . . +

ns(ta+tJ + m&.

As 2tg =

So wy = ny + (n+m)Q. Further w(x+x) = 2n5 = 0 as 2x = 0. Also

w(y+v) = Z(ny+(n+m)b). If n + m is even, this is 2ny which is 0. If n + m is odd, we get either 2b (m odd, n even) or 2(y+p) = 2ta (m even, n odd). Finally w(2t(x+y)) = Ztw(x+y). R2/<2ta>

In either case w(y+y) = 0.

has exponent 2t. So w(2t(x+y)) = 0. This finishes

the proof. This brings us t o the final d. g. near-ring t o consider, namely R3. We first note the annihilator of R3. Lemma 2.24. --

Ann(R3) = A.

This is again obtained directly from the multiplication table. Lemma 2.25. --

/f r a + b E S, rhen 2ra is a relation in R3.

-

Proof. Consider (l+b)(ra+b+ra+b) = ra + b + b + ra + b + b 2ra, -Lemma 2.26. /f {a,ra+b) 5 S, then 2(r+a)a is a relation in R3.

since b(ra+b) = b.

161

D.g. near-rings on the infinite dihedral group Proof. Consider -

(l+b)(a+ra+b+a+ra+b) =

5 + 0 + rg

+

+

21

+

5

0 + rg +

+

+

=

2(r+l)g. This leads t o our final result in this section. Theorem 2.27.

/f either a

E

S or a + b

E

S or {ra+b,(r+l)a+b} 5 S, then (Fl3.5) =

R3/< 2a >. Proof. If a -

E

S, then S must contain an element of the form ra

+

b as well.

By

lemmas 2.25 and 2.26 it follows that 2rg and 2(r+l)g are relations, hence so is 2g. since H.C.F.(r,r+l) = 1. If a + b

E

S, then lemma 2.25 gives the result. Finally if {ra+b,(r+l)a+b}

S, then 2rg and 2(r+l)g are relations, hence so is

25

as before.

5

Just apply theorem 2.6

now.

3. THE UPPER FAITHFUL 0. G. NEAR-RINGS. We will now determine the upper faithful d. g. near-rings for Ro, R1, R 2 and R3 for two sets of generators in each case, namely S = (a.b} and S = (b,a+b}

This i s done as the

determination of upper faithful d. g. near-rings is harder than that of lower faithful d. g. near-rings. For (Ro,(b,a+b}) and (Rl,(b,a+b}), the upper faithful d. g. near-rings are RO and R 1 respectively.

So we only deal with the remaining cases.

The method we use for

finding upper faithful d. g. near-rings is that of theorem 2.2 of [61. A useful result in calculating upper faithful d. g. near-rings is theorem 13.12 of [71,

which says that a free group in any variety is faithful. Hence it follows that if (R,+) lies in a variety, then so does (R',+). We quote this as a lemma.

-Lemma 3.1. Let (R,S) be a d g. near-ring. N(R,+) lies

in a variety, then so does (R ,+).

An application of this to our case is as follows. Since (Ri,+) is metabelian, i. e. has

*

abelian derived group, then so i s (Ri ,+). cases we consider.

*

Also Ri is generated by two elements in all the

So (Ri ,+) will be a homomorphic image of the free metabelian group

on two generators, a group which is easy to handle.

*

In practice this means that to

determine (Ri ,+) we need only determine the orders of the two generators, x and y say, find out what [x,yl := -x

-y+

x + y, is. and what the conjugates of [x.yl are. We write

conjugates in the form xy := -y + x + y. We establish the following notation for the rest of this section.

Let T := (Dm,+)

*

<1>, be the free product of an infinite dihedral group given by (1.1) and an infinite cyclic group generated by 1. Since we will generally be identifying Dar with (Rj,+) for some j, we write a general element of T as w = n i l + r1 + where ni

E

Z, ri

E

. .. +

n s l + rs,

D-, for 1 5 i

5s

and the first and last elements may be zero, s

We remind the reader that theorem 2.2 of endomorphisms of T by 1 . s = s,

ri.s = ris

(3.2)

2

1.

[Sl shows that if we define the elements of S as

S.J. Mahmood and J. D.P. Meldrum

162

S generates

then

R,* in Mo(T). We start with (Ro,{a,b}).

Note that in Ro, we may assume

without loss of generality that w = n l since ris = 0 in a l l cases with Ro. Then (Ro*,S) is given by

Theorem 3.3. Let S = (a,b}.

Ro* =

*

Proof. We

first check that 2b is a relation in RO . But n l .(b+b) = 2nb = 0.

(Ro*,+) = , metabelian.

Hence

a has infinite order and b has order 2, and by lemma 3.1, Ro

Consider

m l a + b + mga + b + .

an ,

,

arbitrary

element

(Ro*,+).

of

2

+ mta + b + mt+la, where mi E Z, t

is

It

of

the

*

is

form

1, and m i and mt+1 may

be zero. Then it i s a relation if

nl(m1a + b +

... +

mta + b + mt+la) = mlna + nb + .

for a l l n. If n is even, this means n(m1 + has

infinite

mlna + b +

order,

...+

this

forces

. ..+

mi +

- m2na

..-

+ .

m l - m2 + . . . - mZq + mZq+l = 0.

have

., .+

+

mtna

nb

+

+

mt+lna = 0.

mt+l)a = 0 for a l l even values of n. As a . + m t + l = 0.

If

n

is

odd,

then

mtna + b + mt+lna = 0. This means that there must be an even number

of b's, i. e. t = 2q and m l n a mla + b +

, ,

..

mZqa + b + r n z q + l a

m2qna + mzq+ina = 0.

Conversely

satisfies

annihilates n l for all n. This is equivalent to

*

C$, m2i

easy

is

it

C i mi = 0

to

check

X i (-1)'mi = 0,

and

= 0,

As before we must

,C,:

that

if

then

it

m2i+l = 0. So the set of

relations for Ro is

We now check what happens to the commutators. All commutators are words of the + m t + l a in which El::

form Cl,,(mia+b)

mi = 0. A standard technique from group theory

shows that words in commutators are precisely words of this form. So a l l the relations in

Ro*,

other than

We now consider [a,bla - [a,bl

are in the derived group.

Zb,

-2a+b+a+b+a+b-a+b+a,i.e.m1 =-2,m2=m3=1,m4=-1.m5=

- [b.al

makes it easy to check that this is a relation. Also we look a t [a,blb

=

1. This

= 0, simply

using 2b = 0. This finally shows t h a t the presentation in the theorem is that of (Roil,+). W e now move o n t o R1. Here again we have only one case, namely S = {a,b}. Theorem 3.4. Let S = (a,b).

Then (Rl*,S) i s given by

R1* = . Proof. We first shows

that

mla + b +

nla

the

is

Let w E T as in (3.2). Consider

a

relation

in

R1

*

.

An

arbitrary

As 2b = 0, this

element

Of

R1

*

is

mta + b + mt+la = u as in theorem 3.3. Consider wu. First we point out multiplication

. . . + nsa + + (n+k)b + . . . +

klb +

ml(wa)

2b

...+

from +

.

Then w.2b = 2fb for some integer f. as all products lie in .

w.2b.

that

*

check that 2b is a relation in R1

table,

we

have

=

ria

rib

=

kib

ksb, wb = (n+k)b, where n = C i ni and k = mt(wa) + (n+k)b + rnt+l(wa).

If

n

+

k

is

say.

xi

ki.

even

So

wa

=

Hence wu = this

gives

D.g.near-rings on the infinite dihedral group (ml+.

. . +mt+l)(wa)

163

= 0. Take n1 = 3, n2 = 1, k l = 1 = k2. Then wa = 3a + b + a + b = 2a.

Hence we must have ( m l + . . . +rnt+l)2a = 0 and so C i mi = 0. Then wu = 0 for all cases in which n + k i s even. If n + k i s odd, this gives ml(wa) + b

...+

f

mt(wa)

b + mt+l(wa) = 0. Take "1 = 1,

f

k1 = 2. Then wa = a and we must have m l a + b +

. ..+

proof

t = 2q.

of

theorem

.

m2 + m4 + . wa

By

ml(Za+b) + b ml

. + mzq = 0.

.

E

+

this

gives

us

Then wu = 0 for n1 = 2,

taking

a l l cases

k l = 1,

we

ml

+

+

As in the

m t + l = 0.

m3 + .

in which

have

. . . + mzq(2a+b) + b + mZq+i(2a+b).

. . . + rnzq+l

f

3.3,

mta + b

..

+ mZqtl = 0,

n + k is odd and

wa = 2a

+

The coefficients

So

b. of

wu

=

b add up to

+ 2q = 2q. Hence wu will always lie in c a > in this case. We rewrite

wu as ml(2a+b) + m2(-2a+b) + . . . + mgq(-2a+b)

+

mgq+l(2a+b). Note that mi(i2a+b) = 0

if mi i s even and is i 2 a + b if mi i s odd. So wu reduces to CiEJ((-1)i-12a+b), where i f J

I:?,+'

if and only if mi i s odd. Since even, it follows that E, =

IJ( is

even, say 2f.

_+1. So wu = ~ 1 2 a- E22a + .

need ~1 - € 2

+

da(C,EJE,(-l)i-')

. .

.-

mi = 0, and we have left out a l l mi such that mi is

..

f

This can be rewritten as CjEJ(E,2a+b), where

~ 2 f - 1 2 a- ~ 2 f 2 a = 2 a ( E 1 - ~ 2 + . . - ~ 2 f ) . So we

~ 2 =f 0. But if wa

4

,

we have wa = da + b, say. Then wu =

= 0. We conclude that the full set of relations i s

where J i s defined by i f J if and only if mi is odd, and j is a dummy variable which enumerates the elements of J. We now check what happens to the commutators. subset of those in (Ro*,+).

relations are in the derived group. Let c := [a,bl. power of d is equal to a power of c. nd - n'c, -a

f

or

The relations in this case are a

Hence (Ro*,+) i s a homomorphic image of (R1*,+).

So all

Consider d := [a,bla. We show that no

If not then nd = n'c say, i. e. we have a relation

n(-Za-b+a+b+a) - n'(-a-b+a+b)

n(-2a+b+a+b+a) + n'(b-a+b+a)

=

n(-a+b+a+b) + a + n'(b-a+b+a). Without loss of generality we may assume that

=

n >

0.

We have to consider separately n' > 0 and n' < 0. First let n' > 0. Then we have a relation m l a + b which q = n + n', m1 = -2, mi = (-l)i for 2 as

can easily

So

be checked.

i

5 2n,

J = (2, . . .,2q+l},

relation m l a + b +

5 i 5 2n-1,

. ..

f

mzqa

+

b in which

mzn = 2, mi = (-l)i for 2n+l

q = n

...

f

mzqa

f

b

Now let n'

-

5 i 5 2q.

n ' - 1, m l

< 0. =

f

mZq+ia in

< i 5 2(n+n')+l,

for 2n

and (-l)i+J = (-l)i+i-l

Z i E ~ ( - l ) i +=j -2q # 0 and this cannot be a relation.

2

f

mi = (-l)i+'

= -1.

Thus

Then we have a

-2,

mi = (-1)'

for

But then all mi for i odd are

negative. Hence C Z , m2i-1 # 0, and this cannot be a relation. So c and d are independent elements. Consider cb = -b

-a-b+

a + b

+

b = -b

-

a + b + a = [b,al = -c. Also, since R1* is

metabelian. it follows that db = ca+b = cbfa = ( - c ) ~= -(ca) = -d. -2a

-

-c

The final consideration is

i s a relation to finish the proof. But da - c = ca+a - c = We a - b + a + b + 2a - ( - a - b + a + b) = -3a - b + a + b f 2a + b - a + b + a.

da, We wish t o show that da

S.J. Mahmood and J.D.P. Meldrum

164

get, using the general form for u, m l = -3, -3+2;1=0,

1-1=0, J = (1,2,4,5}.

1+ 1- 1

-

1 = 0. Hence da

m2 = 1, m3 = 2, m4 = -1,

So ,Z,c~(-l)i+i = (-l)’+’

- c is

So

m5 = 1.

+ (-1)2+2 + (-1)4+3 +

(-lp4

=

a relation.

The next near-ring to consider is (R2.S) with S = {a.b} and {b,a+b}.

We consider

(R2*,{a,b}) first. Theorem 3.5.

Ler

Then (R2*,S) is given b y

S = {a,b}.

R2* = ca,b; 2b. c := [a,bl, d := [a,bla, da = c, cb = -c, db = -d>.

Proof. We first

check that 2b i s a relation in R2*. Let w

E

T as in (3.2). Consider

w(b+b). From the multiplication table wb = nb. Hence w(b+b) E 2 = {0}, for all w We now consider wa.

1

i

1.s.

We can write wa = n l a + k l b + .

Then wa = ml(wa) + nb

+

. ..

..+

E

T.

nsa + ksb, where ria = kib,

+ mt(wa) + nb + mt+l(wa).

From now on the

proof follows a line very similar to that of theorem 3.4 and leads to the same additive structure.

*

Now we turn t o (R2 ,{a+b,b}).

We, perhaps surprisingly, obtain the same additive

structure as before. Theorem 3.6.

Lei S = {a+b,b}.

Wriiex := a + b, y := b.

Then (R2*,S) is given b y

R2* = <x.y; 2y, c := [x,~], d := [x,yIx. dX = c, cy = -c, dv = -d>. Proof. First we show that x has infinite order. that

wx = n l x + k l b + . . . + nSx + k,b.

nl(a+b) + k l b + . . . + ns(a+b) + ksb.

where

We start by calculating wx. rix = kib.

15i

1.s.

So

We see wx

=

Take n1 = 1, k l = 1 to obtain wx = a + b + b = a.

Hence w(fx) = f(wx) = fa is never 0 unless f = 0. Thus x has infinite order. Next we show that y has order 2.

But wy = nb as in theorem 3.5. Hence 2y = 0. We now see that

wu = ml(wx) + nb + . .

.+

mt(wx)

+

nb + mt+l(wx). The rest follows as in the proof of the

two previous theorems. The last d. g. near-ring that needs t o be considered is (R3.S). We look a t two cases, namely S = {a.b} and S = {a+b,b}. as we have mentioned earlier. To maintain continuity, we consider S = {a+b,b} first. Ler S = {a+b,b}.

Theorem 3.7.

Write x := a + b, y := b. Then (R3*,S) is given by

R3* = <x,y; 2y. c := [x,y], d := [x,yIX. dX = c, cy = -c, dy = -d>.

Proof.

We start by looking a t wx and wy: wx = n l x + k l b +

rjx = kib, 1 5 i 5 s. Also

WY

...

+ nsx + k,b,

where

As in the proof of theorem 3.6, this forces x to have infinite order.

= nib + klb + .

. . + nsb +

ksb E .

As in the proof of theorem 3.4, we get 2y

as a relation. The rest of the pattern should be familiar from the last three theorems. This brings us to the last of the upper faithful d. g. near-rings, namely (R3*,{a,b}). Theorem 3.8. Ler S = {a,b}.

men

R3* = ca,b; 2b, [a,bla = [a,b], [a,blb = [b,al>.

Proof. We can see that 2b i s a relation as in the proof of the previous theorem.

In

this case though, we have wa = na from the multiplication table. Again wb = (n+m)b, in the same way.

So wu = mlna

+

(n+m)b +

...

+ mtna + (n+m)b + mt+lna.

needs a slight mOdi?iC8tjOn of the proof of theorem 3.3 to finish this proof.

It now only

165

D.g. near-rings on the infinite dihedral group 4. DEFECTS.

In this last section we find the upper defect, lower defect and defect in the six cases in which the upper faithful d. g. near-ring has been found. Theorem 4.1. Lets = {a,b}.

Then

(i) Q(R0.S) is a zero ring on an infinite cyclic group,

*

(ii) D (R0.S) i s a zero ring on an infinite cyclic group, (iii) D(R0.S) is the direct sum of g(R0,S) and D*(Ro,S).

Proof.

From theorem 2.7, D(R0.S) i s c4a>. i. e. an infinite cyclic group.

From

theorem 3.3 we see that D*(Ro,S) = <[a,b] + 2az. again an infinite cvclic group. Finally we see, again from theorem 3.3, that 4a and [a.b] commute additively.

Hence the additive

structures of the three defects are as stated. The rest follows since all products are zero in Ro. Theorem 4.2.

Lets

= (a.b).

Then

(i)D(R1.S) is a zero ring on an infinite cyclic group,

*

(ii) D (R1.S) i s a zero ring on the direct sum of two infinite cyclic groups, (iii) D(R1.S) is a zero ring on the direct sum of three infinite cyclic groups. Proof. -

From

theorem

4na.4ma = 16nmb = 0.

2.17,

From

(Q(Ri,S).+)

theorem

3.4,

=

14a>.

(D*(Rl,S),+)

Products =

are

zero

<[a,bl + 2a. [a.bla

since +

2a>.

Since a l l products of a and b are equal to b. we see

Consider (-a+b+a+b)(-2a+b+a+b+a).

that the product above i s 24b = 0.

Similarly all other products are zero.

theorem 3.4, (D(RI,S),+) = c[a,b], [a,bIa, 4a>.

Again from

Since 2a commutes with [a,bl, it follows that

(D(Rl,S),+) is the direct sum of three infinite cyclic groups.

As above any product in

D(R1.S) results in an element of the form 2nb = 0. This finishes the proof. Theorem 4.3.

Lets = {a,b}.

Then

(i) D(R2.S) is a zero ring on an infinire cyclic group,

*

(ii) D (R2.S) i s a zero ring on the direct sum of two infinite cyclic groups, (iii) D(R2.S) is a zero ring on the direct sum of three infinite c y c k groups.

*

Proof. From theorem 2.22, (g(Rz,S).+) = C z a r and from theorem 3.5, (D (Rz,S).+) = <[a,b] + 2a, [a,bIa + 2a>. The rest follows as in the proof of theorem 4.2. Theorem 4.4.

Lets = {b,a+b}.

Then

(i) D(R2.S) is a zero ring on an infinite cyclic group,

*

(ii) D (R2,S) i s a zero nng on the direct sum of two infinite cyclic groups, (iii) D(R2.S) is

a zero ring on the direct sum of three infinite cyclic groups.

Proof. From theorem 2.21, (g(Rz,S),+)= <2a>. -

= 4nmb = 0.

All products are zero, since 2na.2ma

From theorem 3.6 we can see that, in the notation of that theorem,

(Dm(R2,S).+)= <2x, c+d>, as can be checked by writing these elements in terms of a and b and using the known structure of Da. theorem 3.6. In (R**,S)

2x commutes with both c and d as we see from

the products are given by x2 = y, y2 = xy = yx = 0. As 2y = 0, it

*

follows that, as before, D (R2.S) i s a zero ring. Finally, again from theorem 3.6, and in the notation of that theorem, (D(Rz,S),+) = <2x, c, d > and as above 2x. c, d a l l commute, and all products are zero.

S.J. Mahmood and J. D.P. Meldrum

166

Theorem 4.5. Lets = {a.b).

Then

(i) Q(R3,S) is a zero ring on an infinite cyclic group, (ii) D*(R3,S) is a zero ring on an infinite cyclic group, (iii) D(R3.S) i s a zero ring on the direct sum of two infinite cyclic groups.

ploof. From theorem 2.27, (lp3,S),+) = < 2 a r . The multiplication table shows that this i s a zero ring.

From theorem 3.8 we get that (DX(R3,S).+) = <[a.bl + 2a>

(D(R3,S),+) = <[a,b], 2a>.

and

Again the multiplication table shows that a l l products are zero,

since 2b = 0 and a l l elements involving b have an even number of b's in them.

Theorem 4.6.

Let

S

= {b,a+b).

Then

(i) D(R3,S) is a zero ring on an infinite cyclic group,

*

(ii) D (R3.S) is a zero ring on the direct sum of two infinite cyclic groups, (iii) D(R3.S) is a zero ring on the direct sum of three infinite cyclic groups.

Proof.

From theorem 2.27 we see that Q(R3.S) is the same as in theorem 4.5. We

now consider theorem 3.7. From this we again see that (D*(R3,S),+)= <2x, c+d> as in the last theorem, and the remainder of the theorem follows as before. In [S] all defects were calculated for some d. g. near-rings.

*

*

In all cases the defects

were zero rings, Q(R,S) = D (R,S) and D(R.S) = p(R,S) 63 D (R,S). Here we have examples in which Q(R.S) is not isomorphic to D*(R,S).

In [51,theorem 3.5, we have an example in

which (Q(R,S),+) is not a n abelian group, hence Q(R,S)

i s not a ring.

But the other

statement still holds. This statement is most unlikely t o be generally true. So a next stage would be t o seek counter examples to this statement.

REFERENCES

S. J. Mahmood. Limits and colimits in categories of d. g. near-rings. Proc. Edinburgh Math. SOC. 23 (1980). 1-7. J. J. Malone. D. g. near-rings on the infinite dihedral group. Proc Royal SOC. Edinburgh 78A (1977), 67-70. J. D. P. Meldrum. The representation of d. g. near-rings. J. Austral. Math. SOC. 16 (1973). 467-480. J. D. P. Meldrum. The endomorphisrn near-ring of an infinite dihedral group. Proc. Royal SOC.Edinburgh 76A (1977). 311-321. J. D. P. Meldrum. Presentations of faithful d. g. near-rings. Proc. Edinburgh Math. SOC.23 (1980), 49-56. J. D. P. Meldrum. Upper faithful d. g. near-rings. Proc. Edinburgh Math. SOC. 26 (1983). 36 1-370. J. D. P. Meldrum. Near-rings and their links with groups. Pitman Research Notes No. 134, London 1985. G. Pilz. Near-rings. 2nd. Edition. North-Holland. Amsterdam 1983.

Near-ringsand Near-fieldsG.Betsch (editor) 0 Elsevier SciencePublishers B.V. (North-Holland), 1987

167

NEAR-RINGS ASSOCIATED WITH COVERED GROUPS C.J.

MAXSON

Mathematics Department Texas A6M University College Station, TX 77843 From the time of Descartes, early in the 17th century, mathematicians have been interested in associating algebraic structures with geometric structures and investigating the interplay. In this paper we continue this line of investigation by associating near-rings to generalized translation spaces and noting how the geometry influences the algebraic structure.

I.

GEOMETRY We start with some geometric background and some definitions. is a translation plane if its group

that an affine plane A

In this case T is

tions operates transitively on the set of points of A. an abelian group with congruence partition

(Ti),

Recall

T of transla-

i.e., each Ti is a sub-

group of T, lTil = ITj[ T = u T Ti n T = ( 0 1 , i # j and T T = T, i i’ j i j i f j. On the other hand let C = {Gal be a non-trivial congruence partition (at least two components) consisting of subgroups Ga this congruence partition of G

a translation plane

of a group G.

taking the elements of G as points, the cosets of the components of lines and setting XG

parallel to

yG

B

if and only if a

known that every translation plane has the form C(G) G

with a suitable partitlon.

From

is obtained by

C(G)

=

p.

C

as

It is well

for some abelian group

More will be said about the equivalence of cer-

tain geometries and groups with special properties later. translation planes see [l], 141

For material on

or 1101. = ( P , L, 1) P called lines,

We now define generalized translation structures. Let Z where P

is a set of points, 1 a collection of subsets of

with an incidence relation defined in P x L x

L satisfying the following axioms:

(Al) Every two p o i n t s

(A2)

L and a parallelism defined in

ILI

22

a, b in P are incident with at least one line;

and for each A

E

L,

IAI

2;

(A3)

Parallelism is an equivalence relation;

(A4)

For each x

E

P.

A

E

1 there exists a unique line containing x

and parallel to A. Further let @

be a one-one map of

incidence structure C

such that O(P)

?’into the collineations of the is a point transitive group of fixed

168

C.J. Maxson We say

point free collineations which map lines onto parallel lines. is a generalized translation structure.

For a generalized translation structure , let 0 ment in P a

in P

denote the ele-

is the identity map in the group @(PI.

such that @ ( O )

0,

there exists a unique map

in @ ( P I such that $,(O)

=

For each a.

With-

out l o s s of generality we take Q(a) = Qa. Note that if every two points determine a unique line then we have a For other work on translation

translation structure studied by Andre [ 2 ] .

structures see the paper by Biliotti and Herzer [ 3 ] and the references

If further we require ( A ( = IBI for each A, B E L then we have a Sperner translation space ((111). We now extend the relationship between translation planes and congruence be a group partitions to generalized translation structures. Let written additively, but not necessarily abelian, with identity 0 . A collecgiven there.

tion

When

C = {H,K,. . . I (C1)

U C

(C2)

VH 6 C ,

H

(G,C)

IH6

is a

if

c c,

C,

and VK C C ,

H f G

the pair

G

H

=

the cover.

or H r i K

K

(OK

=

<

K

then H E K .

=

P(G),

{OI.

By taking P( G ) L(G),

ther. by associating to each a C G

The

If further,

and setting x + H II y + K

tine to verify that E ( G )

H f

partition) and the pair

be a covered group. x C G}

if

(G.C) is called a covered group.

cells of

is called a fibration

group_. Let

+

H # {O},

are called

VH, K

(c3) Then C

C

G

G

=

is a cover of

C

elements of

{x

of subgroups of

II

>

= C,

iff H

is a fibered

(G,C)

L(G) =

satisfies (All

=

K, it is rou-

-

(A4).

Fur-

the left translation La, denoted by

@(G):

a + ha we find that we have a generalized translation structure

(C(G),

Q(G)).

Conversely every generalized translation structure arises in this manner (see ( 1 2 1 ) .

From this we see that a generalized translation structure may be

and we henceforth do s o . considered as a covered group
(HI =

(“1

VH,

K

C

C, and as noted above each translation

plane arises from a group with a congruence fibration.

Thus fibrations, equal

fibrations and congruence fibrations “tighten” the structure.

We now tighten

the structure in an alternate fashion namely by requiring that there be a semigroup of operators. Let

<E,O>

domorphism of C

be a generalized translation structure and recall that an enis an incidence preserving function u: P

+

P,

that is,

169

Near-rings associated with covered groups

for A

E

L,

5B

u(A)

E

L.

The collection end I: of endomorphisms of

a semigroup with identity under the operation of composition. a subsemigroup of (01)

The

end C

identity id

For each u E S

Is

is

with the following properties: and the constant map w(a)

previously identified) for each (02)

I:

Suppose S

a

p

E

=

0 (where 0 has been

are both in S ;

the following diagram commutes

1

1

end E

end C Then we say T =

is a generalized translation space with operators,

henceforth denoted by GTSO. If

is the group with cover giving rise to the GTSO

u

then property (02) ensures that each

E

is an endomorphism of G.

S

then since ~ ( 0 =) 0 , each line through 0 is mapped by through 0. Hence, S

But u into a line

is a semigroup of endomorphisms satisfying

(01)'

The identity endomorphism and zero endomorphism are in S ;

(02)'

For each a U(Gi)

5 Gj

.

E

for each Gi

S,

Conversely if G

T = <E,@,S>

E

C, there exists G

j

is a group with a cover

C

E

C with

and a semigroup of endo-

morphisms satisfying (01)' and (02)' then we obtain a GTSO. GTSO arises in this manner, thus we identify the GTSO

Furthermore every



with the asso-

ciated group with operators and cover .

We now show how to associate near-rings with GSTO's, . First, we consider the

E(G.C)

set

=

{f E End G

I f(H)

that the semigroup of operators plays no role here.) functions E(G,C) tations of E(G,C),

5 H,

VH € C}.

(Note

Under composition of

is a semigroup with identity, called the semigroup of

. We consider the near-ring distributively generated by

denoted by

sider MS(G,C)

= {

dg E ( G , C ) .

f E MO(G)

For our second associated near-ring we con-

I

f(H)

5 H, VH

C

C, f u = u f, Vo € S )

which

is a zero-symmetric near-ring with identity under the operations of function

addltion and composition.

Unfortunately the word "kernel" has been used in

the literature for both of these near-rings. the kernel of



and

M (G,C) S

In this paper we call dg E ( G , C ) In the next

the centralizer of .

section we discuss the centralizer and in Section 111 the kernel. 11.

THE CENTRALIZER OF W e begin this section with the result that centralizers of GTSO's are

very general, Indeed every zero-symmetric near-ring with identity arises in

C.J. M a s o n

170

this manner. The next result, whose proof is due to Peter Fuchs extends the corresponding result in [12] where it was established only in the finite case. Theorem 11.1. [12]. Let N be a zero-symmetric near-ring with identity in which ab

=

1 implies ba

1 , a,b E N. Then there exists a GTSO, T

=

=

such that

M (G,C).

N

S

Proof.

Let G

{(l,O),

(0,l))

=

N x N.

Clearly G

N x N

H C

#

is contained in a maximal one.

denote the set of all maximal N-subgroups of G.

{H )

%

N x N

Hence maximal N-subgroups of

is a generating set).

exist and every N-subgroup C =

is a finitely generated N-group (e.g.,

C

Let

is a cover of

H and if (a,b) C G, then (a,b) C N(a.b) 5 Ha B for some maximal N-subgroup Ha. Therefore 11 Ha = G. For each a C N, let p : G + G be the function defined by p ( u , v ) = (ua,O) and let t: G + G be the function defined by t(u,v) = (v,O). Then pa, t are N-endomorphisms of G. We let S be the semigroup of endomorphisms

G

since Ha 2

implies Ha

a C N) U It).

so

is an N-subgroup of G semigroup of operators. As

1

{id} U {pa

generated by

=

a(Ha)

a

For each for some H

HB

B

S

C

and

Ha E C,

Thus

C C.

u(H,)

is a

S I

Proceeding as in [ 1 2 ] one then shows N

=

Ms(G,C).

with most representation results the above leads to several interest-

ing avenues of investigation.

Here we consider only the problem of character-

izing structural properties of M (G,C)

in terms of the geometry

S

.

(Further representation results are considered in r121.1 For the remainder of this section, all structures are finite and

Convention:

all near-rings are zero-symmetric with identity.

We first characterize when MS(G,C) some further concepts.

Let T

is a generating set if G

=

Syi and

=

finite, generating sets exist.

is a near-field.

be a GTSO. S y i F Syj

For each yi

C

....

if

ifj.

Y, define

I

For this we need

A set

Y

=

{yl,...,y t)

Since G

Ii = n {Ha

yi C Ha) and let I = {sequences (xl, xt) xi c If and F(Yi, F(x. x.)) where F(u,v) = { ( a , B ) E S x S au = Bv}. Further let J’ J i-th projection map on I. We say is kernel dependent if, pi I Ker a P { O } implies pi I Ker a ( s e e [ll] 1. Recall also

I

c

x,

y C G*

a l , . . . , un,

= G \{O)

are connected if there exist w~,...,w~-~E

...,t

tl, a1

x

= tl

a2 w 1

C S

G

I

is Ha C C,

yj) c pi be the for a 6 S, that and

such that

w1 f 0

= t2

w2 # O

u n wn-1 = t n y f O This relation is an equivalence relation on

G

*

and the equivalence classes

171

Neatvings associated with covered groups

are called the connected components.

G is said to be connected if G"

is a

connected component. Theorem 11.2.

Let T

[12].

=

. Then MS(G,C)

is a near-field iff T

is connected and kernel dependent. To obtain definitive structural results one places some restrictions on the semigroup

o€

operators. One considers the situation In which

group of automorphisms (with 0 adjoined).

As

S

is a

one might suspect (from the pre-

vious work on centralizer near-rings ( 1 7 1 ) the orbits of the action and stabilizers of elements in G"

play an important role.

For results in the case of

fibrations see 1141 and for the general results see [51. Next, one considers the situation in which S = U

{O, id}.

We write M,(G,C)

ble and not nilpotent, Ma(G,C)

S

for MS(G,C).

i s a cyclic semigroup,

When

a is not inverti-

is not a simple near-ring.

Therefore one

restricts to the case of a nilpotent operator. Theorem 11.3. 151, 1141.

Let a be a nilpotent operator on

IKer a I - I

(G,C).

connected components.

(11.3.1)

There are

(11.3.2)

There is a unique generating set, Y.

(11.3.3)

If

C is a fibration, Ker a

is contained in a unique cell,

say €Io. When

G

is connected much can be s a i d .

Theorem 11.4. 151, [141. Let a lowing are equivalent.

be a nilpotent operator on

(11.4.1)

Ma(G,C)

is a near-field.

(11.4.2)

Ma(G,C)

is a simple near-ring.

(11.4.3)

M (G,C) <(G,C)

(11.4.4)

(G,C).

The f o l -

is a 2-semisimple near-ring.

2

Z2.

Further, if C (11.4.5)

Ma(G,C)

(11.4.6)

Y I1 Ho = 0 .

is a fibration the above are equivalent to

is a local near-ring.

If G is not connected and a

is a nilpotent operator, necessary and

sufficient conditions in terms of the geometry which characterize when Ma(G,C)



have been determined

is simple. Again €or the case in which

C

is a fibration see [14] and for the general case see 151. We conclude this section with two suggestions for further research, first a rather specific problem and then a more general program. Sug 1:

Characterize 2-semisimplicity in terms of

S = U { O , id).

for

172

C J. Maxson

Sug 2:

For other classes of operators, (e.g.,

inverse semigroups,completely

0-simple semigroups) characterize the structure of MS(G,C) of 111.

THE KERNEL OF

.

We recall from Section 1 that the kernel of

is the near-ring

dg E distributively generated by the dilatations E

=

E(G,C).

When

is a

C

is well-known [ 3 ] , [ 6 1 , [ 7 ] ,

fibration, the structure of the semigroup E

[lo],

in terms

.

[Is].

Theorem 111.1.

If is a finite fibered group then E*

group of fixed point free automorphisms of

G.

is a

= E\{O)

Further, E* is a cyclic group.

A now classical result states that in the case is a translation plane, G

is an abelian group.

finite translation plane, dg E

Thus from the above theorem, if is a finite field [ 3 ] , [ l o ] .

is a



Therefore to

generalize to arbitrary fibrations one is lead to determining the structure of Such a study has been carried out in [ 7 1 and [ B ] .

fibered groups.

For our

particular case we specialize one of the structural results (see also [6]). Theorem 111.2.

If

is a finite €ibered group with

E* f {id)

then G

is a finite p-group for some prime p, of exponent p and of nilpotency class at m s t

2.

Using the above result concerning E and G along with some work of Herzer [6] (recently generalized in [ 1 3 ] ) the following rather surprising result has been obtained. Theorem I I G . [ 1 5 ] . commutative ring. If

E* = {id)

Examples ( [ 7 ] )

If



is a finite Eibered group then dg E

If, further, E*

f

then dg E

{id}

then clearly dg E = Zn

show that in this case n

where

n

is a

is a finite field. is the exponent of G.

need not be a prime.

Theorem 111.3,

together with the classical work of Andre ( [ l ] ) , shows that whether or not G is abelian, whenever



E*

f

in a natural manner.

{id)

there is a field associated wtth the geometry

We note that in the non-abelian case, although

the elements of the associated field are sums of automorphisms, they need not themselves be morphisms. We also mention that i n the abelian case the field dg E applications.

The significance of the field dg E

still unknown. Recently (191, [16]) a study of the near-ring groups

has been initiated.

i n the above studies

G

has geometric

in the non-abelian case is dg E

for finite covered

In general, very little is known.

In fact,

is restricted to be a finite elementary abelian

173

Near-rings associated with covered groups

p-group and the structure of the ring dg E

is investigated.

In [91 several

basic results, several examples and a complete solution to the case G are presented.

In [I61 necessary and

3

P sufficient conditions in terms of an

associated lattice are given for the kernel dg E ring or a semisimple ring.

= (2 )

to be a field or a simple

The surface has just been scratched, there is

still much to be investigated here. REFERENCES Andr6. J., Uber nicht-Desarguessche Ebenen mtt tansitiver Translationsgruppe, Math. Zeitschr., 0 (1954), 156-186. , Uber Parallelstrukturen, 11: Translationsstrukturen, Math. Zeitschr., -/a (1961 ), 155-163. BILIOTTI. M. and Herzer, A . , Zur Geometrie der Translationsstrukturen mit eigentlichen Dilatationen. Abh. Math Sem. Univ. Hamburg 2 (1983), 1-27. Dernbowski, H. P., Finite Geometries, Springer-Verlag, New YorkHeidelberg-Berlin, 1968. Fuchs, P. and Maxson, C. J., Kernels of Covered Groups with Operators (in preparation). Herzer, A., Endliche nichtkornmutative Gruppen mit Partition Il und ftxpunktfreiem n-Automorphismus. Arch. Math. 34 (1980), 385-392. Karzel, H. J. and Maxson, C. J., Fibered groups with non-trivial centers, Result. der Math., 7 (1984). 192-208. , Fibered p-groups. Abh. Math. Sem. Univ. Hamburg (to appear). Karzel, H., Maxson, C. J. and Pil-z,G. F., Kernels of Covered Groups. Result. der Math. (to appear). Luneburg, H., Translation Planes, Springer-Verlag, New YorkHeidelberg-Berlin, 1980. Maxson, C. J., Near-rings Associated with Sperner Spaces, JOUK. of Geom., 20 (1983). 128-145. , Near-rings associated with Generalized Translation Structures. JOUK. of Geom. 24 (19851, 175-193. Maxson, C. J. and Meldrum, J. D. P., D G Near-rings and Rings (submitted). Maxson, C. J. and Oswald, A., Kernels of Fibered Groups with Operators, (submitted). Maxson, C. J . and Pilz, G. F., Near-rings determined by fibered groups, Arch. Math. 44 (1985), 311-318. Maxson, C. J. and Pilz, G. F., Kernels of Covered Groups 11, (submitted). Pilz. G. F., Near-rings, Revised Edition, North-Holland, New York, 1983.

This Page Intentionally Left Blank

Near-ringsand Near-fclds,C. Betxh (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1987

175

KRULL DIMENSION AND TAME NEAR-RINGS J.D.P. MELDRUM and A.P.J. VAN DER WALT Department of Mathematics, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ. Scotland.

Department of Mathematics, University of Stellenbosch, Stellenbosch 7600, South Africa.

A tame near-ring R is one which has a faithful module in which all R-modules are R-ideals. Such a near-ring is very ring-like. In this paper we extend some results on rings with Krull dimension to tame near-rings. Krull dimension i s a generalization of the minimal condition. The main results are as follows. (i) A tame right noetherian near-ring has a possibly transfinite power of its J2 radical zero. (ii) The Jg radical of a tame near-ring does not contain a non-zero idempotent ideal. (iii) The nil radical of a tame near-ring,is residually nilpotent.

1.

INTRODUCTION AND DEFINITIONS. Contrary to what is suggested by their name, general near-rings are in some respects

very far from rings. One such is the fact that none of the obvious generalizations of the Jacobson radical to near-rings i s necessarily nilpotent even in the finite case. This is one reason for introducing tame near-rings, a reasonably large class of near-rings which behave very like rings. They were first introduced by Scott [lo] and [ l l ] and he showed how close to rings they were in their behaviour. In particular tame near-rings with the descending chain condition have a l l three J (for Jacobson) radicals nilpotent. In this paper we generalize results about rings with Krull dimension to near-rings.

Apart from those

results that are essentially lattice theoretic in nature, the proofs of the results about near-rings are not straight generalizations of the proofs from ring theory. This is mainly due to the fact that the ring-like properties lie in a faithful module, whereas the near-ring has the Krull dimension. A (left) near-ring i s a set R with two operations, + and x,y,z

E

R.

., such that

(R,+) is a not

is a semigroup and x(y+z) = xy + xz, Ox = 0, for all

necessarily abelian group, (R,.)

If (G,+) i s a group written additively then Mo(G) = (f : G

+

G, OGf = O G }

is a

near-ring under point-wise addition and composition of functions. A homomorphism from

R into Mo(G) defines G as an R-module. of G such that HR (g+k)r - gr

E

5 H.

A submoduie H of the R-module G is a subgroup

An R-ideal of G is a normal subgroup K of G such that

K for all g E G, k

E

K, r

E

R.

under the right regular representation.

The group (R,+) is a n R-module, denoted RR,

A right invariant subnear-ring

of R i s an

R-submodule of RR and a right ideal of R is an R-ideal of RR. A left invariant subnear-ring of R i s a subgroup S of (A,+)

such that RS

5S

and a left ideal of R is a normal left

invariant subnear-ring. An ideal i s a left ideal which is also a right ideal. An R-module G

J.D.P. Meldrum and A.P.J. van der Walt

176

is monogenic if G = gR for some g

E

G. A monogenic R-module G is of type 0 if it has no

non-trivial proper R-ideals, of type 1 if it is of type 0 and g

E

g R = (0). of type 2 if it has no non-trivial proper R-submodules. J.+(R) as the intersection of all annihilators of R-modules

G implies gR = G or For v = 0,1,2,

of type v.

define

These definitions

follow closely those t o be found in Meldrum [81, or in Pilz [91, except that he uses right near-rings. and his nomenclature for the substructures of R-modules is different.

Most

standard results can be found there. Recall that a tame R-module G is an R-module in which a l l R-submodules are R-ideals.

A near-ring is tame i f it has a faithful tame R-module.

Two large families of

tame near-rings are given by firstly, the subnear-rings of Mo(G) generated by a semigroup S of endornorphisms of G containing Inn G, the inner automorphisms of G, and secondly, the zero-symmetric parts of near-rings of polynomials. We n o w come t o Krull dimension. Defined originally for commutative rings, it has been extended to arbitrary rings where it has been used extensively (Gordon and Robson [2]). In this context it is essentially a condition on the lattice of substructures generalizing the descending chain condition. Let G be an R-module.

Denote the Krull dimension of G by kd(G). Then kd(G) = - 1 i f and only

if G = (0). If a is an ordinal and kd(G) + a then kd(G) = a if there is n o infinite descending

chain

G = Go

2 G1 1.G2 2 . ..

of

R-ideals

of

G

such

that

kd(Gi/Gi+l) + a

for

i = 0.1.2,. . _. In Gi/Gi+l we define the Krull dimension relative to the R-ideals of G/Gi+l.

Note that kd(G) = 0 corresponds t o the descending chain condition on R-ideals. that R has Krull dimension if the R-module RR has Krull dimension.

need not necessarily have Krull dimension. Finally for an ideal I of the near-ring R w e define lo = R, I 8 + l= 18.1, for limit ordinals

a and 10 =

I, 16+1 =

n;=, (18)"

and l a =

We say

Any given R-module la =

n ~I8 < ~

nB
In the next section w e gather together results about Krull dimension and powers of ideals needed later. In the last section we present our results about tame near-rings with Krull dimension.

2. PRELIMINARY RESULTS.

In the applications we need a number of results about modules with Krull dimension from Gordon and Robson 121 and Lenagan 171. The difficulty in using these results is t o unravel those results that arise from the properties of the lattice of substructures from those which use results about rings. Here we list the results we need with indications of proof where necessary.

Lemma 2.1.

(121 Lemma l.l(i)). L e t G bs a tame R-module with Kfull dimension, H an

R-ideal of G . Then kd(G) = sup{kd(G/H), kd(H)} if either side exists Note that w e need tameness since R-ideals of H are not necessarily R-ideals of G. Indeed the result is no longer true if G is not tame.

Example 2.2. The wreath product G of two infinite cyclic groups satisfies the maximal condition on normal subgroups but contains a normal subgroup H which is the direct sum of an infinite number of infinite cyclic groups. Take R t o be the integers with 1 acting as

177

KruN dimension and tame near-rings

the identity map on G. Then G has Krull dimension (since it satisfies the maximal condition on R-ideals, see lemma 2.4) but H does not (see lemma 2.6). Lemma 2.3.

Let G b e an R-module with KruN dimension

Then every homomorphic

image o f G has Krull dimension less than o r equal to the Krull dimension of G. See proposition 1.2(i) in [2]. This result follows because the lattice of R-ideals of a homomorphic image of G is a sublattice of the lattice of R-ideals of G, and it does not need tameness of the R-module. Lemma 2.4.

Every R-module with the maximal condition on R-ideals has Krull

dimension This result can be proved by the same methods as proposition 1.3 of [2]. Before we generalize proposition 1.4 of [21, we need the concept of, and a result about uniform dimension. We say that the R-module G has finite uniform dimension if there is a fixed (finite) bound on the number of summands in a direct sum of non-zero summands. Lemma 2.5. Let G be a tame R- moduJe which does nor contain an infinite direct sum

of nan-zero R-ideals

Then G has finite uniform dimension

Proof. We first show that any non-zero R-ideal of G contains an R-ideal of G which is uniform, i.e. a n R-ideal with the property that any two non-zero R-ideals of G contained in it intersect non-trivially.

Let H be a non-zero R-ideal of G. If it is uniform, we are home.

If not there exist K1, L1 R-ideals of G contained in H such that K1 n L 1 = {O}.

So K1 0 L1

< H. If either K1 or L1 are uniform, we are home. If not we obtain K1 0 Kg 0 L2

5H

where K2, L2 are non-zero R-ideals of G contained in L1. Since G does not contain an infinite direct sum of non-zero R-ideals, this process must stop and it can only stop when we have a non-zero uniform R-ideal. The second step is to show that there is a finite number of uniform R-ideals of G whose sum i s direct and is an essential R-ideal of G, i.e. it has non-trivial intersection with every non-zero R-ideal of G. By above, G has a uniform R-ideal H i . If H i is not essential, then there is a non-zero R-ideal K1 such that H1

n K1 = {O}.

Then K1 contains a uniform

non-zero R-ideal H2 and H i Q H2 is direct. This process must stop after a finite number of steps as before, for the same reason. This gives us the second step. The last step is to obtain the fixed bound on the number of non-zero direct summands. The method used is a version of the Steinitz Exchange Theorem used with finite dimensional vector spaces. At first we consider direct sums consisting of uniform R-ideals. Let H i G I . . . @ Hn be a direct sum of uniform R-ideals which i s essential. To be able to use this sum for the exchange process, we need to show that if K is an R-ideal of

G which intersects each Hi non-trivially, then K is essential in G. This we now do. For i = 1,.

.@

. . , n, define

Ln is direct and L

5 K.

Li = K n H i , a l l non-zero by hypothesis. Hence L = L1 C3. We show that L is essential in G. Let g

E

G - ( 0 ) and let gR

denote the R-ideal of G generated by g, since G i s tame. Since H i Q . we have gRn(H1 Q . . . @ H ),

# {O}.

Thus there exists g1 = h i +

...

. .. +

. Q Hn

is essential,

hn where hi

E

H,,

g1 t gR, and some hi # 0, without loss of generality h i # 0. Because H i is uniform, and L1

J.D.P. Meldrum and A.P.J. van der Walt

178

So we have h l r l

is a non-zero R-ideal in HI, L1 i s essential in H i . Hence h l R n L 1 = {O}. = I1 # 0. Then g1r = h'l + h'2 +

have 0 # h'l

E

. .. +

where h'l = I1 # 0. If h'2,

h',

L c\ gR and we are home. If h'2,

. . ., h'n

..

., h,'

are all zero we

are not a l l zero, then without loss

of generality h'2 # 0. We now repeat the process to get h'2R n L 2 = (0) and there exists r2

E

...+ . . . + h",

R such that ( h ' l + h'2 +

1'1 + 12

L n g R . If h"g +

E

h',)r2

= 1'1 + 12

= 0

+

h"3 +

... +

h",.

Then 1'1

+

12 # 0 and

we are home. In any case we eventually obtain a

non-zero element of gR c)L after at most n steps. Thus L is essential. Now suppose that MI,

. . .,

Mm are non-zero R-ideals of G such that M1 8 . . . @ Mm

is a direct sum. We show that m

5 n,

which will give us our result. M2 @ . . . @ Mm i s

not essential since it intersects M i trivially. Hence (Mq I3. . . @ Mm) f~Hi = (0) for some i,

. . I3 Mm i s direct. Similarly . . @ M, i s not essential and after renumbering the Hi's if necessary, the sum @ M3 @ . . . @ M, is direct. Proceeding in this way we obtain that m 5 n.

without loss of generality i = 1. Hence the sum H i 8 M2 @ . H1 @ M3 8 3 . H i @ H2

This proof is a fairly straightforward adaptation of the proof of the corresponding result in Chatters and Hajarnavis [l]. lemma 1.9. We come now to the generalization of proposition 1.4 of Gordon and Robson [21. Lemma 2.6. --

A tame R- module with Krull dimension has finite uniform dimension

This can be proved as in [2]. We now look at critical R-modules. kd(G) =

a

We call a non-zero R-module G a-critical if

and the Krull dimension of any proper homomorphic image of G is strictly less

than a. A critical R-module is an R-module which i s a-critical for some ordinal a. Lemma 2.7. --

Any non-zero

R- module with Krull dimension has a critical R- ideal

The proof is similar to that of proposition 2.1 of [2]. We now present a technical lemma, generalizing Proposition 1 of Lenagan [7], which we will need later. Lemma

28.

Let

G

be

0 c H1 < H2 < . . . such that G R-ideals G = Go

a

module

= u"i=1

with

Hi.

an

ascending

chain

of

R-ideals

If there is an infinite descending chain of

> G1 > . . . such that for all i,j

then G does not have Krull dimension

See Proposition 1 of I71 for the proof. We now come to the question of the length of well-ordered descending chains of R-ideals in modules with Krull dimension.

This generalizes work of Gulliksen [3] and

Krause [41.

-Lemma 2.9. Let G be an R- module with Krull dimension a. Then a 1 limit (B; G has Further, if a < w, then

a well-ordered descending chain of R-ideals of length at least wB}. any R-module G with the propelty that a = limit

(8; G has a well-ordered descending

chain of R- ideals of length at least we) has Krull dimension a.

Proof.

We use transfinite induction on a. Consider the case a * 0.

Then G has

Krull-dimension 0 if and only if G has the descending chain condition on R-ideals.

That

179

KruN dimension and tame near-rings

holds if and only if every well ordered descending chain of R-ideals in G i s of finite length, i.e. 0 = limit (6; G has a well-ordered descending chain of R-ideals of length at least Now assume that the first result is true for all ordinals y and

consider

a

well-ordered

descending

u = w 6 . n l + . . , + nr, where n1, , . ., nr

5 a.

t o show that 6

E

2

of

R-ideals

to8}.

Let G have kd(G) = a in

G

of

length

Z, and the exponents of w are ordinals. We need

6 > a.

So suppose that

Let G ,

u. Since u

well-ordered descending chain of length for each m

chain

< a.

be the u " . m t h

= w B . n l , and

6 > a,

term of the

w e can find Gm

1. Then Gi/Gi+l has a well-ordered descending chain of R-ideals of G of

length wa inherited from G. By the induction hypothesis the Krull dimension of Gi/Gi+l relative to G is not less than a. for each i 2 0. By definition kd(G) f a. This finishes the proof, by induction, of the first statement. We now prove the second statement by induction. The case a = 0 has already been done. As a is finite in this case, we have limit (6; G has a well-ordered descending chain of R-ideals of length at least u6} = max

(6; G

has a well-ordered descending chain of

< a.

Suppose G

has Krull dimension a. Then kd(G) 1.a implies that G has a well-ordered

descending

R-ideals of length at least u6}. We assume that the result is true for y chain of R-ideals of length at least w", lemma, we have

a

= max

(6; G

using the induction hypothesis. Using part of the

has a well-ordered descending chain of R-ideals of length

at least uB}. Now suppose that G has the property that a = max (5; G has a well-ordered descending chain of R-ideals of length at least w6}. infinite descending chain of R-ideals.

Let G = Go

2 G1 2 G2 2 . . .

b e an

If more than a finite number of the factor modules

Gi/Gi+l are not of Krull dimension < a (relative t o G) then an infinite number of these factors have a well-ordered descending chain of R-ideals of length at least these wa.w

chains = uatl.

to

G, we

get

a

well-ordered

descending

This cannot happen by hypothesis.

factor modules Gi/Gi+l are not of Krull dimension C kd(G) + a by the induction hypothesis.

chain of

ua. By lifting

length at

least

So at most a finite number of the

a

and thus kd(G) = a. since obviously

This finishes the proof of the second part of the

lemma by induction. The final set of preliminary results concerns powers of ideals. We list them without proof because they are easy t o prove and the proofs parallel exactly those for rings in Krause and Lenagan [51, after account has been taken of the left-right switch in the definition.

Note that with near-rings, powers of ideals are just subsets o f the near-ring,

not ideals in their own right.

Lemma 2.10. Let I be an ideal of the neor-ring

R.

Then the following hold

5 lat6 for a// ordinals a, 6. (ii) (I")" 5 l a S nfor all ordinals a, positive (iii) la 5 lug for all ordinals a. (i) la.lS

integers n.

J.D.P. Meldrum and A.P.J. van der Walt

180

3. TAME NEAR-RINGS WITH KRULL DIMENSION. Here we extend some results concerning rings with Krull dimension due t o Lenagan

[S] and [7] and Krause and Lenagan [5]to tame near-rings with Krull dimension. Many of the proofs are suitable amalgams of proofs for the ring case taken from the papers mentioned above, together with one or two special features. We consider three topics: the transfinite powers of the J2 radical in tame near-rings with the ascending chain condition on right ideals, the non-existence of idempotent ideals contained in J2(R), where R i s tame with Krull dimension and finally the nil radical in tame near-rings with Krull dimension. For the first of these three topics we consider the following situation.

Let R be a

right noetherian tame near-ring, i.e. a tame near-ring with the ascending chain condition on right ideals. It will have Krull dimension by lemma 2.4, say kd(R) = a. By lemma 2.9 any well ordered descending chain of right ideals has length strictly less than ua”.

Let G be

a faithful tame R-module. Lemma &l.

With the notation given above, let g

E

G . Then the R-module gR has a

well ordered descending chain of R-ideals which reaches (0) and such that aach factor is 2- primitive By proposition 3.4

Proof.

isomorphism i s given by gr

+

of

Pilz 191. we have gR

AnnR(g) + r.

‘R

R/AnnR(g) where the

By the ascending chain condition on right

ideals, we can find a right ideal L1 of R maximal subject to containing AnnR(g). Then Ll/AnnR(g) is a maximal right ideal in R/AnnR(g) and so gL1 i s a maximal R-ideal in gR. Since gR i s tame as the submodule of a tame module, we deduce that gR/gLI i s 2-primitive. 2-primitive.

Now repeat the process with gL1 to obtain gL2 such that gL1/gL2 i s Continuing transfinitely if necessary we obtain the result that we want.

Corollary 3.2.

The conclusion of lemma 3.1 holds for any monogenic tame R-module

for a right noetherian near-ring R.

Lemma 3.3. that J2‘

With the same hypotheses as lemma 3.1,there exists an ordinal

5 AnnR(gR)

proOf. Given g

u

such

for a / / g E G. E

G , we construct the well-ordered descending chain of R-ideals

whose existence is guaranteed by lemma 3.1, say

Since each factor i s 2-primitive, we have

for all ordinals A. ordinal

The case all p C

We will prove by transfinite induction that (gR)JzU = (0) for some

u by showing that (gR)J2‘ 5 gRA for all ordinals A.

X.

If

X

= 1 has already been pointed out. So assume that the result is true for

A = p

+

1 for some p, then (gR)J2’ = (gR)JZp. J2

hypothesis and gRp. J2 gRi =

nN<xgRp

1.gRp+l

and J2’ =

5

= gRA by the remark above. If

n,
J2p.

Let gr

E

gR.,

X

gR, x E J2’.

J2 by t h e induction is a limit ordinal, then Then for all p

< A,

181

KruN dimension and tame near-rings x

E

J*p

(gr)x

E

and

by

nu<),gR,,

the

= gRA.

induction

hypothesis

This is true for all gr

E

5 gR,,.

Hence

Hence (gR)J2A

5 gRA,

(gr)x E (gR)JzP

gR, x

J2 A .

E

which is sufficient t o complete the induction argument, hence the proof of the result. Corollary

Proof.

g . Jq

w"+l

< Ann(GR).

By lemma 2.9, any well-ordered descending chain of right ideals in R has

< Ann(gR)

length strictly less than ua+'. Hence J2

for a l l g

This proves the corollary.

-Theorem 3.5.

Let R be a tame right noetherian near-ring.

E

G by the lemma.

us+i Then J2(R) = (0) i f R

has an identity and has square zero otherwisa

proOf. If R has an identity, then GR = G, and since G i s faithful, it follows from corollary 3.4 that J2

*a+%

*u+l

faithful, we have (J2

,oc+l Otherwise RJ2

= (0).

annihilates G and since G is

)2 = (0).

Theorem 3.6. Let R be a tame right noetherian near-ring.

Then ( J Z ( R ) ) ~is nilpotent if

R has an identity, and (J2(R))a+1 has square zero otherwise Proof. The second part follows from lemma 2.10 and theorem 3.5. The first pan follows from lemma 2.10 and lemma 2.9 as in the proof of theorem 3.5 once we observe that by lemma 2.9, J2u = (0) for some

u

= ua.n

+ .

. . for

some positive integer n.

We now turn to the non-existence of idempotent ideals within the J2 radical. There

are two results of this type, one where the module has Krull dimension, the other where the near-ring has Krull dimension. Let R be a tame near-ring with a faithful tame R-module G. Let G have Krull dimension.

~Definition 3.7. The critical socle series of G is defined by So(G) = (0). Sk+i(G)/Si(G) is the sum of a l l critical submodules of G/Sk(G) and

s~ =

u,,<x

S,(G)

for limit ordinals

A.

By lemma 2.7 it follows that SA+1(G) > Sl(G) for all ordinals

A

and hence that

G = Sa(G) for some ordinal a. Theorem 3.8. Let R be a tame near-ring with a faithful tame R-module H with Krull dimension Then J2(R) does not contain a non-zero idempotent idear!

Proof.

Let I

5

J2(R), I be an idempotent ideal.

8-critical submodules of H with 8

5 kd(H).

We show that I annihilates all

Note that there is no distinction between

submodules and R-ideals in H, since H is tame. The cases 6 = -1,0 are both trivial since 0-critical modules are just the 2-primitive submodules of H. Now assume that LI = (0) for all modules with kd(L) < 8. Then if M is a 6-critical submodule of H and M' is a non-zero submodule of M, we have kd(M/M')

< 6.

Hence (M/M')I = (0). i.e.

M' 2 MI. Thus MI will

generate the unique minimal submodule of M if it is not zero. Since I 5 J2(R), it follows that (0) = MI.1 = MI2 and hence ld

5

Ann M, i.e. MI = (0) since I = ld.

This

proves the result by induction. Corollary 3.9.

If R is a tame near-ring with a faithful tame R-module with Krull

dimension then, for any ideal I sequence or I is nilpotent

< Jz(R),

either I > l2

> . . . is a

strictly decreasing

J.D.P. Meldrum and A.P.J. van der Walt

182

Proof.

Otherwise J2(R) would contain an idempotent ideal contrary t o the theorem.

We now turn t o the case in which R has the Krull dimension. Theorem 3.10.

Let R be a tame near-ring with Krull dimension. Then 4 ( R ) doss not

contain a non-zero idempotent idear!

Proof. Let G be a faithful tame R-module.

Let g E G, I

J’(R), I an idempotent ideal.

Then by proposition 3.4 of Pilz [9] we have gR = R R/AnnR(g) and as R has Krull dimension, so does R/AnnR(g) by lemma 2.3.

theorem 3.8 it follows that gR.1 = (0). holds for all g

E

5 G, hence gR is Thus RI 5 AnnR(g) and

Also gR

5 AnnR(G).

G, and so I*

a tame R-module.

so I’

By

5 AnnR(g).

This

As G is faithful, it follows that l2 = (0). This

proves the result. Corollary

3.11.

contained in J2(R).

Let R be a tame near-ring with Krull dimension. Let I be an ideal Then either the sequence I

> I’ > . . . is strictly decreasing or

I is

nilpotent

Proof. Otherwise J2(R) would contain an idempotent ideal contrary t o the theorem. We come finally t o the nil radical of a tame near-ring

with

Krull dimension.

Unfortunately we cannot obtain the nilpotence of the nil radical as happens for t h e ring case. But we do obtain something.

~Lemma 3.12. Let R be a tame near-ring with Krull dimension and let G be a faithful tame R-module, with g E G . andC is nil then C2

Write A for AnnR(g).

If (C+A)/A is a critical right ideal o f R/A

5 A.

Proof. Let (3 : R

+

5 G by proposition 3.4 of Pilz [91. gR 5 G and G is tame. Let C be a

RIA where as before R/A =R gR

By lemma 2.3, RIA has Krull dimension, and is tame as

nil right ideal of R, such that CO is critical. We can find such an ideal unless all nil right

So without loss of generality w e may

ideals of R are contained i n A, by lemma 2.7. assume that CO # (0). Let cO But c” = 0 f o r some n

tame.

follows that (ce)m-’ CB.CB

E

>

CO, cO # 0. Then (cC)O is an R-ideal of R B since RO 0. So there exists a least m

f 0 and (cO)m-’

>

annihilates cO. Thus AnnC(cO)

Ce/AnnCe(CB) = C + A/AnnC+A(ce)

and

AnnC+A(Ce)

IS

1 such that (ce)m = 0. It

1.(A.cm-’}.

2 {A,cm-’}

Hence

> A.

Thus

kd(cO .CO) < kd(CO) since C B is critical. Hence as CO is critical, we have cO .CO = 0, i.e. (CB)2 = (0) and C2

5 A.

The fact that a non-zero submodule of a critical tame module

with Krull dimension has the same Krull dimension follows from lemma 2.1.

_ Lemma _ _ _ 3.13. _ Let R be a tame near-ring with Krull dimension. Let N be the nil radical of R. If N is not nilpotent, then there exists a chain of right ideals 0 = A0 such that each AiB is nilpotent but

A = AnnR(g), g

E

url AiO

< A1 <

is not nilpotent, where O : R

-c

R/A for

G, a faithful tame R-module

Proof. By hypothesis and using lemma 3.12. we deduce that there is a non-zero right ideal CB with square (0).

By Zorn‘s Lemma w e may choose a maximal right ideal

contained in N whose square is zero in R/A. Call this right ideal A l . Assume that w e have chosen

4,

then choose Ai+l so that Ai+lO/A,e is a maximal right ideal contained in

NB/AiB whose square i s zero.

Obviously each AiO is nilpotent.

If XO =

u

AiB is

nilpotent. then XO = AnB for some n, which is impossible by hypothesis and lemma 3.12.

183

Krull dimension and tame near-rings

Lemma 3.14.

Let R be a tame near-ring with Krull dimension and faithful tame

module G. Let N ‘be the nil radical of R. f o r every g E G, there exists n = n(g) such rhat

Nn(g) C AnnR(g).

Proof. Assume the result does not hold. Then by contained i n N

,

lemma 3.13, there is a right ideal X

where X0 i s not nilpotent and 0 i s defined as in lemma 3.13. We also

have X = u z 1 Ai and AiB is nilpotent for each i 1 1 .

Since NB and X8 have Krull

dimension, using lemmas 2.1 and 2.4, we can apply lemma 2.8 and deduce that for some n. j we have (XB)”

5 (XB)”+l

+

Aj0.

Hence (X0)”

1.A,B

since the powers of X0 form a

strictly decreasing sequence (corollary 3.1 1). Thus XB is nilpotent, a contradiction. Theorem 3.15.

Let R be a tame near-ring with Krull dimension, N the nil radical of

R. ThenNW= {O}.

Proof.

N~

Let

5 fl gEG AnnR(g)

G =

be

a

faithful

(01 since G i s

tame

R-module.

By

lemma

3.14,

faithful.

ACKNOWLEDGMENT. The work on this paper was done while the first named author was visiting the Department of Mathematics of the University of Stellenbosch. Financial assistance of both the Council for Scientific and Industrial Research and the University of Stellenbosch is gratefully acknowledged.

REFERENCES

Chatters, A.W. and Hajarnavis, C. R. Rings with chain conditions (Pitman Research Notes in Mathematics, 44. London. 1980). Gordon, R. and Robson. J.C. Krull dimension Mem. Amer. Math. SOC. 133 (1973). Gulliksen. T.H. A theory of length for noetherian modules J. Pure and Appl. Algebra 3 (1973). 159-170. Krause, G. Descending chains of submodules and the Krull dimension of noetherian modules J. Pure and Appl. Algebra 3 (1973), 385-397. Krause, G. and Lenagan, T.H. Transfinite powers of the Jacobson radical Comm. Alg. 7 (1979). 1-8. Lenagan, T.H. The nil radical of a ring with Krull dimension Bull. London Math. SOC.5 (1973). 307-31 1. Lenagan, T.H. Reduced rank in rings with Krull dimension Ring theory (Proc. Antwerp Conf.), 123-131. (Lecture Notes in Pure and Appl. Math. 51, Dekker. New York. 1979). Meldrum. J.D.P. Near-rings and their links with groups (Pitman Research Notes No. 134, London, 1985) Pilz. G. Near-rings (North-Holland, Amsterdam, 1977 (2nd Ed. 1983)). Scott, S.D. Near-rings and near-ring modules (Doctoral Dissertation), Australian National University) 1970. Scott, S.D. Tame near-rings and N-groups Proc. Edinburgh Math. SOC. 23 (1980). 275-296.

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Near-rings and Near-fields, G.Betsch (editor) Q Elsevier Science PubMers B.V.(North-Holland), 1987

185

SOLUTION OF AN OPEN PROBLEM CONCERNING 2-PRIMITIVE NEAR-RINGS

J H MEYER and ANDRIES P J VAN DER WALT

Department of Mathematics, University of Stellenbosch, Stellenbosch 7600. South Africa*.

ABSTRACT We construct a 2-primitive non-ring with identity which will clarify an open problem posed in 1971. In this note we solve an open problem about 2-primitive near-rings raised by Betsch [ l ] in connection with his theorem 4.8, which reads as follows:

Let R be a r i g h t near-ring with i d e n t i t y 1 which i s not a ring and which i s 2-primitive on the R-module G. I f R contains an idempotent e of rank 1, then R contains a minimal l e f t ideal or we have e# 1

(+I

(By the rank of an element

and

r E R

eG) = 0 .

Ann ( G R

we mean the cardinality of the set of non-

zero orbits of Aut G on rG.) R The problem is to find an example to show that the exceptional case can indeed occur.

(+I

We shall show that matrix near-rings over some near-fields

furnish examples of this phenomenon. Matrix near-rings were introduced in Meldrum and Van der Walt [ 2 1 .

For the

convenience of the reader we provide the pertinent definitions: For a natural number over the near-ring {fr,:Rn+Rnl

R

n,

we define the n

as the subnear-ring

1
where

X

n m a t ~ xnear-ring

of M(Rn)

Mn.ln(R)

generated by

fr.
...,

r

> =

17 t. = rr. and tk = 0 if k # i. Here Rn * I n denotes 8 (R,+), and n-vectors are written with pointed brackets and are 1 thought of as column vectors. The elements of Mn(R(R) are called n X n 17

< tl,t2,

. ..,

tn >,

with

matrices over R and will usually be denoted by upper case letters. In particular, if R

happens to be a (right) near-field, then

zerosymmetric (right) near-ring with identity

1

1

fll + f22

+

Mn(RI

... +

f1 nn'

is a See [ZI

for more details. In matrix ring theory, the matrices with zeroes in all the rows except in a fixed row, play an important role. near-rings:

If

1


We will generalize this concept to matrix 1. m n, we call a matrix of the form fki,Uj, with J j=l

1

*Financial assistance by the Council for Scientific and Industrial Research is gratefully acknowledged.

J.H. Meyer and A.P.J. van der Walt

186

and U E Mn(R) for all 1 < j < m, a k-th row m a t r i x . 3 j tells us more about the construction of such matrices:

r. E R

Lemma 1

Lemma 1 Let 1 < k < n. If u i s any matrix, then fziU i s a k-th rotJ m a t r b for a l l r E R, 1 < i 6 n. Furthermore, there e x i s t s an expression for fr,u which c o n s i s t s e n t i r e l y of functions of the form ki

fs ( s E R), kj

apart

porn parantheses and operators. Proof From the definition it follows that fr.U is a k-th row matrix. Now, ki

where

[x; p,q]

denotes f h .

1

Then, for each

j <mi,

we treat

r.. [rij; k,k. .IV. = fkR,, Vij 11

13

in the same

11

0

way, e t cetera. The next lemma is quite useful:

Lemna 2 If

i s a minimal l e f t ideal of

L

is an invertibZe then LV := {LVJL E Ll is aZso a

-1

element of M ~ ~ R ) i.e. , v &sts, minimal l e f t ideal of Mn,(R).

mnl,(R)and

V

Proof It is routine calculation to check that LV is a left ideal. Suppose it is not minimal.

Then we have a proper subset .$

left ideal properly contained in L,

properly contained in

LV.

But then

of

L

(sV)V-' =

such that .$V is a

s

is a left ideal

a contradiction.

0

Now let F be an infinite near-field which is not a field. The example we are going to construct works for any n 2 2, but for the sake of convenience, we stick to the case n = 2.

141,

R

is 2-primitive on

Thus, let R := M (F). As was pointed out in 2 2 G := F The next proposition is fundamental in

.

this paper:

Proposition 1 AutRG G

(F

Io},.)

where the elements of

F

103 act on

by multiplication on the r i g h t .

Proof It is easy to check that < a,B > @ ( A ) = < a A , B A > morphism, suppose

A

'$: F

{Ol

is a monomorphism.

+

defined by

To see that it is an epi-

is an R-automorphism of

since, if < 1,0 > A = < 0,B >,

Then ci f 0 ,

AutRG

G, and that < l,O>A = c a , @ >. 1 then < 1,0 > A = (fll< 1,0 >)A

1

1

f (< 1,0 28) = f l l < 0,B > = < 0,O > which implies B = 0 . But only the 11 zero-element < 0 ' 0 > can be mapped onto < 0,O > by an R-automorphism. Thus,

=

if < y,6 >

is any element of

(fTl < 1,O > ) A

+

(f:l

6 + f21 )A = 6 f21(< 1,0 >A) =

G, < y,6 >A = (fTl < 1,0 >

< 1,0 > ) A = fy ( < 1,0 > A )

11

+

187

Solution of an open problem

'

fY1 < a,B > + fZ1 < a,B > = < y a , h > = < y,6 > @ ( a ) . Thus, O(a) = A ,

a E

with

{o}.

F

0

From this proposition it follows that a full set of orbit representatives of Aut

is given by {< l,h >Ih E F] U {< 0,1 >, < 0 , O > I . Now consider 1 fll. It certainly is an idempotent in R and, since 1 = {< h,O >lh E F], fll is of rank 1. Also, fil f I, the identity

G

R

on

G

the matrix f1 G 11 matrix.

At this p i n t we choose a specific F , field

arising from Q(x),

(F,+,o)

namely the infinite Dickson near-

the field of rational functions over the

rationals, by defining multiplication as follows:

where d = := degree (p(x)) - degree (q(x)). q (x) or [ 5 1 for more details. We need the following:

Lemna

See

[3],

example 8.29,

Let F be the i n f i n i t e near-field as defined above and suppose i s a f i r s t row matrix. Then u< 1,0 > = < -,R(x) O > f o r some s (XI poZynomiaZs R(x) rmd S(x) i n Qtx]. Furthemore, there i s an i n f i n i t e sequence of functions {niliEN c_ F such that 3

U E IM2(F)

P. (XI

f o r polynomiaZs

and Q .(x) where

P ,( x )

Pi(x)

i.e.

d(-)

QiW

G k (say) for a l l

such a m y that

{d(qi)}

.

i E N

Qi (XI

1 i s bounded from

Also, we may choose the

q.

above,

in

i s not bounded from below.

h o o f The infinite sequence of functions we have in mind here is t h e set 1 1 { ~ j ~ =f o~r ,m E N large enough. Note that d(-) = -i, so that X

X

1

{d(T)lY%

is not bounded from below.

We use the following induction argument

X

to justify our claim: matrices of the form

and

U< 1

1, > ~ = < 0 X

is fixed for all that

+

Since

x fll

is a first row matrix, it can be built up from

U

and

fY,(h,y E F),

a -1,0 s(x) ' xi

i s 1.

>

for all

Now suppose V

by lemma 1.

Note that d(-) r(xJ s (x) are first row matrices such

i 3 1.

and W

J.H. Meyer and A.P.J. van der Walt

188

for all

i

> m3

(say) where m

3

> m1'

Pi(x) Since d(-)6 qi (x)

N

kl and

d(g(x h(x

+

+

+

+

i))

g(x + ; i+ i)pi(x) a

kj (say), such that

d(

h(x

+

+

i, we must have

is independent of

i)

< k3

for all

i

>

1.

i)qi(x)

go Actually, we also have to consider the case '"::f

V(V f I), but since we can g (x)

assume

to be a first row matrix, by lemma 1 ,

fh(x) V = 0. If 12

V = I, then

go

go 12

V

v

= fh(x)

12

and we have considered this case already at the beginning of our

189

Solution of an open problem Taking lemma 1 into consideration, the result follows.

induction process.

0

We will use lemma 3 to prove the next proposition, from which it will follow that Ann (G R

Proposition

1

f G) = 0, showing that the exceptional case 11

Let

2

can occur.

(+)

be t h e i n f i n i t e near-field as i n lemma 3.

F

Suppose

{ B I , B 2 , ..., Bm 1 i s a f i n i t e s e t of o r b i t s of AutRG on G. Then m

U 8 ) = 0. i=l i

Ann (G R

Proof

Bi

We may assume that the

are non-zero.

Consequently, an orbit

< 1,0 >, < 0,1 > or < 1 , x >

may have representatives:

8

for a non-zero h E F.

Now suppose m ig18i) f 0 .

AnnR(G

l<j G m

and U

B

such that U B . f 0 for a certain 1 for infinitely many orbits BT. We have to consider

U E MZ(F)

Then there is a matrix = 0

three cases: (i) 8. is generated by < 1,0 >. I Suppose U< 1,0 > = < a,B > with (say) a # 0, -1 then fyl u< 1,0 > = < 1,0 > and also,

-1 fyl U< 1.1 > = < O , O

for almost

?

-1 h E F, i.e

all

U< 1,x > = < 0,O >

fyl

E F.

a finite number of

according to lemma 1.

A E F, except maybe

for all

-1 Furthermore, f y l U

is a first row matrix,

So, without loss of generality, we may suppose

is a first row matrix such that U< 1,0 > = < 1,0 >

< 0,O >

for almost all

of functions

for all

i

different

>

1.

ni,

for almost all

But since

rl..

for infinitely many

So

B.

I

Pi (x) and such that d(-)< k (say) Qi (x) belong to different orbits for

in Q[x]

< l,rli >

we must have

Pi(X)

d(rli ) = -d(rli)

By lemma 3 there is an infinite sequence Pi (x) such that U< l,n > = < 1 + erli,O > for i Qi (X)

and Qi(x)

- - -1

h E F.

-

‘Qi’iEN

polynomials Pi(x)

U

U< 1,h > =

and

u<

1,Q. > = < 1 1

+

Pi (x)

Q~(X)O‘it0

>

=

< 0,o >

This implies that

-1

-rli

(inverse with respect to

o

)

i.

This is clearly a contradiction, since Pi (x) ( ) can be made arbitrarily large, while d

cannot be generated by

Q, (x)

< 1,0 >.

k.

J.H.Meyer and A.P.J. vun der Walt

190

i s generated by < 0,l >.

(ii) 8

1

Then, in the same way as (i), we can find

a matrix V E M2(F) such that V< 0,1 > = < 1,0 > and V< 1,x > = < O , O > for almost all A E F. But then, if U = V(f 112 + f21 1 + f22), 1 we have and U< l,h > = < 0,O >

U < 1,0 > = < 1,0 >

E F, which

for almost all

is impossible by (i).

(iii) 8

i8

1

< 1,x > f o r a mn-zero

generated by

A E

F.

As before, we can

such that V< 1 , x > = < 1,0 >

find a matrix V E M2(F)

and

V< 1,C > = < 0,O > for almost all 5 E F. But then, if 1 x 1 U = V(fll + f21 + f22), we have U< 1,0 > = < 1,0 > and U<

l,C >

>

= < 0,O

5 E F, a contradiction by (i).

for almost all

Thus, we must have m

U B

Ann (G R

-

By a result in 111, 55 (pp 89 A :=

Aut G , R

i=l i

) = 0.

0

2 there is a mapping m E MA(F ) ,

go),

where

such that

rn< X , O > andrnc

A,B

>

= < X,O >

for all

X E

0,O >

for all

h,@E F ,

= <

F

B

# 0.

Hence, by proposition 2 , cur example also shows that there are functions in 2

M (F ) A

which are not matrices.

In the next result we shall see that, although there are minimal R-subgroups

A

in this near-ring, for example ideals in R.

{fll

6 + f211X,6 E

there are no minimal left

F),

This implies that not only does the exceptional case

(+)

occur

in our example, but it occurs exclusively.

Proposition

3

Let

be t h e same near-ring as before.

R

Then R

does *lot

contain a minimal left i d e a l .

Proof matrix in

Suppose

f.

f

R

is a minimal left ideal of

and that

U

is a non-zero

According to theorem 4 . 3 [ 11, any non-zero element of

rank 1 and consequently, rank finding an element X

U =

1.

f

W e shall produce a contradiction by

in a minimal left ideal such that rank X

> 2.

Since

non-zero, it must be non-zero on an infinite number of orbits of Aut

R

according to proposition 2 .

u< and

Thus, we can choose a,@ E F '

@

on

>

l

=

< y o X , & o h >,

with

A E

F

{O}.

a

U<

1,cC > = < Y,6

UT< 1,a-l0 B > = U< 1,B > = < Y O

A,&.

h >.

Now UT E LT, which is a minimal left ideal according to lemma 2 , because -1 1 a-1 T = f l l + f22 exists. Let us define the matrix W as follows: -1

w = +

f2* 2d-l

if

6 # 0

U

is

G,

such that

f l l + f22. Then

UT< 1,l > = and

IO),

G

1,a > = < y,& > # < 0 , o >

U< 1,B

Let T be the matrix

has

Solution of an open problem Then, if

5

:= a

-1

o B , we have 5

# 1

while

wuT< 1,l >

=

w< y,6 > = < l,x >

1,5 >

=

w< y o X , 6 0 X > = < X,xoX >.

and wuT<

Moreover, W T E LT. Now consider the matrix 1 + f12 1 + f21 5 v = fll The inverse of V

v-1

=

191

+

1 2. fZ

is

(1-0-1 1 fll (€11

f;; )

+

-1 f-c(l-c) 1 22 (fll + f;;)

+

+

1

f22

which can be easily verified by showing that W-l= V-lV = I. According to 1-a 2, hV is a minimal left ideal of R and furthermore WIppv< 0,l > = WUT< 1,l > = < l,x > and WIpI11< 1,0 > = WU"< 1,c > = < X,xoX > . Let X = Y ( 1 + WTV) - Y, where Y = f1l l + f22. x Then X E LTV, and

x<

0,l > = Y ( < 0 , l > + < 1,x = Y<1,1

Clearly, < 1,(x + 1)'

-

x >

and

+

x >

-

< l,(x

of Aut G on XG which implies rank R any minimal left ideals.

X

>)

-

Y< 0 , l >

< 0,x >

+

1)x > 2.

belong to different orbits

Consequently, R

does not have 0

REFERENCES Betsch, G., Some structure theorems on 2-primitive near-rings, Colloquia Mathematica Societatis Jdnas Bolyai, 6 . Rings, modules and radicals. Keszthely (Hungary), 1971. [ 2 1 Meldrum, J.D.P. and Van der Walt, A.P.J., Matrix near-rings. To appear in Archiv der Mathematik. [31 Pilz, G . , Near-rings (North Holland, 1977). Van der Walt, A.P.J., Primitivity in matrix near-rings. To appear in [4] Quaestiones Mathematicae. [5] Zemmer, J.L., Near-fields, planar and non-planar, The Math. student, 32 [ 11

(1964)

.

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Near-ringa and Near-fsldr, C. Betrch (editor) 0Elsevier Science Publishere B.V. (North-Holland), 1987

193

ARE THE JACOBSON-RADICALS OF NEAR-RINGS M-RADICALS ? Rainer MLITZ Institut fur Angewandte und Numerische Mathematik, TU Wien Wiedner HauptstraRe 8-10 A-1040 GIien Let U be a universal class of right near-rings (i.e. a class o f near-rings satisfying the law (x+y)z = xz+yz which is closed under taking homomorphic images and ideals). It is well known that a Kurosh-Amitsur-radical on U is a map N -t fN q N ( c means "ideal of") defined on U and satisfying : (pl) (pN+I)/I c p(N/I) for all I 4 N , (P2) P(NlPN) = 0 ( ~ 3 ) P(PN) = PN (p4) pI = I , I Q N imply I E ~ N. (pl) and ( p 2 ) define the Hoehnke-radicals and a Hoehnke-radical is a KuroshAmitsur-radical i f and only if it satisfies pN = 0 o) (0 # I a N PI # I ) (p5) (see [31 and e.g. [61). The Jacobson-radicals J2 and J3 are Kurosh-Amitsurradicals (at least for zero-symmetric near-rings), Jo and J1 are not because o f the lack o f ( p 3 ) - see [41. Y

3

-

Recently, the concept of Kurosh and Amitsur has been generalised to that of M-radicals (which include considerably many interesting factorisations of various structures) see [51. For our near-ring case, M will be a binary relation

-

on the universal class U satisfying the following conditions: (MI) A M N * A # 0 is a subnearring of N , ( ~ 2 ) N M N for all N # 0 in U , (M3) A M N * A M (pN or A = 0 for every homomorphism cp defined on N , (M4) A M N , I q N and A c I imply A M I . An M-radical is then a Hoehnke-radical satisfying (p5M) pN = 0 c) (A M N * pA # A ) . Notice that ( p l ) ,(p2) and (p5M) always imply (p4M) p A = A , A M N - A c p N . Direct application of the above yields Proposition 1 : If M i s any r e l a t i o n s a t i s f y i n g (MI) t o (M4) and containing the r e l a t i o n of being a nonzero ideal, t h e following assertions are equivalent: (1)

p

i s a Kwosh-Amitsur-radical

which is M-strong ( i . e . f u Z f i l s (p4M)) ;

R. Mlitz

194

(2)

p i s both an M-radical and a Xurosh-Amitsur-radical

;

.

p i s an &radical fuZfiZZing (p3) (3) N o t i c e t h a t P r o p o s i t i o n 1 i s v a l i d f o r Q-groups

.

A r e c e n t r e s u l t by Anderson, K a a r l i and Wiegandt ( I l l ) t h e n y i e l d s

C o r o l l a r y 1 : For zero-symmetric near-rings,

J 2 and J3 are M-radicals for

every reZation M s a t i s f y i n g (MI) t o (M4) and situated between the r e l a t i o n s of being a nonzero ideal and that of being a nonzero right-invariant subgroup of ( i . e . a subgroup A of N s a t i s f y i n g AN c A I .

N

By t h i s c o r o l l a r y , P r o p o s i t i o n 2.7 i n [51 y i e l d s C o r o l l a r y 2 : F o r zero-symetric near-rings and every r e l a t i o n M with (MI) t o (M4) situated between that of being a nonzero ideal and that of being a right-

invariant subgroup (# 01, the radical classes R2 and R3 of J2 and J3 are closed under M-extensions : A M N , A E R , N / 4 > E R mp N E R

.

(where denotes the ideal of N generated by A I

As e x h i b i t e d i n [5J, f o r a s s o c i a t i v e and a l t e r n a t i v e r i n g s . t h e M - r a d i c a l s w i t h r e s p e c t t o t h e r e l a t i o n o f being a nonzero s u b r i n g a r e e x a c t l y t h e s t r o n g Kurosh-Amitsur-radicals ; thus, f o r every r e l a t i o n M w i t h ( ~ 1 )t o (M4) s i t u a t e d between t h a t o f being a nonzero s u b r i n g and t h a t o f b e i n g a nonzero i d e a l , t h e M-radicals f u l f i l (p3). T h i s means, t h a t t h e sum R o f N c o i n c i d e s w i t h pN

.

N o f a l l r a d i c a l M-subobjects

P

I n general, we o n l y know t h a t R N i s c o n t a i n e d i n p N ; P thus, i n [51, besides t h e general case, “approximable r a d i c a l s “ (pN i s generated

as an i d e a l by a r a d i c a l M-subobject) and “attainable r a d i c a l s ” (pN = R N) were studied

. All

P

t h e n a t u r a l examples t u r n e d o u t t o be a t t a i n a b l e . I n t h e f o l l o w i n g

a n e a r - r i n g example o f an approximable, b u t n o t a t t a i n a b l e M - r a d i c a l i s g i v e n Example : F o l l o w i n g

[51, P r o p o s i t i o n 4.3

,a

.

class S o f near-rings i s the

semisimple c l a s s o f an M - r a d i c a l i f and o n l y i f N E S i s e q u i v a l e n t t o t h e prop e r t y t h a t every M-subobject o f N has a nonzero homomorphic image i n S t h e r e i t i s e a s i l y seen t h a t f o r any r e l a t i o n f i e s Nc M N

M

.

From

w i t h ( ~ 1 )t o (M4) which s a t i s -

f o r a l l non zero-symmetric n e a r - r i n g s N (where Nc denotes t h e con-

s t a n t p a r t o f N), the c l a s s S o f a l l zero-symmetric n e a r - r i n g s i s t h e semisimple c l a s s o f an M-radical on t h e c l a s s o f a l l n e a r - r i n g s

.

N a t u r a l examples o f such

r e l a t i o n s M a r e those o f being a nonzero subnearring resp. a nonzero ( i n v a r i a n t )

. The corresponding r a d i c a l i s g i v e n by pN = and i s t h e r e f o r e approximable . Suppose now t h a t p i s a t t a i n a b l e ; t h e n (by P r o p o s i t i o n 1) p i s N-subgroup

a

Kurosh-Amitsur-radical whose semisimple c l a s s i s homomorphically c l o s e d and contains a l l near-rings with zero-multiplication

.

By a r e s u l t o f Betsch and K a a r l i

(121, Theorem 3.3) S has t o be t h e c l a s s o f a l l n e a r - r i n g s , a c o n t r a d i c t i o n It follows t h a t o i s not attainable

.

.

195

Are the Jacobson-radicals of near-rings M-radicals? The above example shows t h a t (pl),(p2) q u e n t l y , t h e r e may be some hope t o f i t

and ( p 5 M ) do n o t i m p l y (p3); conse-

Jo and J 1 i n t o a w e l l - d e s c r i b e d (see 151)

r a d i c a l t h e o r y by t h e use o f a s u i t a b l e r e l a t i o n M M-strongness ( i . e . dical

.

. As

we a l r e a d y know, t h e

( p 4 ~ ) )i s a necessary c o n d i t i o n f o r a r a d i c a l t o be an M-ra-

I t i s s t r a i g h t f o r w a r d t o see t h a t (p4M) i s e q u i v a l e n t t o t h e M-regulari-

t y o f t h e corresponding semisimple c l a s s S , i . e . t o :

N E S

,A

M N

*A

has a nonzero homomorphic image i n S

.

P r o p o s i t i o n 2 : The semisimple classes of J o and J 1 are M-regular f o r every reZation M s a t i s f y i n g (MI) t o (M4) and contained i n the r e l a t i o n of being a non-

.

Consequently, the radical classes of Jo and J 1 are M-strong zero l e f t ideal f o r these r e l a t i o n s M Proof :

.

L e t L be a nonzero l e f t i d e a l o f a J1-semisimple n e a r - r i n g ; t o show

t h e l e f t - i d e a l - r e g u l a r i t y o f S1 (corresponding t o J1),

i t s u f f i c e s t o show t h a t

t h e r e i s some n o n t r i v i a l L-group o f t y p e 1. To t h i s aim t a k e an a r b i t r a r y N-group G o f t y p e 1

. LG

= 0 i m p l i e s L G AnnNG

. Since

the i n t e r s e c t i o n o f the

N - a n n i h i l a t o r s o f a l l N-groups o f t y p e 1 i s J1(N) = 0, t h e r e must be some G s a t i s f y i n g LG # 0

. In

such a group G, t h e r e i s some g w i t h Lg # 0, hence Lg = G

( s i n c e Lg i s a normal N-subgroup i n t h e simple N-group G); moreover, t h i s shows t h a t G i s a s t r i c t l y c y c l i c L-group. Thus, i f I H i / i E I l

i s a c h a i n o f normal

L-subgroups o f G w i t h Hi # G f o r a l l i E I,we have LHi = 0 f o r a l l i E I,and thus LH = 0 f o r t h e u n i o n H o f t h e Hi

.

mal L-subgroup K i n G s a t i s f y i n g LK = 0 type 1

.

By Z o r n ' s Lemma, t h e r e i s a maximal nor-

.

G/K i s t h e n a n o n t r i v i a l L-group o f

F o r Jo t h e p r o o f i s e s s e n t i a l l y t h e same ( i n t h i s case, t h e Hi

# G do

n o t c o n t a i n L-generators o f G, i m p l y i n g t h a t K does n o t c o n t a i n L-generators o f G)

.

T h i s p r o p o s i t i o n g i v e s some hope f o r 3, and J 1 t o be M - r a d i c a l s f o r "good"

. However,

relations M

a t l e a s t f o r J1, t h i s hope i s destroyed by t h e f o l l o w i n g

P r o p o s i t i o n 3 : Let p be a Hoehnke-radical on a universal c l a s s U of near-

.

rings If U contains a t l e a s t one near-ring N s a t i s f y i n g pN # 0 and p(pN) = 0 then p i s not an M-radical on L1 f o r any r e l a t i o n ifl s a t i s f y i n g ( b l l ) t o (M4)

.

Proof : Suppose t h a t p i s an M - r a d i c a l ; t h e n by (p5M), pN # 0 i m p l i e s t h e e x i s t e n c e o f some A M N w i t h pA = A i n pN

.

Now, ( M 4 ) i m p l i e s A M pN

assumption

. Because o f

, yielding

p(pN)

(M3) and ( p l ) , A i s c o n t a i n e d

# 0 i n contradiction t o our

.

C o r o l l a r y 3 : For most of the universal claeses U df near-rings (including that of a l l zero-symmetric near-rings with conanutative addition and a l l larger universal c l a s s e s ) , J 1 i s not an M-radical f o r any r e l a t i o n M s a t i s f y i n g (MI) t o (M4)

.

,

R. MIitz

196

Proof : F o l l o w i n g K a a r l i ([41), i t s u f f i c e s t h a t , f o r some f i n i t e group G w i t h a proper nonzero subgroup H, t h e c l a s s U c o n t a i n s t h e n e a r - r i n g o f a l l mappings from G i n t o G which preserve 0 and H (which t h e n f u l f i l s t h e r e q u i r e -

.

ments o f P r o p o s i t i o n 3 above)

F o r Jo t h e s i t u a t i o n i s more complicated: u n t i l now t h e r e i s o n l y one example ( g i v e n by K a a r l i [41) showing t h a t J o i s n o t idempotent

. Unfortunately, t h i s . Nevertheless, i t

example does n o t f u l f i l t h e requirements o f P r o p o s i t i o n 3 m i g h t be used t o o b t a i n a p a r t i a l answer f o r Jo

. To

c o n s t r u c t h i s example,

K a a r l i s t a r t s w i t h a c y c l i c group A o f o r d e r 4 ( g e n e r a t o r : a),

and t h e group

B o f a l l {O,lbsequences w i t h a f i n i t e number o f nonzero e n t r i e s . On t h e d i r e c t sum G o f these groups he considers t h e (zero-symmetric) n e a r - r i n g o f t r a n s f o r mations S generated by a t r a n s f o r m a t i o n so s a t i s f y i n g 2so(a+b)f! B f o r a = a. ( f o r more d e t a i l s see [41) . B i s t h e n an S-subgroup o f G, S/(B:G)S # 0 i s 2 1 - p r i m i t i v e , (B:G)S = 0 , i m p l y i n g Jo(S) = (B:G)S . F u r t h e r , K a a r l i considers t h e (zero-symmetric) n e a r - r i n g T o f a l l t r a n s f o r mations t on G which s a t i s f y :

[ o tg =

I

f o r g e 6

a(t)ao

1 B(t)ao

t

or

g = o

b ( t ) E 2AtB

E 2A

f o r a l l g E BLC

,

f o r a l l g E CLCOI

where C denotes t h e subgroup o f B c o n s i s t i n g o f a l l elements w i t h an even number o f nonzero e n t r i e s

.T

i s shown t o be 0 - p r i m i t i v e

.

The n e a r - r i n g N d e f i n e d on t h e d i r e c t sum o f t h e a d d i t i v e groups S+ and Tt by t h e f o l l o w i n g m u l t i p l i c a t i o n has S as i t s J o - r a d i c a l : (s+t)(s'tt') = with k(s',t) =

1

0

SS'

t

k(s',t)so

f o r s ' G f! B

B(t)so

+

or

f o r s'G E B

tt'

s' = 0

, s' #

0

.

Now, i f L i s any J o - r a d i c a l l e f t i d e a l i n N, by P r o p o s i t i o n 2 i t i s c o n t a i n e d i n Jo(N) = S and hence i n J o ( S ) = (B:G)s some t E T w i t h B ( t ) = 2 (stt)(mtt')

-

, we

( s + t ) t ' = sm

.

I f we t a k e some m # 0 i n (B:G)S and

obtain

+

2so ;

. Thus, - ( s + t ) t ' i s n o t i n (B:G)s and ( I ~ : G ) ~ c o n t a i n s no nonzero l e f t o f N . It follows t h a t N i s a near-ring without Jo-radical l e f t - i d e a l s i s n o t Jo-semisimple . Using t h i s example, we o b t a i n :

sm belongs t o t h e i d e a l (B:G)s o f S, b u t 2so(a+b) f B f o r a = a, (stt)(m+t') ideal which

P r o p o s i t i o n 4 : For most of the universal classes U of near-rings ( i n c h -

ding t h a t of a l l zero-symetric near-rings with c o m t a t i v e addition and a l l larger universal c l a s s e s ) , Jo is not an M-radical f o r any r e l a t i o n M s a t i s f y i n g

(MI) t o (M4) und contained i n the r e l a t i o n of being a nonzero l e f t ideal

.

197

Are the Jacobson-radicals of near-rings M-radicals?

The q u e s t i o n whether Jo i s an M - r a d i c a l f o r some n a t u r a l r e l a t i o n M remains thus open. I n connection w i t h P r o p o s i t i o n 3 i t would h e l p knowing t h e s o l u t i o n o f the following

Is t h e r e a (zero-symmetric) n e a r - r i n g N ( w i t h commutative a d d i -

Problem :

t i o n ) s a t i s f y i n g Jo(N) # 0 and J o ( J o ( N ) ) = 0 ?

REFERENCES 1

2

3 4 5

6

Anderson, T. , K a a r l i , K. and Wiegandt, R. , On l e f t s t r o n g r a d i c a l s o f near-rings, Manuscript . Betsch, G. and K a a r l i , K. , S u p e r n i l p o t e n t r a d i c a l s and h e r e d i t a r i n e s s o f semisimple classes o f n e a r - r i n g s , C o l l o q u i a Math.Soc.Janos B o l y a i 38 (Radical Theory, Eger 1982), 47-58, North-Holland Pub1 .Camp., Amsterdam/ Oxford/New York , 1985 Hoehnke, H.-J. , Radikale i n allgemeinen Algebren, Math.Nachr. 32 (1966), 347-383 K a a r l i , K. , On Jacobson t y p e r a d i c a l s o f n e a r - r i n g s , Acta Math. Hung., t o appear . MBrki, L., M l i t z , R. and Wiegandt, R. , A general Kurosh-Amitsur r a d i c a l t h e o r y , Manuscript . M l i t z , R. , R a d i c a l s and semisimple classes o f Q-groups, Proc. Edinburgh Math.Soc. 23 (1980), 37-41

.

.

.

SUMMARY

.

An example o f a non idempotent &radical o f near-rings i s given The behaviour of the Jacobson-radicals of near-rings with respect t o the concept of M-radical is studied I t turns out that J 2 and J are M-radicals f o r s e u e m l natural relations M, t h a t on most of the uniuersaj classes of near-rings J is not not an M-radical f o r relations M contained in that of being a nonzero ? e f t ideal and t h a t J1 i s in general not an M-radical f o r any r e l a t i o n M

.

.

This Page Intentionally Left Blank

Near-rings and Near-blds, C. Betsch (editor) 0 Elsevier Science Publishers B.V (North-Holland), 1987

199

ON MEDIAL NEAR-RINGS

S i l v i a P e l l e g r i n i Manara # ABSTRACT We study t h e n e a r - r i n g s N i n which x y z t = x z y t f o r a l l x , y , z , t ~ N and we c a l l them medial. We prove t h a t a l l such n e a r - r i n g s have a s u b d i r e c t s t r u c t u r e very s i m i l a r t o t h a t o f t h e s t r o n g l y I.F.P. nearr i n g s "61). I n t h e p a r t i c u l a r case o f medial r e g u l a r n e a r - r i n g s , we observe t h a t they g e n e r a l i z e t h e 8-near-rings o f L i g h [ 5 ] and t h e sem i - r i n g s o f Subrahamanyam [ 131 Moreover we o b t a i n a necessary and s u f f i c i e n t c o n d i t i o n f o r a nearr i n g t o be medial, r e g u l a r and s u b d i r e c t l y i r r e d u c i b l e . Afterwards we have s t u d i e d t h e s e t o f n i l p o t e n t s elements o f a medial n e a r - r i n a . We f i n d c o n d i t i o n s f o r t h e s e t Q(N) o f t h e n i l p o t e n t elements o f N t o be an i d e a l . I f N i s a medial r i n g , Q(N) i s an i d e a l and i t c o i n c i des w i t h t h e r a d i c a l s o f N. I n t h i s case N/Q(N) i s a s u b d i r e c t sum o f i n t e g r a l domains.

.

1. INTRODUCTION The i d e n t i t y " x y z t = x z y t " has been developed i n t h e t h e o r y of quasi-groups:

an

impulse t o t h e i r i n v e s t i g a t i o n goes back t o t h e ' 4 0 s w i t h Toyoda's theorem (see Toyoda, On axioms o f l i n e a r f u n c t i o n s , Proc. Imp. Acad. Tokyo, 1941, 17). We study t h e n e a r - r i n g s N i n which x y z t = x z y t f o r a l l x,y,z,t medial.

For n e a r - r i n g s , t h e

t y " and i t s study

f i t s into

E N and we c a l l them

m e d i a l i t y i s a c o n d i t i o n o f " p a r t i a l commutativithe

set

of

studies o f the near-rings w i t h

p r o p e r t i e s o f p a r t i a l commutativity. We see t h a t t h e medial n e a r - r i n g s general i z e t h e 8 - n e a r - r i n g s o f L i g h [ 5 l a n d t h e semi-rings o f Subrahamanyam [ 13

1

.

The necessary and s u f f i c i e n t c o n d i t i o n f o r a n e a r - r i n g t o be medial, r e g u l a r and s u b d i r e c t l y i r r e d u c i b l e p u t s i n evidence t h e connection between t h i s s t r u c t u r e and t h e weakly commutative n e a r - r i n g s . I n t h e study o f t h e s e t o f n i l p o t e n t elements, t h e m e d i a l i t y works i n a s i m i l a r way as t h e I.F.P.

We f i n d

c o n d i t i o n s f o r t h e s e t Q ( N ) o f n i l p o t e n t elements o f N t o be an i d e a l and show t h e connection t o t h e various prime i d e a l s o f N. I f N i s a medial r i n g , Q(N) i s an i d e a l , and i t c o i n c i d e s w i t h t h e r a d i c a l s o f N. I n t h i s case t h e med i a l i t y presents i t s e l f again as a c o n d i t i o n o f weak c o m m u t a t i v i t y : i n f a c t #Address:

S i l v i a P e l l e g r i n i Manara, F a c o l t B d i I n g e g n e r i a d e l l ' U n i v e r s i t a d i Brescia, V i a l e Europa 39. 25060 BRESCIA. Work c a r r i e d o u t on b e h a l f o f M.P.I.

S.Pellegrini Manara

200

N/Q(N) i s a s u b d i r e c t sum o f i n t e g r a l domains.

2. PRELIMINARIES N w i l l i n d i c a t e a l e f t n e a r - r i n g ; f o r t h e d e f i n i t i o n s and t h e fundamental nomention. An element e E N i s idempo-

t a t i o n s we r e f e r t o [ l o ] w i t h o u t e x p l i c i t e 2 t e n t i f e = e; i f x e N we i n d i c a t e A (N) = d r i n g N = N + N "mixed" i f No# { O ) and Nc# o c i m p l i e s xzy = 0 f o r a l l x , y , z ~ N , s t r o n g l y I.F.P.

I01

N / x =

and r e g u l a r i f W x a N 3

x ' x a r e i d e m p o t e n t elements f 4 ]

t y ~ /N x y

.A

=

0 1 and we c a l l a n e a r -

n e a r - r i n g N i f I.F.P.

if xy=O

I.F.P.

i f e v e r y homomorphic image i s

XX'X.

In a r e g u l a r n e a r - r i n g x x ' and

.

We call a near-ring N medial if xyzt = xzyt for all x,y,z,t in N. 3.

SUBDIRECLY IRREDUCIBLE MEDIAL NEAR-RINGS P r o p o s i t i o n 1. A weakly commutative $ near-ring is medial.

I n f a c t if N i s weakly commutative x y z t

=

zxyt

=

x z y t and N i s m e d i a l .

P r o p o s i t i o n 2. Let N be medial:

1. If e # 0 is an idempotent element of N, A (el is an ideal of N. d 2. I f N has a left identity, it is wakly commutative.

k

k k

3. For all x, Y E N and k integer, ixyl = x y

.

1. We know t h a t A ( e l i s a r i g h t i d e a l o f N. For ncN and y6A (N) we have: d d eny = eeny = eney = 0 and A ( e ) i s a l e f t i d e a l . d 2. I f u i s a l e f t i d e n t i t y o f N, xyz = uxyz = u y x z = y x z , f o r a l l x , y , z ~ N and N i s w e a k l y commutative. 3. T r i v i a l . P r o p o s i t i o n 3. I f N is a mixed medial near-ring, N

N N = {O 1 and NoNc = c o I f N i s m e d i a l , V XEN, WnotNo, Onn

=

OOnn

=

OnOn

0

0

o f N. I t f o l l o w s t h a t N N = I 0 1 and N N = N c o o c c

.

0

=

is an ideal of N,

On

= 0

0 and N

0

i s an i d e a l

P r o p o s i t i o n 4. The homomorphic images of a mediaZ near-ring are medial. The p r o o f i s t r i v i a l . The c o n s t a n t and t h e z e r o - n e a r - r i n g s

a r e m e d i a l . I n t h e f o l l o w i n g we w i l l

e x c l u d e t h e s e t r i v i a l cases. Theorem 1. If N is a non trivial medial subdirectly irreducible near-ring,N

satisfies one of the following properties: §

a. n e a r - r i n g N i s w e a k l y commutative i f x y z = y x z

f o r a l l x,y,z E N , [ 1 0 1

.

20 1

On medial near-rings

1. N is zero-symmetric, simple and each non-zero

idempotent is a Left identity;

2. N is zero-symmetric and the intersection of the non-zero ideals of N is non-

zero and without non-zero idempotents; 3. N is mixed and when the intersection I of the non-zero ideals of N contains an

idempotent, then I = N

.

L e t N be medial, s u b d i r e c t l y i r r e d u c i b l e and non t r i v i a l . I f N i s simple and zero-symmetric, N s a t i s f i e s 1.:

i n f a c t , i f i t has an idempotent e, Ad(e) = I 0 1

because A ( e ) i s an i d e a l . Therefore e ( e n - n) = 0, W n e N , i m p l i e s e l e f t i d e n d t i t y . L e t N be zero-symmetric b u t non simple and l e t P be t h e i n t e r s e c t i o n of t h e non zero i d e a l s o f N. Now P # t O 1 and we suppose t h a t

P c o n t a i n s an idempotent

e. I f t h e r e i s an element n E N such t h a t en # n, we have A d ( e ) # 1 0 1

,

hence

n - e n E A ( e l . Therefore P i s c o n t a i n e d i n Ad(e) and t h u s e E A ( e ) , t h a t i s d d 2 = e, a c o n t r a d i c t i o n . Otherwise e i s l e f t i d e n t i t y c o n t a i n e d i n P and P = N

e

because N i s zero-symmetric,

b u t t h i s i s excluded. Therefore P d o e s n ' t c o n t a i n

idempotents and N s a t i s f i e s 2. Suppose N i s mixed

. We know t h a t

No i s an i d e a l

o f N and so N i s non simple. I f P i s t h e i n t e r s e c t i o n o f t h e non zero i d e a l s o f N, we have P c N

0

. If P has an idempotent e #

0, i t cannot be en # n f o r n E N,

because

otherwise e E A ( e ) and t h i s i s excluded. Therefore, i f P has an idemd p o t e n t e # 0, t h i s element i s l e f t i d e n t i t y , P = N and N s a t i s f i e s 3. 0

C o r o l l a r y 1. A non trivial, medial near-ring is isomorphic to a subdirect sum of near-rings that satisfy one of the properties of Th. 1.

I t f o l l o w s immediatly from Prop. 4 and Th. 1. I n f a c t each n e a r - r i n g i s i s o -

morphic t o t h e s u b d i r e c t sum o f s u b d i r e c t l y i r r e d u c i b l e n e a r - r i n g s ( s e e [ 2 1 ) .

4.

SUBOIRECTLY IRREDUCIBLE MEDIAL REGULAR NEAR-RINGS A z e r o - n e a r - r i n g i s medial b u t n o t r e g u l a r , a c o n s t a n t n e a r - r i n g i s medial

and r e g u l a r ;

i n t h e f o l l o w i n g , an MR-near-ring i s a medial r e g u l a r

n e a r - r i n g , and i t i s non t r i v i a l i f i t i s n o t constant. I n general a r e g u l a r n e a r - r i n g i s n o t s t r o n g l y I.F.P.,

t h i s happens i f N i s

medial: i n f a c t : P r o p o s i t i o n 5. An MR-near-ring is strongly I.F. P. L e t N be an MR-near-ring and I an i d e a l o f N: i f x y e I then f o r a l l n E N , xny = x x ' x n y = x x ' n x y c I . P r o p o s i t i o n 6. Let N be a non trivinl MR-near-ring; then

202

S. Pellegrini Manara

1. N has no nilpotent elements. 2. A (xi is an ideal of N, for alZ Z E N. d 3. I f x = x x ' x t N , then A (xx') = { O ) [ A I x ' x ) ={OI] i f f x x ' [ x ' x l is a LefL d d identity of N. k 1. If n i s n i l p o t e n t , n = 0 f o r some i n t e g e r k. Now n = n n ' n i m p l i e s n ' n = k k k ( n ' n ) = n ' n = 0 (Prop. 2.3) and hence n = 0.

A ( x ) , xny = x x ' x n y = x x ' n x y = 0. d 3. I n f a c t x x ' n = x x ' x x ' n [ x ' x n = x ' x x ' x n 1 and hence x x ' ( n 2. For n

t

N, and y

[x'x(n - x'xn) = 0

1

E

.

I f A (xx') = i 0 )[Ad(x'x)

d i d e n t i t y . The v i c e v e r s a i s t r i v i a l .

=

101

1,

XX'

-

xx'n)

=

0

[ x ' x ] i s a left

P r o p o s i t i o n 7. Let N be a r.on trivial subdirectly irreducible MR-near-ring: 1. I f z is a non zero idempotent of N, A ( z l # ( 0) i f f z is constant and in

d

.

this case A l z ) = Iv d 2. If x is not constant, and x = xx'x, then xx' and x'x are not constant and non-zero idempotents and A (x) =

d

3.

0 1=

A

d

(xx') = A fx'x). d

N has a left identity.

1. I f z i s constant, A ( z ) # I 0 1 . Viceversa,let Ad(z) # { O 1 and A = n Ad(h) d w i t h A ( h ) # I 0 1 . Since N i s s u b d i r e c t l y i r r e d u c i b l e , t h e r e i s some W E A , w # 0. d Then a l s o W ' W E A (ww'w = 01, hence A i s an i d e a l . I f t h e r e i s an element 0 # YEN, 0, we g e t A (w'w) # { 0 I and so w ' w w ' w = 0, b u t t h i s i s excluded d Therefore f o r a l l Y E N ( y # 0 ) w'wy # 0 and A d ( w ' w ) = { O 1 . From

such t h a t w'wy (Prop. 6.1). Prop. 6.3,

=

w ' w is a l e f t i d e n t i t y . So f o r a l l idempotent z o f N such t h a t

A ( z ) # { O 1 and f o r a l l Y E N we have zy = z ( w ' w ) y = ( z w ' w ) =~ Oy. I n p a r t i c u l a r d i f y = z, z z = z = Oz,and z i s c o n s t a n t . F i n a l l y , i f z i s constant,A ( z ) ? N . i f d 0' y t A ( z ) , zy = 0 = Oy and y t No. d 2. L e t x=x +x be a n o t c o n s t a n t element o f N. I f x x ' i s c o n s t a n t , we o c have x = x x ' ( x + x ) = x x ' x t x w i t h x x ' x = 0 because o f N N = ( 0 1 Thereo c o c 0 c o f o r e x must be constant, a c o n t r a d i c t i o n . I f x x ' = 0, we have Ox = x and x i s

.

again constant. I n a s i m i l a r way we prove t h a t x ' x i s not c o n s t a n t . F i n a l l y from (XI { O ) = A (xx') = A (x'x). d d d 3. This f o l l o w s from t h e Th. l.,keeping i n mind t h a t now N i s r e g u l a r and

t h e above p o i n t 1.,A

cannot be o f t y p e 2 (see above p o i n t 2. and Prop. 6.2) Theorem 2. A non-trivial subdirectZy irreducible MR-near-ring with a right distributive element h is a commutative field.

203

On medial near-rings Obviously h i s zero-symmetric and so Ad(h) Under our hypotheses, AS(h) i s an i d e a l o f N

= {

. We

0 1

. Let

A ( h ) = I y e N / yh=Ol. S

suppose A ( h ) # t 0 1 and we S

A = n A ( x ) ( A ( X I # I 0 1 ) . Since N i s s u b d i r e c t l y i r r e d u c i b l e , d d AS(h) n A = L # t 0 } I f w e L we have wh = 0 and t h e r e f o r e Ad(w) # t 0 1 and w

let

.

constant, b u t w e A ( x ) = N and t h i s i s excluded. Consequently A ( h ) = { 0 ) . Mo0 S d reover N i s zero-symmetric: i n f a c t f o r a l l a € N, ( 0 a ) h = 0 = OaOh = 0 and

A ( h ) = { O 1 i m p l i e s Oa

=

S

and x # 0,

we g e t

0. I f N i s zero-symmetric i t i s i n t e g r a l : i f xy = 0

x'xy

=

0 (where x = x x ' x ) and x ' x i s a zero-symmetric

idempotent element, t h e r e f o r e y = 0. Also

-

(n

n x ' x ) h = nh

n

=

=

xx'xyxx'

-

nx'xh

=

N has i d e n t i t y : i n f a c t

0 because x x ' i s l e f t i d e n t i t y , b u t A ( h ) = S

{

0 1 and so

n x ' x f o r a l l nE N. In our case N i s medial and so f o r a l l x, Y E N , xy = =

xx'yxxx'

=

y x . So N i s a commutative f i e l d .

Theorem 3. A non trivial MR-near-ring is isomorphic to a subdirect sum of subdirectzy irareducible near-rings that are fields or near-rings with left identity in which each non constant element x = x x l x is such that xx! and z'x are left identities. I t f o l l o w s f r o m Th. 2, Prop. 7, keeping i n mind t h a t each n e a r - r i n g i s i s o -

morphic t o t h e s u b d i r e c t sum o f s u b d i r e c t l y i r r e d u c i b l e n e a r - r i n g s ( [ 21 ) . I f a n e a r - r i n g i n which each elemnt i s power o f i t s e l f ( [ 1 1

,[ 7 1 )

i s called

MP-near-ring, we observe t h a t an MP-near-ring i s r e g u l a r and t h e r e f o r e Th. 3 app l i e s . The MP-near-rings g e n e r a l i z e t h e 8 - n e a r - r i n g s o f L i g h [ 5

1 because

a wea-

k l y commutative n e a r - r i n g i s medial and t h e 8-near-rings Consist o f idempotent elements. I n t h i s way t h e r e s u l t s o f [ 5 We g i v e now a (see [ 8

characterization

1

of

are c o r o l l a r i e s

to

Th. 3 ( s e e [ 7

I).

s u b d i r e c t l y i r r e d u c i b l e MR-near-rings

I).

Theorem 4. A uear-ring N is a non trivial subdirectly irreducible MR-nearring iff: 1. it is a zero-symmetric, weakly commutative, N-simple and integral near-ring; 2. it is a weakly commutative near-ring,

with N

0'

regular, minimal ideal,

N -simple, integral; with N unique N-subgroup of N. Moreover the right c' annihilator of each non-constant element is zero.

L e t N be a non t r i v i a l s u b d i r e c t l y i r r e d u c i b l e MR-near-ring.

It s a t i s f i e s t h e

hypotheses o f Th. 1, b u t n o t t h e p r o p e r t y 2 because N i s r e g u l a r . I n f a c t , if

I i s t h e i n t e r s e c t i o n of t h e non zero i d e a l s o f N, f o r i € 1 , t h e s i s some i ' c I

204

S. Pellegrini Manara

such t h a t ii' and i ' i a r e non zero idempotent i n I and t h i s i s excluded. Theref o r e , i f N i s zero-symmetric i t i s simple and a l l i t s non-zero idempotents a r e i s a weakly commutative n e a r - r i n g (Prop. 2.2) and f o r a l l

l e f t identities; N

X E N , A ( x ) = { O l (Prop. 6.2).Hence N i s i n t e g r a l . Moreover N i s N-simple: l e t d R be an N-subgroup o f N; we have R N c R . I f X E R, t h e r e i s some X ' E N such t h a t

xx'x

=

x with xx'

being a l e f t i d e n t i t y o f N, belonging t o R. Then N C R .

V i c e v e r s a , i f N i s a weakly commutative n e a r - r i n g i t i s medial (Prop. 1 . ) and obviously

i t i s s u b d i r e c t l y i r r e d u c i b l e because i t i s simple. Moreover i t i s

r e g u l a r : i n f a c t f o r a l l a e N aN i s some

YE

N because now N i s N-simple and hence t h e r e

=

N such t h a t ay = a and f o r a l l Z E N , ayz

i n t e g r a l . Now W aEN, 3 a ' e N / aa'

=

=

az hence y z

=

z,because N i s

y and a a ' a = y a = a i m p l i e s t h a t N i s r e g u l a r .

Otherwise, f r o m t h e Th. 1. N i s mixed and No i s t h e minimal i d e a l o f N. We N t N . L e t R be a proper non-zero N-subgroup o f N and R n N # R; o c C then t h e r e a r e a c R \ N and a ' such t h a t aa' i s a non zero and non constant i -

know t h a t N

=

C

dempotent o f N (Prop. 7 . 2 ) .

Therefore A d ( a a ' ) = t 0 1

Nc. L e t us prove t h a t N 0

RCNc and so R = N

C

i s regular. If

0

group o f N i s N

i s a l e f t i d e n t i t y and

I f now RNcR, a a ' c R and aa'N

N i s weakly commutative (Prop. 2.2). which i s excluded.Therefore

, aa'

and i t cannot be xN

= (

XE

. The

No, xN

=

NcR

unique p r o p e r N-subgroup i s N because t h e unique N-sub-

0 ) because x i s non c o n s t a n t . Moreover

h + h such t h a t x ( h + h ) = y. o c o c Such y e x i s t s because xN = N, b u t now x ( h t h = y i m p l i e s xh t h = y and so o c o c y = 0. Therefore N i s N -simple, i t i s an i n t e g r a l zero-symmetric weakly comxN = N 0

C

0'

i n fact: for a l l Y E N

0

0

there i s h

=

0

= 10) , d f o r a l l n o t constant elements x. Viceversa, i f N i s a weakly commutative medical

m u t a t i v e n e a r - r i n g . From t h e above p o i n t 1, N i s r e g u l a r .

Finally,

0

n e a r - r i n g w i t h a minimal i d e a l ,

A (x)

i t i s s u b d i r e c t l y i r r e d u c i b l e . We prove t h a t

N i s r e g u l a r . If n = n + n i s a t y p i c a l element o f N and 0 # n ' e N 0 0' o c nn' # 0 because Ad(x) # I0 ) i f f x i s a c o n s t a n t element. Therefore ( x t 0 o = N because t h e unique N-subgroup o f N i s N and so t h e r e i s Y E N such C'

then x )N = c that

ny = n + n and f o r a l l Z E N we have nyz = nz t h a t i s yz = z. Otherwise t h e r e o c i s Y ' E N such t h a t n y ' = y and t h e r e f o r e n y ' n = n. I f n i s a zero-symmetric e l e ment t h e r e i s some n ' such t h a t n t h e a s s e r t i o n is tri v i a1

=

nn'n, so N

.

0

i s r e g u l a r ; i f n i s constant

C o r o l l a r y 2. The ideals of a non trivial subdirectly irreducible MR-nearring N are of the form N + H, with H being an idea2 of N C

I f N has i d e a l s , i t i s o f t y p e 2 of Th. 4.

.

Now i f I i s one o f i t s i d e a l s ,

205

On medial near-rings

i t i s of t h e form I = N +I w i t h I an i d e a l o f N .If I = Io+ I I = InN I =InN o c C C c' 0 0' c c Now N i s minimal, s o I = N and I cannot be an i d e a l b u t It i s normal i n N 0

0

0

.

C

and t h e n It i s normal i n Nc. C

Viceversa if I

I = N t I i s an i d e a l o f N. I n f a c t f o r C' o c n t i t i - n - n = n t ( n t i - n ) t ((11 t i c 0 c - c - 0 0 c 0 c c c - n ) - n - ( n t i - n ) 1 t ( n t i - n 1 = n f i w i t h 7 E I because 1; i s c 0 c c c c c c o c c c normal i n Nt Therefore It i s a normal subgroup o f Nt. Moreover N I = N(N t I c ) = I and nEN, n

i s

C

i s an i d e a l o f N

-

n = n

t

i

=

I and f o r a l l n, n ' c N and

t 0

C'

= NN

t

= (n

t

o

I LN

0

I

t

c o c i ) ( n ' t n ' ) - n(n' o c o

n') c

t

=

(n

i t

i ) n ' - n' o c

t

I : (n

-

t

i ) n ' - nn' =

nn' E N c I . 0

0-

C o r o l l a r y 3. A non trivial subdirectly irreducible MR-near-ring is a right duo-near-ring. § I f N i s o f t y p e 1 o f Th. 4, t h e a s s e r t i o n i s t r i v i a l . L e t N be o f t y p e 2 of

I,

i c

=I01

n

t

i ) n ' - nn'

For n

=

0, i n '

0

=

=

I nN # 0

{

0)

C

N I L I and t h i s i s excluded ( N = N

0

0

5.

t

, J i s r i g h t i d e a l o f N and so J

a r e t h e sum o f N

N and f o r

i ) ( n ' t n ' - n(n; t n ' ) = ( n t i ) n ' - n n ' E N n I = C 0 0 0 o c 0 and (Prop. 7 11, i c N C , t h a t i s I L N ; b u t t h e n I i s

(n

=

a l e f t i d e a l ,too,because

J

nNo= { 0 1 we have f o r n, n ' E

4 and I a r i g h t i d e a l o f N. I f

t h e Th

0

0

i s m i n i m a l ) . Therefore

. The r i g h t

ideals of N

and o f a normal subgroup o f Nc,and these a r e a l s o l e f t i d e a l s .

NIL RADICAL OF MEDIAL NEAR-RINGS We w i l l denote by N' t h e m u l t i p l i c a t i v e semigroup of N; if K i s a subset of N

and n is an element o f N, we w i l l We w i l l

denote

write

t h e prime r a d i c a l s o f t y p e v (

P (N), P (N), see f o r i n s t a n c e [ 9

1

P(K)

2

1,

[ 12

=GKAd(k) v

=

o,

and L(K)=N\(Ku P ( K ) ) .

1, 2 ) w i t h Po(N),

1.

P r o p o s i t i o n 8. If K is a maximal subsemigmup of N . , N is the disjoint union of

K, P ( K I and L I K ) . F i r s t o f a l l , P ( K ) n K =$because K i s maximal: i n f a c t i f y c K w i t h xy =

XE

P ( K ) n K, t h e r e i s

O E K and t h i s i s excluded. By d e f i n i t i o n L ( K ) n ( K u P ( K ) )

=

@

and t h e r e f o r e p r o p o s i t i o n holds. Theorem 5. If N is medial and K is a maximal subsemigroup of N', then for each

y

E

L(KI there is some x E K such that xy is nilpotent iff N is zero-symmetric.

L e t ' s suppose t h a t f o r

Y E

L(K) there i s

§ A r i g h t duo-near-ring (see[ 5 are ideals.

1)

XE

K such t h a t xy i s n i l p o t e n t . From

i s a n e a r - r i n g i n which a l l r i g h t i d e a l s

S.Pellegrini Manara

206

Prop. 8 we w i l l prove t h a t L ( K ) , K and P ( K ) d o n ' t have constant elements. L f K I c No:

L e t y be a constant element o f L(K); from t h e hypotheses t h e r e i s some x

E

K with

xy n i l p o t e n t , b u t f o r t h e constant element y we g e t xy = y f o r a l l x t N, a c o n t r a t y and xy a n i l p o t e n t element, w i t h x t K . Then t h e r e i s an o c k k k i n t e g e r k , such t h a t ( x ( y o t y,)) = ( x y o t y c ) = 0. We o b t a i n ( x y t y ) = By +y = o c o c = 0 w i t h R element o f N. Now Byo€ No because N i s an i d e a l of N, y E Nc b u t

d i c t i o n . Let y = y

C

0

N n N = o c

K n N =$ c

(

0 1 and t h e r e f o r e yc= 0, so L ( K ) has o n l y zero-symmetric elements.

:

L e t k be a c o n s t a n t element o f K; from t h e Prop. 3, A d ( k ) l N

0

= @

and A d ( k ) n L ( K )

=

, b u t t h i s i s absurd because o f t h e c o n d i t i o n L ( K ) c N o above.

PIKIn N =

4:

I n f a c t , t h e elements o f P(K) are r i g h t a n n i h i l a t o r s o f some element o f K, and i f a constant element y belongs t o P(K), we have xy Viceversa, l e t N be zero-symmetric,

. Since y

nerated by K L J t y l i n K there are Put x1x2.

".

x .s t a i n x y ' l x yl* 1 2 k = il+ i2t

x =

~

...

x 2, , 1' x; o b v i o u s l y

... xsyi'

... t

=

y, f o r a l l x t N.

L(K) and M t h e subsemigroup o f N ' ge-

K, M c o n t a i n s K p r o p e r l y . Therefore O E M . Now i xs such t h a t xlyi1x2yi2 xsy = 0.

. ..

XE

x1x2

K. Applying t h e m e d i a l i t y k - 1 times we obilti2t...i 5 = 0, t h a t i s xyk = 0 where

...

SY

is. I f xyk = 0, a l s o x k - l x y k

Therefore xy i s n i l p o t e n t and

6.

Y E

=

=

xkyk = ( x y l k

and xy # 0 because of Y

=

o

(Prop. 2 . 3 ) .

4 P(K).

ZERO-SYMMETRIC MEDIAL NEAR-RINGS

I n t h e f o l l o w i n g our n e a r - r i n g s w i l l be zero-symmetric.

If N has subsemigroups in N', we define Q(NI = N\ mi&

U .€A

K a , where{ K' 1 is the f a =€A

of a l l maximal subsemigroups of IY-; if N has no multiplicative subsemi-

groups in N ' , we p u t Q1N) = N .

P r o p o s i t i o n 9. The set & ( N I is the set of a l l niZpotent eZements o f N. I f N has no subsemigroups i n N \ I O } Q(N) = N. I f N has subsemigroups i n N', subsemigroups i n N'.

, each element i s n i l p o t e n t . We have l e t { K a I a E h be t h e f a m i l y o f maximal

Since each Ka d o e s n ' t c o n t a i n n i l p o t e n t elements, Q ( N )

c o n t a i n s t h e n i l p o t e n t elements o f N.

Hence

i f x c Q ( N ) , x i s n i l p o t e n t be-

cause otherwise i t would generate a subsemigroup o f N ' contained i n a maximal subsemigroup K.

,

a contradiction.

207

On medial near-rings

We recall that: P2(N) 1 P ( N ) 1 1 (see [12 1 Def. 3.1, 3.2, 3 . 3 , 3.4) (a

1

PO(N)

In [ 1 1 lit is proven that P O ( N ) has only nilpotent elements. Therefore: Po

( 6 )

(N)

Q(N)

Theorem 6. If N i s a medial near-rings t h e n ( Y

1

P

-2

( N l 2 P (Nl 2 & I N ) 2 P o f N l 1

We have to prove that each prime ideal of type 1 elements o f N: from of

(a )

and ( 6

)

it will follow

( y

contains the nilpotent 1. Let P be a prime ideal

type 1 and .a a nilpotent element with order of nilpotency k . If aoP P , the-

re is t0E N such that a 1 = a0t0a0 f P; otherwise a0c P since P is prime of type 1. In this way we are able to construct two sequences of elements t a and It I n n such that a = a t a f o r n = 1, 2, ... and an f P. Moreover each a # 0, n n-1 n-1 n-1 n because a = a t a = 0 implies a E P and this is excluded. The element ak n n-1 n-1 n-1 n-1 k . “contains“ the element a 2 times; applying the mediality k-1 times, we 0 have a = 0, a contradiction. Therefore ac P. k Corollary 4. If N i s a medial near-ring i n which each prime i d e a l of t y p e 0 i s a prime i d e a l o f t y p e 1, t h e n &(hi) i s an i d e a l .

Corollary 5 . If N i s medial r i n g ,

Q ( N ) is an i d e a l .

7. MEDIAL R I N G S Theorem 7. If N i s a medial r i n g t h e n N\K subsemigroup K of N’

i s an i d e a l of N , for each maximal

.

I f y l , y are elements of N \ K , there are two elements m

m EK, such that 1’ 2 2 mlyl and in2y 2 are zero or nilpotent. Therefore there are two integer nl”l and n >1, such that (m y In’= (in y 0. If (m y )‘I= 0, applying n times the me2‘ 1 1 2 2 1 1 diality, we obtain ( m m y Inlt1 = 0. If (rn2y2ln2= 0, applying n times the me1 2 ln 2 diality we obtain ( m m y ) 2= 0. Let now n + n t n + 1. One of the terms in the 1 2 2 1 2 n . expansion of (mlm2y1 - m m y ) i s o f the form (m m y 1’1 (m m y )i2 (mlm2yl)i3 1 2 2 121 1 2 2 ... ( m 1m 2y 2 i where i 1+ i 2t ... t i k = n. Applying the mediality, we have: i ( m rn y 1’1 ( m m y 2 ... (in m y lik= ( m m y (m1m 2y 21’2. NOW i l s n l + 1 or 121 1 2 2 122 121 ‘i2bn and this term i s zero. Therefore the element m m (y - y ) is nilpotent 2 1 2 1 2 and y - y E N\K. Moreover, for all Y E N \ K and n c N, it is easy to verify that 1 2

-.-

ny and yn belong to N \ K .

208

S.Pellegrini Manara Theorem 8. I n a medial r i n g N , an i d e a l P i s minimal prime of t y p e 2 i f f

P = N

\

K f o r some m m i m a l subsemigroup K i n N'

.

x o r y belong t o N \ K because o t h e r w i s e x y t K and

If x y N~ \ K ,

L e t P = N'K.

t h i s i s excluded. Therefore N \ K i s an i d e a l o f t y p e 2. Moreover i t i s minimal of t y p e 2. We suppose t h a t H i s a prime i d e a l o f t y p e 2 contained i n N \ K . we prove t h a t

y t N\K:

YE

H. If

YE

from Th. 5 t h e r e i s some

N\K,

xy i s zero o r n i l p o t e n t . I f xy = 0, then x y H~ and t h e r e f o r e prime o f t y p e 2; b u t now x F H because x t K and then (xy)

S E

X E

XE

K such t h a t

H since H i s

H. I f xy i s n i l p o t e n t , s-1 H f o r some i n t e g e r s . Since H i s prime o f t y p e 2, xy t H o r ( x y ) E H.

A f t e r a f i n i t e number of steps

4H

y e H and s i n c e x

Y E

we o b t a i n i n any case x y H~ and t h e n

X ~ Y KE

Hence K

because otherwise x y e P and s i n c e P i s prime o f t y p e 2,

is

a

subsemigroup o f N',

subsemigroup K ' o f N'. f o r e N x K ' c P . So P

X E

H or

i t i s ~ E H Therefore . H = N\K.

Viceversa l e t P a minimal i d e a l prime o f t y p e 2. We p u t K = N \ P . also

Let

Now N \ K '

X E

YE

K,

P o r y e P.

t h e r e f o r e c o n t a i n e d i n a maximal

i s a minimal i d e a l prime o f t y p e 2 and t h e r e -

and P

N\K'

=

and

If x,

=

N\K'.

P r o p o s i t i o n 10. I f N i s a mediaZ r i n g , N\K

i s a minima2 prime idea2 of

type 1 , for a maximal subsemigroup K of N'

From Th. 7.,

N\K

of t y p e 2 and t h e r e f o r e prime o f t y p e 1.

i s prime

L e t us suppose t h a t a prime i d e a l H o f t y p e i s c o n t a i n e d i n N \ K . therefore there i s me i n t e g e r s , ( x y )

X E

S

=

K

LetyE N\K,

such t h a t xy i s zero o r n i l p o t e n t . I n any case, f o r so-

0. Applying t h e medial p r o p e r t y s times,we o b t a i n ( x n y )

= 0 f o r a l l n EN.

From Th. 6

, H

contains

the

s + l--

n i l p o t e n t elements

o f N, t h e r e f o r e x n y e H, f o r a l l nE N. As H i s prime o f t y p e 1, xNyc H i m p l i e s x c H o r y t H , s i n c e x # H, y e H and N \ K i s a minimal prime i d e a l o f t y p e 1. Now we are a b l e t o prove: Theorem 9. I n a medial r i n g ,

&fly)

From t h e d e f i n i t i o n , Q(N) = N\*:I\K,

= P in) = P f ~ ) . 2 1 =.:AN\

K a ) and t h e r e f o r e f r o m t h e Th. 8,

Q(N) i s t h e i n t e r s e c t i o n o f t h e prime i d e a l s o f t y p e 2. But P ( N ) c n ( N \ K , 2 =€A and t h e r e f o r e Q(N) = P 2 ( N ) = P 1 ( N ) . Theorem 10. A medial r i n g N

)

without n i l p o t e n t elements is a commutative r i n g .

We observe t h a t i n a medial r i n g N w i t h o u t non zero n i l p o t e n t elements, f o r 2 a l l x, Y E N , y x y = ( y x I 2 . I n f a c t f o r a l l x, Y E N xyxy = xxyy because N i s med i a l and t h e r e f o r e xyxy

-

xxyy = 0, t h a t i s x ( y x

-

x y ) y = 0. Since N has no non-

209

On medial near-rings z e r o n i l p o t e n t elements, x ( y x

-

xy)y = 0 i m p l i e s y x ( y x

I t ' s easy t o prove t h a t f o r a l l x, Y E N , n i l p o t e n t elements,xy

-

x y ) = 0.

L

( x y - y x ) = 0 and s i n c e N i s w i t h o u t

= y x and N i s a commutative r i n g .

C o r o l l a r y 6. If N is medial, N/&(NI is a subdirect sum of integral domains. Trivial.

REFERENCES H.E. B e l l , Near-rings i n which each element i s power o f i t s e l f , B u l l . Aus. Math. SOC. 2 (19701, 363-368. G. Gratzer, U n i v e r s a l Algebra, Van Nonstrand 1968. H.E. Heatherly, Near-rings w i t h o u t n i l p o t e n t elements, Publ. Math.Debrecen, 20 (19731, 201-205. H.E. Heatherly, Regular n e a r - r i n g s , J . I n d . Math. SOC. 38 (19741,345-354. S. Ligh, The s t r u c t u r e o f a s p e c i a l c l a s s o f n e a r - r i n g s , J . A u s t r . Math. SOC. 13 (19721, 141-146. S. Ligh, A s p e c i a l c l a s s of n e a r - r i n g s , J . A u s t r a l . Math. SOC. 18 (1974), 464-467. S. P e l l e g r i n i Manara, Sui q u a s i - a n e l l i m e d i a l i i n c u i ogni elemento e potenza d i se stesso, R i v . Mat. Univ. Parma, ( 4 ) ll (1985) S. P e l l e g r i n i Manara, Sui q u a s i - a n e l l i m e d i a l i r e g o l a r i , ( t o appear) S. P e l l e g r i n i Manara, Sul r a d i c a l e n i l d i q u a s i - a n e l l i m e d i a l i , ( t o appear) G. P i l z , Near-rings, North-Holland, Amsterdam, N.Y.(1977). D. Ramakotaiah, R a d i c a l s f o r n e a r - r i n g s , Math. Z. 97, 45-46. 0. Ramakotaiah, G.K. Rao, I.F.P. Near-rings, J . A u s t r a l . Math. SOC. ( 2 7 1 , (19691, 365-370. N.V. Subrahamanyam, Boolean semirings, Math. Ann. 148 (19621, 395-401.

SUMMARY

I quasi-gruppi mediali sono stati a lungo studiati; noi introduciamo nei quasi-anelli la 'xyzt = xzyt' per ogni x, y, z , t del quasi-anello e chiamiamo Mediali, in analogia a quunto visto nei quasi-gruppi, i quasi-anetli con questa propriet6. Nei quasi-anelli La Medialit6 si presenta come una condizione di "commutativitd parziale" e il suo studio si inserisce nella serie di studi di quasi-anelbi con propriet6 di commutativitci parziale. In generale dimostriamo che tali quasi-anelli hanno una struttura sottodiretta molto simile a quella dei quasi-anelli fortemente I.F.P. Se, in particolare sono anche regolari, sono isomorfi ad una somma sottodiretta di quasi-anelli sottodirettamente irriducibili che sono campi o quasi-anelli con identitci sinistra in cui ogni elemento non costante x = xx'x B tale che x'x ed xx' sono identitci sinistre. In questo mod0 vediamo che i quasi-anelli mediali generalizzano i 8-quasi-anelli di Ligh ed i semianelli di Subrahamanyam. Otteniamo anche una condizione necessaria e sufficiente affinchd un quasi-inello sia mediale regolare e sottodirettamente irriducibile. Questo risultato wette in evidenza le analogie fra questa struttura ed i quasi-anelli debolmente commutativi. Infine abbiamo studiato l'insieme degti elementi nilpotenti di un quasi-anello mediale. Anche qui la medialitci agisce in mod0 simile alla I.F.P. Inoltre troviamo legami tra 1 'insieme degZi elementi nilpotenti e i vari radicali primi e condizioni affinch2 tale insieme sia un ideate.

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Near-ringsand Near-fElds, G.Betsch (editor) 0 Elsevier Science Publishers B.V.(North-Holland), 1987

21 1

NEAR-RINGS AND NON-LINEAR DYNAMICAL SYSTEMS Gunter F. PILZ I n s t i t u t f u r Mathematik, Johannes-Kepler-Universitat Linz A-4040 Linz, A u s t r i a ABSTRACT

In t h i s paper we study l a r g e classes o f n o n - l i n e a r systems a d m i t t i n g a t r a n s f e r f u n c t i o n which completely describes t h e i r input-output behaviour. Many concepts which are widely b e l i e v e d t o be p a r t i c u l a r t o l i n e a r systems t u r n over t o t h i s c l a s s o f "separable" systems. These are systems X = (Q,A,B,F,G) i n which t h e map F,G can be separat e d as F(q,a) = a(q) + a(a), G(q,a) = y ( q ) + 6 ( a ) , where Q,A,B a r e groups, a,y maps and a,6 homomorphisms. I f we study separable systems w i t h i d e n t i c a l i n p u t - and output groups then i t turns o u t t h a t t h e p a r a l l e l - ( t h e s e r i e s - ) connections o f systems correspond t o the a d d i t i o n ( t h e composition, r e s p e c t i v e l y ) o f t h e i r t r a n s f e r functions. Hence these systems form a near-ring ( t h e "non-linear analogue o f a t o p a r a l l e l - and series connections. Many systemr i n g " ) w.r.t. t h e o r e t i c questions can be studied i n t h e framework o f near-rings. I n p a r t i c u l a r , we study feedbacks, r e a c h a b i l i t y questions, i n v e r t i b i l i t y w i t h delay L and close w i t h a discussion o f t h e r e a l i z a t i o n problem o f separable systems. D e f i n i t i o n 1: A ( d i s c r e t e , dynamical , t i m e - i n v a r i a n t ) nyntem ( o r aLLtomaton) I i s a quintuple I = (Q,A,B,F,G), where Q i s a s e t ( o f n R a t e ~ ) ,A a s e t ( o f LnpU%), B a s e t ( o f o u * p u t ~ ) ,F a f u n c t i o n Q x A + Q ( t h e n t a t e .thanbLtLon d u n d o n ) and G a f u n c t i o n Q

x

A

-

B ( t h e ou*plLt d u n d o n ) .

The d e s c r i p t i o n o f I i n D e f i n i t i o n 1 i s u s u a l l y c a l l e d the " l o c a l descript i o n " o f X. I n order t o o b t a i n t h e " g l o b a l " d e s c r i p t i o n , we do n o t consider a s i n g l e i n p u t , b u t a s e r i e s o f i n p u t s i g n a l s aiy

which e n t e r the system " a t time i E Z " . Hence w e ' l l consider i n p u t sequences (ai)i z . It i s generally assumed t h a t the i n p u t s d o n ' t come i n "since e t e r n i t y " ; so we assume t h a t t h e r e e x i s t s an index k € Z such t h a t ai = O f o r a l l i < k . Sequences o f t h i s type are u s u a l l y c a l l e d "formal Laurent series": D e f i n i t i o n 2: For any set X containing 0, l e t L(X) be t h e s e t o f a l l sequences (xi)i

, for

which t h e r e i s some k E Z such t h a t xi = O f o r a l l i < k.

The elements o f L(X) a r e c a l l e d (60hmaL) Lament h e h i e 6 o f elements o f X. I f X i s a f i e l d then L(X) i s j u s t the q u o t i e n t f i e l d (see e.g. r i n g X [ [ z l l o f ,jotunaL powetr h e h i e s (xo,x,,

...)

[ I l l ) o f the

( w i t h xi E X ) . I f we d e f i n e

212

G. Pilz

z:= (0,l ,O,O,...)

then u s i n g t h e usual a d d i t i o n and m u l t i p l i c a t i o n o f power

t 1

series, we can w r i t e X [ [ z l l =

1

L(X) =

xiz1

I

and i n the same s p i r i t we w r i t e

xiziIxiEX>,

i20 k c Z , xi EX}.

i>k Hence z i s n o t a "symbol" o r "indeterminate", as f r e q u e n t l y believed. The r e p r e s e n t a t i o n o f s e r i e s (xi)

i s u s u a l l y c a l l e d t h e z-hnbdozrn

by sums B i z '

o r discht-te Laplace t ~ n ~ d o l u nIt . was n o t r e a l i z e d from t h e beginning t h a t t h i s transform i s ( a t l e a s t a l g e b r a i c a l l y ) merely a t r i v i a l i d e n t i t y . t h e r e i s a l s o no need whatsoever t o w r i t e xaiz-i

(Actually,

i n s t e a d o f raizi.)

I n t h i s context, t h e i n t e r p r e t a t i o n i s as f o l l o w s . A t a c e r t a i n "time" k E

Z

(hence k can be negative), t h e system i s i n s t a t e qk when t h e f i r s t i n p u t ak a r r i v e s . The system produces an output bk=G(qk,ak) and changes i t s s t a t e qk i n t o qk+l

a r r i v e s , and so on.

= F(qkyak). Then ak+l

L e t us remark t h a t f o r continuous systems, t h e sequences (ak,ak+l which can be considered as f u n c t i o n s from

Z into A

-

,...) -

have t o be replaced by a

c o n s i s t e n t and causal f a m i l y o f i n p u t f u n c t i o n s from some time s e t T i n t o A.

t, as i n D e f i n i t i o n 1 , i s c a l l e d f i n e m i f Q,A,B f i e l d K and F,G a r e l i n e a r maps on t h e product space G can be decomposed i n t o l i n e a r f u n c t i o n s a:Q

+

a r e v e c t o r spaces over some Q x

Q, B:A

A.

+

Q,

I n t h i s case, F and y:Q +

B and 6:A

-+

B

A. If the v e c t o r spaces i n question a r e f i n i t e dimensional, a,B,yy6 a r e u s u a l l y r e presented by matrices. such t h a t F(q,a) = a ( q ) + B(a), G(q,a) = y ( q ) + 6 ( a ) h o l d f o r a l l (q,a) E Q

x

I t i s n o t t r u e , however, t h a t these decompositions are o n l y p o s s i b l e f o r

l i n e a r systems. We a r e going t o

i n t r o d u c e "separable" systems now. They a r e

much more general than l i n e a r ones, a l l o w h i g h l y n o n - l i n e a r t r a n s i t i o n and output f u n c t i o n s , b u t we can do w i t h these systems most t h i n g s we can do w i t h l i n e a r ones. D e f i n i t i o n 3: I (as i n D e f i n i t i o n 1) i s c a l l e d oepmabbe i f Q,A,B

a r e groups

( w r i t t e n a d d i t i v e l y , b u t n o t n e c e s s a r i l y a b e l i a n ) and i f t h e r e a r e maps a:Q -. Q, y:Q +

B, and homomorphisms 8:A

+

Q, 6:A

+

B such t h a t F(q,a) = a ( q ) + B(a),

G(q,a) = y ( q ) + 6 ( a ) h o l d f o r a l l q E Q , aEA. We then denote Z by (Q,A,B,a,p,y,6) o r simply by (a,B,r,6). C l e a r l y , each l i n e a r system i s separable. I f Q = A = B = ( R , + ) , = q2 + s i n q t 3a, G(q,a) = eq

- 1 + na

determine

F(q,a)

=

a n o n - l i n e a r separable system.

213

Near-rings and non-linear dynamical systems

Separable systems f i t i n t o t h e classes o f non-linear systems described by D.L.

Casti (131, p. 19).

The system M(G) o f a l l maps from an ( a d d i t i v e ) group G i n t o i t s e l f i s a "near-ring" w . r . t . pointwise a d d i t i o n and composition. A neat-hing N i s a gener a l i z e d r i n g i n which we do n o t assume t h a t a d d i t i o n i s commutative and i n which j u s t one d i s t r i b u t i v e law ( f o r t h i s paper: (a+b)c = actbc) i s assumed t o hold. We say t h a t n E N i s zetto-nymm&,ic

i f nO = 0 holds;

N i s zeho-nymmethic i f every

n c N has t h i s property. Mo(G) : = { f € M ( G ) l f ( O ) = O ) i s an example of a zerosymmetric near-ring. See [14] f o r t h e theory o f near-rings. For d i s c r e t e systems ( o r automata) i n which the s t a t e s e t Q i s a group, t h e r e i s a n a t u r a l associated n e a r - r i n g

N(z);

i t i s t h e subnearring o f M(Q)

generated by a l l f u n c t i o n s fa ( a E A ) w i t h f a ( q ) =F(q,a) and by t h e i d e n t i t y f u n c t i o n i d (because we d o n ' t want the system t o change i f no inputs a r r i v e ) . For separable systems, N(Z),

i s generated by i d and a i f Q i s abelian. This

construction o f a " s y n t a c t i c near-ring" i s the obvious e x t e n t i o n o f the monoid o f an automaton (see e.g.

[lll)i n t h e presence o f a group s t r u c t u r e on Q. N(,)

i s t y p i c a l l y n o t a r i n g , n o t even i n t h e case o f l i n e a r systems o r automata (see [61). I n t h i s paper, however, we show t h a t separable systems i t s e l f form a nearr i n g by means o f s e r i e s / p a r a l l e l connections o f these systems. For l i n e a r systems we do get a r i n g , see 1161. I n f a c t , our treatment f o l l o w s p a r t l y t h e ideas i n Sain's book [16].

I n order t o perform t h i s c o n s t r u c t i o n , we have t o

represent a system 1 by i t s g l o b a l input-output map .,f

This i s always possible

f o r separable systems (see below). Even if A i s "only" a group, we may w r i t e (ak,a ktl,...)

( w i t h k € Z ) as

1 aizi. I t i s one o f t h e fundamental ideas o f a l g e b r a i c systems theory t h a t i zk t h e s h i f t - t o - t h e - l e f t operator ( " u n i t d e l a y " ) (ak,aktl ) (ak,akt, .,) ( w i t h ak now a t t h e ( k - 1 ) s t place, etc.) j u s t corresponds t o the m u l t i p l i c a t i o n o f zaizi by z-'. Hence we denote the s h i f t operator by 2. Note t h a t L(A) (see D e f i n i t i o n 2 ) i s an a d d i t i v e group, too, under component-wise a d d i t i o n ; i n B f a c t , L(A) i s a subgroup o f A

,...

-.

,.

.

The map a:Q

-+

Q can be extended t o a map ( a l s o denoted by a) from L ( Q ) i n t o Also, we can extend 0 t o a map from L(A)

L ( Q ) by a(.l qizi) := 1 a(qi)zi. ilk iLk i n t o L ( Q ) , and so on.

214

G. Pilz P r o p o s i t i o n 1: For a separable system I: = (Q,A,B,~,B,Y,~),

- a + f : L ( Q ) + L ( Q ) i s always b i j e c t i v e . Also, - a +

f

t h e map

i s zero-symmetric i f f a i s

zero-symmetric.

1

1

qizi) = 1 -a(qi)zi + qiz i-1 = q k z k - l + ilk irk i>k qizi) = (-a+ Z ) ( , q/zi) i m p l i e s t h a t + 1 (-a(qi)+qi+,)zi. Now ( - a + f)( it k ibk ibk q.z1 and q;zi b o t h have t h e f i r s t non-zero c o e f f i c i e n t a t t h e same index n

Phaob: We have (-a+;)(

1

1

(say) , and qn = q,; -a(qn) + qn+l = -a(qA) + qA+l , -a(qn+l 1 + qn+2 = a(qA+l) + t q;\+2y... i m p l i e s qi = q i f o r a l l i. Hence - a + : i s i n j e c t i v e . I f qiziEL(Q) itk then i t i s easy t o see t h a t ( - a + i ) ( 1 q;zi) = 1 qizi if t h e q; a r e given i n it k ibk d u c t i v e l y as q i : = 0 and q;+l :=a(q;) + qi f o r i > k. This shows t h a t -a+; i s b i jective.

I f a ( 0 ) = 0 , i t i s c l e a r from the formula f o r -a+? t h a t -at;

zero element

x ozi

o f L ( Q ) i n t o i t s e l f . Conversely, i f (-a+Z)(

then we g e t a ( 0 ) = 0.

x 0zi)

maps the =

I 0zi

0

Now we a r e i n a p o s i t i o n t o g i v e a formula which d i r e c t l y r e l a t e s t h e output sequence o f a system t o i t s i n p u t sequence. Observe t h a t a l l c a l c u l a t i o n s have nothing t o do w i t h l i n e a r i t y . A l l we need i s s e p a r a b i l i t y . Theorem 1: I n a separable system

x

= (Q,A,B,a,a,y,6),

hold, i f t h e f i r s t non-zero i n p u t a r r i v e s a t time k E

t h e f o l l o w i n g formulas

Z i n which t h e system i s

i n s t a t e 0: (i)

1

qizi

1

bizi

= (-a+i)-l

(ii)

1

aizi)

= [y(-a+f)-lp+6](

1

aili)

it k

ihk

mod:

B(

il k

ilk

( i ) Since qi+l

= a(qi) + B(ai)

we g e t

1

qi+lzi

i>k = z

+ B(.I aizi). But i 1 qi+lzi+l ilk ibk = z(a( 1 qizi) + whence -a( 1 qiz i>k i>k = B( 1 aizi). I f q k = O then we can w r i t e (-a+z)( i>k P r o p o s i t i o n 1 , -a+; i s b i j e c t i v e , hence t h e r e s u l t .

(ii) f o l l o w s from (i), s i n c e bi=y(qi)+6(ai), + 6(xaizi).

qi+,zi,

: = y ( - a + Z ) - l ~ + 6 from L(A) i n t o L(B) i s c a l l e d t h e Observe t h a t f,

1

q i z i1 +

i>k hence

b i z i = y(.x qizi)+

hence

D e f i n i t i o n 4: I n a separable system I: = (a,B,y,d),

f,

= a(

it k

i >k

the f u n c t i o n ~ J L ~ F I A ~ C&ztLon VL

o f X.

i s zero-symmetric i f t h i s a p p l i e s t o a and y (by Proposi-

215

Near-rings and non-linear dynamical systems

t i o n 1 and t h e f a c t t h a t each group homomorphism (hence B and 6) i s automatic a l l y zero-symmetric). fx completely characterizes t h e input-output behaviour o f 1 i f X s t a r t s i n

s t a t e 0. I f 1s t a r t s i n a s t a t e q # 0 which i s reachable from 0 by means o f an we simply s t a r t a t time k-r, and then i n s t a t e 0. Since

i n p u t sequence al...ar,

i t does n o t make much sense t o s t a r t from non-reachable s t a t e s , f x " c h a r a c t e r i -

zes" X i t s e l f , We c a l l

x

zeho-nqmrnc&tic

Now we connect two systems tion

x1 tt z2

xi

=

i f t h i s a p p l i e s t o f,.

(9, ,Ai ,Bi ,Fi ,Gi)

r e q u i r e s B1 = A2. Then X1

( i = 1,2). The nenied cannec-

X2 : = (Q1 x Q2,A1 ,B2,F,G)

with

F( (4, ,q2) ,a) = (F1(ql ,a) ,F2(q2,G1 (4, ,a) 1 and G( (ql ,q2) ,a) = G2(q2 ,GI (ql ,a) 1. The pm&d connection E l / t2 works w i t h A1 = A2 =: A, B1 = B2 =: B and gives Xl//x2:=(Q1

x

Q2,A.B,F',G')

G ' ( (ql ,q2) ,a) = GI (ql ,a)

w i t h F'((ql,q2),a) +

I f Xi = (Qi ,Ai ,Bi ,ai ,Bi ,yi ,ai) F((ql

,a) = (al(ql

= (Fl(ql,a),F2(q2,a))

and

G2(q2 ,a). a r e separable then i n El

1 + ol(a),a2(q2)

+

tt

E~ we g e t

b2(y1(q1 1 + 6 1 ( a ) ) ) =

is

= (al(ql),a2(q2)+

B2ul(q1))+ (Bl(a),B26,(a)). Note t h a t a + (Bl(a),B26,(a)) a homomorphism. S i m i l a r l y , G((ql,q2),a) =y2(q2) +a2(y1(q1) +61(a)) = = (y2(q2)+62yl(q1))+626,(a). Hence El ++I2is again separable. I n I l / / X 2

we

g e t F ' ( (ql .q2) ,a 1 = (al (4,) + ( a ) ,a2(q2) + b2(a) 1 = (al (4,) ,a2(q2) 1 + ( a ) .b2(a)), and G'((qlyq2),a)=y,(ql)+61(a)+y (q ) + 6 2 ( a ) . I f B i s abelian, t h e l a t t e r 2 2 can be w r i t t e n as ( y l ( q 1 ) + y 2 ( q 2 ) ) + ( 6 1 + 6 2 ) ( a ) , and 6 , + 6 2 i s again a homomorphism. I n summary, we have shown P r o p o s i t i o n 2 : I f El and X2 a r e separable systems, t h e same a p p l i e s t.o tl *t2 and ( i f t h e output groups a r e abelian) t o X 1 / / X 2 - I f X1 and x2 a r e zerosymmetric then t h i s a l s o a p p l i e s t o

x1 ttz2 and rx1//x2.

Theorem 2 : If1 1 + are separable then (i)

fq//x2 = ftl

( i i ) fXl*x2

=

+

fX2 ( i f t h e output groups a r e a b e l i a n )

fx2 a fxl

Phood: These formulas a r e already motivated by t h e d e f i n i t i o n s o f and

tl*x2.

xl//rx2

The equations bi+l = y ( q i ) + 6 ( a i ) etc. o n l y r e l a t e t h e i n p u t a t time

i t o t h e output a t t h e same time i. But i n p u t s a t time i a l s o have i n f l u e n c e on l a t e r outputs ( w e ' l l i n v e s t i g a t e t h i s i n t h i s paper as w e l l ) , and t h i s i n f l u e n c e

i s described by .,f

So we g i v e d i r e c t proofs. Note t h a t t h e p r o o f o f Proposi-

216

G. Pilz

217

Near-rings and non-linear dynamical systems

I o f separable systems a t

Now l e t us assume t h a t we have a f a m i l y

hand which a l l have t h e same a b e l i a n group A as i n p u t and o u t p u t group. Under t h e s e assumptions we g e t t h e f o l l o w i n g C o r o l l a r y t o Theorem 2. C o r o l l a r y : The s e t S o f t r a n s f e r f u n c t i o n s o f those systems which can be constructed from t h e

zi

( i E I ) by means o f s e r i e s and p a r a l l e l connections

c o i n c i d e s w i t h t h e seminear-ring generated by {f, I i E 11 i n i i M(L(A)). I f A i s o f f i n i t e exponent t h e n S i s t h e n e a r - r i n g generated by t h e 's.

f 'i

T h i s h o l d s because i f e i s t h e exponent o f A t h e n eA=O, whence ef,

i s the

zero map, and so ( e - l ) f t = -fz can be o b t a i n e d by (e-1) p a r a l l e l connections o f

Note t h a t t h i s a p p l i e s i n p a r t i c u l a r t o f i n i t e groups A. O f course, S = t h e n e a r - r i n g generated by I f ] iE I} i f I c o n t a i n s w i t h each ti = (a,B,y,d) 'i a l s o t h e system -ti = (ayBy-y,-6) which y i e l d s as can be e a s i l y seen the

-

-

t r a n s f e r f u n c t i o n f - z =-fE. Also, t h i s can b e achieved b y " e x t e r n a l " devices which t r a n s f e r elements o f A i n t o t h e i r negatives. Hence i t makes sense t o impose t h e f o l l o w i n g Conventicxn f o r t h e r e s t o f t h i s paper:

I i s a family o f separable

systems which have t h e same a b e l i a n i n p u t - and o u t p u t group A and which cont a i n s w i t h each

xi a l s o -xi.

l i E I> i s a s u b n e a r - r i n g o f M(L(A)). If each fI i s zero-symmetric, 'i i SO i s < f I i E I> (as a s u b n e a r - r i n g o f Mo(L(A))). I f each f i s an endomor'i 'i phism on L ( A ) then < f / i E I> i s even a r i n g . Hence < f

'i

I f we i d e n t i f y a separable system 1 with i t s t r a n s f e r f u n c t i o n fy ( a s desc r i b e d e a r l i e r i n t h i s paper), we can r e s t a t e o u r r e s u l t s as f o l l o w s .

218

G. Pilz

Theorem 3: L e t S ( A ) be t h e s e t o f a l l separable systems w i t h a b e l i a n i n p u t -

,* )

and output groups A. Then ( S ( A ) , J /

i s a near-ring.

By standard means o f u n i v e r s a l algebra (see e.g.

[41),
I i E I>

=

I

Mf,

I . . . ’fXi ) t E T n ) , where Tn i s t h e s e t o f a l l terms (polynomials) i n il n t h e v a r i e t y o f a l l near-rings.

=

I i E I > , we t r y t o f i n d a more e x p l i c i t e ‘i formula f o r t h e a c t i o n o f fx. This can be achieved by the f o l l o w i n g I n order t o l e a r n more about
D e f i n i t i o n 5: I f X = (a,B,y,6), t h e n - t h Mahhov nymboL ( w i t h n 2 - 1 ) [a,B;nl i s the map from L ( A ) i n t o Q defined i n d u c t i v e l y by

[~,B;-II [a,~;n+11

1

aiz’)

:= o,[~,B;oI(

1

aizi)

:= a ( [ a , ~ ; n ~ (

i>k i>k

aiz’)

1

aizl))+B(ak+n+l)

i>k

=: a [ a , ~ ; n l

1

Note t h a t [a,p;nl

1

i2k

aizi

.I

i>k

aiz

i

:= a ( 0 ) + B(ak)

+

B(ak+n+l)

o n l y depends on ak,

i>k w r i t e [a,Q;nl

(ak,

...,ak+ n ).

...,a k+n .’ h

Obviously we g e t

Proposition 3: I f a i s zero-symmetric then each [a,B;nl and [a,B;nl

ce we can a1 0

aiz i = a ( a ( . . . a ( O ( a k ) + B ( a k + l ) . . . ) ) +

i s zero-symmetric

B(ak+n-l))+B(ak+n).

ibk

Since [a,B;O]

(a,)

d e f i n i t i o n o f [a,b;n]

= a ( 0 ) + p(ak) i s the s t a t e qk+l

Proposition 4: I f a separable system

r =(a,p,y,6)

s t a t e q k = 0 and i f the i n p u t sequence (aklak+l,...) states qk+l ,qk+*,.

( i f qk = 0 ) , we g e t by the

the following

.. such t h a t ,

f o r n E IN

qk+n = [ a , ~ ; n + l l (ak,

...,a k+n-1 )

From t h i s , we g e t the desired formula f o r f,: Proposition 5 : I f f x (

1 ibk

aizi)

s t a r t s a t time k i n the a r r i v e s then

=

1 ibk

bizi

then

x

assumes new

219

Near-rings and noelinear dynamical systems

Note t h a t Theorem 2 could a l s o be proved by i n d u c t i o n using t h i s formula. Now we i n v e s t i g a t e , which f u n c t i o n s f E M(L(A)) can a r i s e as f = ft f o r some separable system

x = (a,p,y,6).

The determination o f a,p,y,6

from f i s then

known as " r e a l i z a t i o n problem". I t i s u s e f u l t o e s t a b l i s h t h e f o l l o w i n g Notation: Lk(A) : = {

1

aizi /ai E A).

i>k

u Lk(A). Furthermore, l e t , f o r B,CEX, (B:C) be t h e s e t k€Z o f a l l f u n c t i o n s f from X t o X such t h a t f ( C ) c B . This coincides w i t h t h e O f course, L ( A ) =

"noetherian q u o t i e n t s " as d e f i n e d i n 1.41 o f [141. Since f z ( L k ( A ) ) E L k ( A ) f o r each separable system t, we g e t Proposition 6: L e t ti ( i E I ) be separable systems.

(ii)N(L(A)) i s a near-ring. l i E I> i s a subnear-ring o f N(L(A)).

( i i i )
'i (iv)

I f each ti i s zero-symmetric then i s a subnear-ring o f t h e 'i zero-symmetric p a r t No(L(A)) o f N(L(A)).

I f t i s zero-symmetric then i t makes no d i f f e r e n c e i f we consider

1

1

aizi) o r f ( aizi) w i t h ai = O f o r i < k . Hence i n t h i s case ft can be izk iEZ regarded as a member of If E M(Az I f ( i ) = 0 f o r a l l ic k, where k i s some i n t e ger (depending on f ) ) .

f(

Notation: I f (A,+) t h e s e t of a l l f',

i s an a b e l i a n group and (Q,+) any group, l e t N(A,Q) be

where I = (a,8,yY6)

i s any separable system on (A,A,Q).

Let

n(A,Q) be t h e subnear-ring cN(A,Q)> generated by N(A,Q) i n t h e n e a r - r i n g N(A) o f a l l separable systems on (A,A,Q) a l s o by f

f o r some Q. I f

x

i s as above, we denote fx

a, 8 ,Y $6'

Recall t h a t we g e t N(A,Q)cfi(A,Q) S N ( A ) SN(L(A)) SM(L(A)). t h i s chain (except t h e f i r s t one) a r e always near-rings.

A l l members o f

220

G. Pilz 0 1 ,... ,...

Now i f f E M ( L ( A ) ) should c o i n c i d e w i t h t h i s ,f we can w r i t e f = ( f ,f ) such t h a t , f o r a l l 1 aiziEL(A), f(Taizi)=(fo(Eaizi),f 1( x a i z i ) ). T h i s imposes q u i t e a number o f r e s t r i c t i o n s on t h e f u n c t i o n s fn (n E

No )

and hence

on f i t s e l f . By comparison o f components, we see t h a t

( 0 ) f o ( . l aizi) = y ( 0 ) + 6 ( a k ) . Hence fo i s an a f f i n e f u n c t i o n depending o n l y on i>k ak, and we can s o l v e f o r 6, s i n c e 6 = f g = t h e zero-symmetric p a r t o f fo. Also, y(0) =f: i s a l r e a d y f i x e d a t t h i s stage. (1)

fl(.l aizi)=u(a(0)+

~ ~ a k ~ ) + 6 ( a k + l ) = u ( a ( O ~ + ~ ( a k ) ) ~ Hence f~~ak+l~.

iSk

1 t h e r e must be a f u n c t i o n g EM(A) such t h a t f ' (

1

aizi)

1 = g ( a k ) + f:(aktl).

i>k We can always s o l v e if a ( 0 ) B I m 8 , and a l s o i n case a ( O ) E I m p i f we have g 1 (a,) =f:(ak) =y(O) f o r each ak such t h a t p ( - a k ) = - p ( a k ) =a(O).

.. .. .. .. .. . . . . . ( n ) The f u r t h e r equations determine a&,y moreandmore, and show t h a t each fn i s a f u n c t i o n j u s t depending on t h e f i r s t n + l c o e f f i c i e n t s ak,...,ak+n

1

aizi.

I n f a c t , t h e r e must be f u n c t i o n s gn : A

-,A

in

such t h a t

irk

"In p r a c t i c e " i t t u r n s o u t t h a t a,p,y

(and 6, anyhow) a r e a l r e a d y determined

a f t e r c o m p a r a t i v e l y few steps. T h i s i s t h e r e a l i z a t i o n procedure o f c a l c u l a t i n g out o f a given f = f

We i l l u s t r a t e t h i s i n t h e case Q = Z , . Z2, p:A + ZZ,, y : Z + A, 6:A -,A, and t h a t y(0) and 6 a r e f i x e d i n s t e p ( 0 ) as 6 = f i , y(0) =f:. I n s t e p ( 1 ) we g e t 1 y c o m p l e t e l y u n l e s s (case ( i i ) ) y(a(O)+p(a,))=g ( a k ) . T h i s f i x e s (case (i)) a(0) = 0, p = 0 ( = zero map) ( i n which case we must have g 1 (a,) = f: f o r a l l a k ) . a,p,y,6

a,B,~,6'

Recall t h a t i n t h i s case a:Z2

4

Note t h a t i n case ( i f a and p may remain "hidden" i f e.g. y i s c o n s t a n t ( t h i s i s o n l y p o s s i b l e i f each gn i s c o n s t a n t ! ) . T h i s case w i l l appear i n t h e Example below. I n case ( i i ) , a ( 0 ) . p and 6 a r e f i x e d , and we g e t t h a t y(O)= - 0 - f c = g n , so gn must be c o n s t a n t ; t h e n y ( 1 ) can (and need) n o t be determined. We see: Remark: The e q u a t i o n f = f i s s o l v a b l e i f f f E N(A). I n t h i s case, y(0) a d3 ,Y ,6 and 6 a r e always u n i q u e l y determined, w h i l e t h i s m i g h t n o t be t h e case f o r a,B, and t h e o t h e r values o f y ( q ) ( Q i s thought t o be known and f i x e d ) . l i n e s a f t e r Theorem 7.

C f . also the

221

Nearrings and non-linear dynamical systems

How b i g i s N(A,Q ? Although L ( A ) , hence M(L(A)), and a l s o N ( L ( A ) ) i n Propos i t i o n 6 a r e always i n f i n i t e ( u n l e s s A = EO}), we g e t P r o p o s i t i o n 7: N A,Q) i s f i n i t e i f A and Q a r e f i n i t e . T h i s f o l l o w s from t h e f a c t t h a t t h e r e a r e o n l y f i n i t e l y many f u n c t i o n s a,B, y,6 which determine separable systems on (A,A,Q).

Z,)

We now determine N(Z,,

i n d e t a i l ; t h i s w i l l a l s o show how one can

proceed i n more general cases. Example 1: I n o r d e r t o d e s c r i b e N : = N(Z2,ZZ2 ) , we need t h e f o l l o w i n g f o u r

).

f u n c t i o n s o f M(L(Z,)

- -1

z := z

j :

( " u n i t s h i f t t o the r i g h t " )

1

aizi

1 1

aizi

+

s

:

+

lzi

1 1

(ak+aktl

i21

i2k

i :

1

i>k

ilk

aizi+

t

... +ak+i-l)zk+i

("summation o p e r a t o r " )

i z k+ i

i20

i>k

Now B and 6 can each be t h e zero map 0 o r t h e i d e n t i t y map i d , w h i l e a and y can be 0,l

( = c o n s t a n t map w i t h values l ) , i d , o r 1 + i d . I t t u r n s o u t t h a t y

plays a c r u c i a l role. I f y = O t h e n each f

Ify

=l

a ,B ,Y .6

= 6

then f a , B , 1 , 6 = j + 6

( i n particular, fo,O,O,o=O, ( i n particular,

1 = f0 $0,I

fo,O,O,id=idEN) € N)

9 0

I f y = i d , t h e n i t i s easy t o see t h a t f O ,B , i d ,6 = z o p + a

fl,B,id,6=Zo f i d ,B , i d ,6

( i n p a r t i c u l a r , z =fo,id,id,OEN)

(ItBIt6

= S o

fit6

( i n p a r t i c u l a r , S = fid,id,id,O€N)

fl+id,B,id,6 = i + S 0 B + 6

IfY

= l t

fa,~,l+id,6'1t

( i n p a r t i c u l a r , i = fl+id,O,id,O€ N ) fa,~,id,6'

This, and t h e f a c t t h a t B,6€ I O , i d l shows t h a t N i s generated by O,id,l,z,S,

i. Since i d t i d = O , S O l = i , we can even say t h a t

222

G. Pilz

Note t h a t N(Z2, Z 2 ) i s n e i t h e r c l o s e d w . r . t .

a d d i t i o n ( s i n c e e.g.

composition ( z o z $ N ( Z 2 , Z 2 ) ).

z + S ( N ( Z 2 , Z 2 ) ) , nor w . r . t .

We now g i v e a l i s t o f t h e f u n c t i o n s i n N(Z2$Z 2 ) , along w i t h t h e i r values i n t h e f i r s t t h r e e components (which shows t h a t a l l o f them a r e d i f f e r e n t ) .

ak t 1

ak

t

1

0

i d z o l t i d ak * 'k+l*

ak ak+l

akt2-

aktl

ak akt 1 ak+l

1

i t i d ak aktl+ akt2

1

z+id

S t id

ak a k + ak t l

ak aktaktl

a k + l t ak+2 a k t a k t l

zo (jtid)tid

'k a k t aktl +

akt2

'kt2'

i+ S + i d ak

+ 1 ak + ak+l

aktl

'

+

1

aktak+ltak+2

ak * aktl* aktE'

Hence we g e t t h e f o l l o w i n g C o r o l l a r y : N(Z2,

Z,) has 28 elements and m(Z2, Z2 ) can be generated by

f o u r elements. Hence every separable system c o n s t r u c t i b l e by s e r i e s / p a r a l l e l connections o f separable systems on (Z2, Z2, Z2 ) can be o b t a i n e d by means o f s e r i e s / p a r a l l e l connections from

xz = (O,id,id,D), zs = (id,id,id,D),

xid

= (O,O,O,id)

, Z,

= (0,0,1

,O),

which a r e systems on (Z2, B 2 , Z,).

In o r d e r t o d e s c r i b e a more general s i t u a t i o n , we need t h e f o l l o w i n g

Nearrings and non-linear dynamical systems

223

D e f i n i t i o n 6: For n E N o and A = B = Q = K = a f i e l d , l e t

...

1

aizi -,~ o , ~ ~ , B ; o I ~ ~ s ~ , ~ ~ , ~ ; I) , I ~ ~ ~ ~ , ~ i>k (where t h e exponent n denotes t h e one w . r . t . t h e m u l t i p l i c a t i o n i n F) be t h e

M ~ s: =

~ , ~ ; ~ I ~ ~

n - t h h4mk.o~map on K. Theorem 4: L e t K be a f i n i t e (Galois) f i e l d and A = B = N(K,K) always contains id,z,l,S,i

(i)

Q = K. Then

o f the preceding Example.

(ii) IfN(K,K) i s contained i n t h e n e a r - r i n g generated by f u n c t i o n s fi (iE.some index set I)then i ( K , K ) = < f i I i ~ I > . ( i i i ) N(K,K)

i s always f i n i t e , a(K,K)

always i n f i n i t e .

where p = c h a r K and hk:x

f o r a l l x E K),

(iv)

N(K,K) =,

(v)

k being a p r i m i t i v e element (i.e. a generator o f t h e c y c l i c group (K*, The n e a r - r i n g h(K,K) f u l f i l l s t h e ascending chain c o n d i t i o n (ACC) on

+

kx

.)I.

subnear-rings.

Phood: ( i ) i s c l e a r since id=fo,O,O,idy z = fO,id,id,O' 1= f O , O , l ,id ,id,O' i = S o l . ( i i ) I f N ( K , K ) c < f i l i c I}S[(K,K) then =R(K,K) = < f i l i c I > .

,O'

S=fid

(iii) f o l l o w s from Proposition 3. ( i v ) a(K,K) i s generated by N(K,K), w i t h a,yEM(K),

B,dEHom(K,K)

i.e.

the s e t o f a l l functions y(-a+z)-lp+d,

(which i s K-isomorphic t o K). I f we take y = O , we

g e t a l l f u n c t i o n s i n Hom(K,K), and hence a l l y(-a+,?)-lO a r e a l s o i n R(K,K).

Now

a f i n i t e f i e l d i s known t o be "polynomially complete" (see [ t o ] o r [ I l ] ) , h e n c e

,.,.ykp-l

t h e r e a r e ko,kl

E K such t h a t y ( x ) = ko + klx + k2x2+

a l l xEK. Hence we can generate a l l y(-a+;)-lO

...

+ k xP-' f o r P-1 1 a d d i t i v e l y from kxn(-a+z)- 8 ,

which i n t u r n can be obtained as t h e composition o f x n ( - a + i ) - ' a w i t h t h e homomorphism hk:x [a,B;lI'(s)

1 aizi i n t o (O,[a,~;Oln((s), izk .), which means t h a t xn(-a+Z)- 8 =Mn. F i n a l l y , a l l elements i n

+

,..

kx. Now xn(-a+?)-'g sends s =

Hom(K,K) can be obtained from a s i n g l e map hk:x

-,kx,

i n which k i s a p r i m i t i v e

element i n K. ( v ) f o l l o w s from t h e f a c t t h a t a(K,K) i s f i n i t e l y generated. Remarks: ( i ) A s i m i l a r , b u t more complicated r e s u l t as f o r f i n i t e f i e l d s (Theorem 3 ) holds f o r f i n i t e vector spaces. I t does n o t seem t o be too w e l l known t h a t each map (we need i t f o r a,y)

-.

f r o m a f i n i t e vector space Q (over a

f i e l d F ) i n t o i t s e l f i s o f t h e form

,...

,...

,. ..

,...

q = (4, ,qn) (p, (ql ,qn) ,pn(ql ,s,f). where t h e pi a r e polynomial functions over K and n=dimFQ. This i s b a s i c a l l y due t o t h e polynomial completeness o f f i n i t e f i e l d s (see [ l o ] ) .

224

G. Pilz

( i i ) I f A i s a f i e l d , N(A) can be considered as n e a r - r i n g o f mappings on a f i e l d , namely on t h e q u o t i e n t f i e l d o f t h e i n t e g r a l domain A [ [ x ] ] o f f o r m a l power s e r i e s . Theorem 3 ( i v ) i n d i c a t e s t h e appearence o f a d d i t i o n , m u t t i p l i c a t i o n , as w e l l as composition. The s u i t a b l e frame f o r t h e s e c o n s i d e r a t i o n s is t h e one o f "composition r i n g s " , whose s t u d y was i n i t i a t e d i n [12] and [ l ] . ( i i i ) I f N(A) i s a r i n g and generated by fx ( i E I ) then every element on i N(A) ( = e v e r y system on (A,A,Q) f o r some Q ) can be o b t a i n e d by a p a r a l l e l connection o f (one o r s e v e r a l ) s e r i e s connections o f t h e xi.

This i s not the

case any more i f N(A) i s a " p r o p e r " n e a r - r i n g ( t h e n one has t o t a k e p a r a l l e l connections of s e r i e s o f p a r a l l e l o f

... c o n n e c t i o n s ) .

( i v ) I f N = N(A), o r N = B(A,Q), can be generated by f u n c t i o n s which a r e e i t h e r c o n s t a n t o r homomorphisms ( t h i s was t h e case f o r

[(Z,,72,) i n o u r

Example!) t h e n N i s an a b s t r a c t a f f i n e n e a r - r i n g (see [141, ch. 9 c ) ) . I n t h i s case, No i s a r i n g and Nc i s a module o v e r No. A f f i n e n e a r - r i n g s a r e t h e "most simple" ones among t h e p r o p e r n e a r - r i n g s . ( v ) Several o t h e r n e a r - r i n g t h e o r e t i c concepts have meaningful i n t e r p r e t a t i o n s i n t h e c o n t e x t o f systems t h e o r y . L(A) i s , i n a n a t u r a l way, an N = N ( A ) o r N = N(A,Q) - group ( = ( n e a r - ) module). A subgroup L of L(A) i s an N-subgroup

i f fz( xaizl)

E

L whenever f x E N and xaizl

monogenic N(A)- o r N(A,Q)-group i s o f t h e form xbizl

=fx(zaizl)

E L . Note t h a t L(A) i s never a

( t h e r e i s no xaizi f o r some f,cN(A)

a r e b a s i c a l l y subsets of N which a r e closed w . r . t .

w.r.t.

such t h a t each

x bizi

cL(A)

o r EN(A,Q)). I d e a l s I on N p a r a l l e l connections, and

p a r a l l e l connections w i t h a r b i t r a r y systems i n N. Elements i n N(A,Q) can

be n i l p o t e n t : take'e.9.

A = R = Q = Z 4 , 6(a) =2a, f = f 0 ,o ,o ,6' Then f which means t h a t t h e s e r i e s composition o f t h e corresponding system i t s e l f sends e v e r y i n p u t sequence i n t o t h e zero sequence (O,O,..

o

x

f

=0, with

.). The

appearance o f n i l p o t e n t ( o r n i l ) N-subgroups i n d i c a t e s t h a t t h e r a d i c a l o f N(A) o f N(A,Q) i s non-zero "141, e.g. 5.45). We now t u r n t o separable systems w i t h a ( s e p a r a b l e ) feedback. L e t T E S ( A ) have a hepahabee 6eedback 6unotion

@:QxA-,

A, ( q , a ) + $ ( q ) + @ ( a ) ,

q a homomorphism. Ifwe encode t h e s t a t e q E Q and t h e o u t p u t a E A t o @(q,a) and add i t t o t h e i n p u t , we g e t a new system . ' 1 see.

The f o l l o w i n g r e s u l t i s easy t o

225

Near-rings and non-linear dynamical systems

P r o p o s i t i o n 8: I f EES(A) has maps a.B,y,d

and i f 0 i s a separable feedback,

(a(q,a) = + ( q ) + q ( a ) , then t0 i s again i n S(A) and has maps a+!3Q,, p$, y + d $ , €I$.

I f E and 0 a r e zero-symmetric ( i . e .

+(O) = 0 ) then t h i s a l s o a p p l i e s t o . ' 1

Hence t h e c l a s s o f separable systems i s closed w . r . t

parallel/series

connections and separable feedbacks. We now t u r n t o questions o f r e a c h a b i l i t y . L e t A* be t h e f r e e monoid over A;

so A* c o n s i s t s o f a l l f i n i t e s t r i n g s o f elements o f A ( i n c l u d i n g t h e empty sequence A).

I f a = a l a 2...anEA*,

we w r i t e n =

151. I f z i s any system (separable

o r n o t ) such t h a t t h e s t a t e s e t i s a group ( Q , + ) , we l e t Ni(z) of

M(0)

N(E ,)

:=

generated by {f;laEA*,

.u

~ i } Then . N0(x)
be t h e subgroup

... and

are groups, too. I f q E Q then Ni(Z)q =: Reachi(q)

Ni(E)

consists

'wl

of a l l s t a t e s which can be reached from q by " ( m u l t i p l e ) i n p u t s t r i n g s o f

length Si".

C l e a r l y , each Reachi(q)

Reacho(q)cReachI(q)c

... holds.

t h e bubgtloup tleachable &om

i s a subgroup o f

u

Q and

i s called iEJN q. If ( Q , + ) f u l f i l l s t%e ascending chain c o n d i t i o n Reach,(q)

:=

Reachi(q) = N,(E)q

(ACC) f o r subgroups (which i s e q u i v a l e n t t o say t h a t (Q,+) i s f i n i t e l y generat e d ) then t h e r e i s some n E INo

such t h a t Reachn(q) =Reach,(q).

One might be i n t e r e s t e d how f a r one can g e t i n p r e c i s e l y i steps. L e t := CfaqI

Rchi(q)

(a1 = i } .

Then i t i s n o t t r u e f o r separable systems t h a t

(as i t i s f o r l i n e a r ones):

Rchi(q)
Example 2: L e t Q = A = ( Z , 2 y + ) , F ( q , a ) = ( q + l ) 2 + 4 a . Then Rchl(0) = tF(O,a) l a € Z12 1 = C1 + 4 a ( a Z,2 ~ 1 = 11,5,91 and Rchl(l) = Rchl(5) =Rch,(9) = C0,4,81. Rchl(0)nRch2(0)

Hence Rchl(0)$Rch2(0),

and we have even

=0 !

The t r o u b l e came from t h e f a c t t h a t t h e system i n Example 2 was n o t zerosymmetric: Proposition 9: L e t E be separable and zero-symmetric, and l e t Qi =Rchi(0). Then, f o r each i E INo

,

( i1

Qi+l

(ii)

Q o ~ Q...E , z Rch,(O)

= a(Qi)

+

I m B ?a($) :=

U

QisReachm(0).

iEINo ( i i i ) Qi+l (iv)

=a(Qi)+Q1

I f B i s an epimorphism then Q, = Rchoo(0) = Reachoo(0) =Q.

226

G. Pilz

Roolj:

Qi+,

(i)

= {F(O,a)l

la1 = i + 1)

=

= {F(F(O,Z),a)la"EA*,

= a(Qi

(ii)

1+ I m B

= i , aEA} =

la"l

= a({F(O,Z)lZEA*,

2a(Qi

= i l l + {B(a)laEAl =

1

We show Qi5Qi+,by i n d u c t i o n on i. Q o = {O), Q, = {a(O) + B(a) l a E A} = I m b 5 (Q,+) , s i n c e B i s a homomorphism and E i s zero-symmetric.

So

QosQ1. Now

l e t Qi-,5Qi. From ( i ) we g e t

Q i + l = a ( Q i ) + I m p ~ a ( Q i - , ) + 1 m B =Qi. ( i i i ) f o l l o w s f r o m ( i ) and ( i i ) . ( i v ) i s a consequence o f ( i i i ) and o f Q, = I m 0. Separable feedbacks do n o t change t h e r e a c h a b i l i t y behaviour: Theorem 4: L e t

x

be a system, and @ a feedback on E. Suppose t h a t E and @

a r e b o t h separable and zero-symmetric w i t h $J b e i n g an epimorphism. Then Rchi(0), and hence Rch,(O)

and Reach-(O)

coincide i n

x

and i n I@,

.

@ @ Ptoolj: L e t Rchi(0) =: Q.1 i n X and =: Q. in X We proceed a g a i n by i n d u c t i o n . 1 For i = O we g e t Q o = I01 =a!. L e t Q. =a:. From P r o p o s i t i o n 8 and P r o p o s i t i o n 9 1 @ ( i ) we g e t Qi+, = ( a + B@)Qf+ I m BJ,= ( a + B@)Qi + B W I ) = a(Qi) + B@(Qi) + B(A) =

= a(Qi)+ I m b=Qi+,,

the r e s u l t .

s i n c e B @ ( Q i ) c B ( A ) , which i s a subgroup o f (Q,+). Hence

0

The l a s t c o n s i d e r a t i o n s had t o do w i t h t h e s y n t a c t i c n e a r - r i n g s N(I)

(lines

a f t e r D e f i n i t i o n 3 ) ; c f . a l s o [151. Now we t u r n t o t o p i c s which i n v o l v e t h e n e a r - r i n g s N(A) = I f , l x E S ( A ) l I f a f u n c t i o n fI

i n "A)

o f t h e "second type". i s ( l e f t - , r i g h t - ) i n v e r t i b l e ( i n N(A)) t h e n t h e r e

i s a separable system X' such t h a t

fI+xl

= f Pf = i d ,,(f 1 ,

=f;fZI

=id,

r e s p e c t i v e l y ) , where i d i s t h e i d e n t i t y f u n c t i o n on t h e group L(A) o f a l l formal L a u r e n t s e r i e s on A. F o r many a p p l i c a t i o n s , however, t h i s i s t o o r e s t r i c t i v e and one r e q u i r e s a weaker k i n d o f " r e v e r s i b i l i t y " : D e f i n i t i o n 8: f EN(A) i s &wem56le w i X h delay L i f t h e r e i s a system 1' such t h a t f,f,= z , where z L ( r a i z i ) = zaizi+L.

f

Hence Z* tl i s a separable system whose t r a n s f e r f u n c t i o n maps (ak,ak+, ,...)

..

i n t o (bk = 0, ,,b, = 0,. ,b k + L - a = O ~ b k + L = a k s bk+L-l = a k + l s - . . ) . i n v e r t i b l e w i t h delay L = 0 i f f i t i s l e f t i n v e r t i b l e .

Also.

fx i s

227

Near-rings and non-linear dynamical systems

I n v e r t i b i l i t y , as defined so f a r , o n l y says something i f X s t a r t s i n s t a t e 0. I n order t o o b t a i n a more general s e t t i n g we f o l l o w 1161 by d e f i n i n g a s l i g h t l y more general form o f i n v e r t i b i l i t y : D e f i n i t i o n 9: 1 i s i n u e h t i b t e luith d&g

L dhom the d r n e q € Q i f t h e r e i s

a f u n c t i o n f which associates t o each sequence bkybk+, sequence sky...

. . ,b k+n+L a

such t h a t E, s t a r t e d i n s t a t e q, responds bk,...ybk+n+L

t o t h e i n p u t sequence ak 'k+n+l

,...,bk+,,.

,...yak+n,xk+n+, ,...,x k+n+L ( f o r some a p p r o p r i a t e

"k+ntL)'

y"'

L i k e i n a good f a i r y - t a l e , t h r e e wishes w i l l be f u l f i l l e d : we can r e s t r i c t

us t o k = 0, q = 0, and n = 0. The t h i n g w i t h k = 0 i s simply a consequence o f timeinvariance: i t does n o t matter a t which time k we s t a r t our engine. We s h a l l assume k = O from now on. The remaining two wishes are l e s s t r i v i a l and seemed a t f i r s t t o be p a r t i c u l a r t o l i n e a r systems. Theorem 5: L e t 1 be separable w i t h maps a,B,y,6

( i n s h o r t : t =(aY!3,y,6)),

and q E Q . ( i ) t i s i n v e r t i b l e w i t h delay L from q

...,bL) -,a. w i t h fE(a,,xl p r i a t e x,,x2 ,... . (bo ,

,x2,...)

( i i ) t i s i n v e r t i b l e w i t h delay L from q

there i s a f u n c t i o n f,: = ( bo y . . .

99

,bLy...)

BL+'

-D

A,

f o r some appro-

E i s i n v e r t i b l e w i t h delay L from 0.

Pmod: (i) e:

4

i s c l e a r ( t a k e k = q = O i n D e f i n i t i o n 9).

L e t bo,...,bn,...,bn+L

be given. We have t o show t h a t we can determine

...,an such t h a t we get G(q,aoa ,...anxn+,...xn+L) xn+, ,. .. ,xntL. We proceed by i n d u c t i o n on n . a,,

I f n = O we choose G(O,ao) =b;,

b i :=y(O)- y ( q ) + bo.

=b b

We get a. = f o ( b i , b ,

,..

.bn+L f o r some

,...,bL)

such t h a t

which means y(0) + 6(ao) =y(O) - y ( q ) + bo, whence 6(ao) = - y ( q ) + bo.

Hence G(q,ao) = y ( q ) + 6(ao) = b o y as desired. For 1 2 i 5 n, l e t b i :=y(O)-y(F(q,a,a aiel)) + bi. With ai =fo(b;,bi+, , b i + L ) we g e t ai-,))+biy from which we get G(O,ai) = b ; , hence y ( 0 ) + 6 ( a i ) = y ( O ) - y ( F ( q , a = G(q,a o...ai)y the l a s t output i n G(q,a ,...ai), as C(q,a ,...ai) = y(F(q,a ai-,))+6(ai) =bi. For i > n we can do t h e same, f o r an a r b i t r a r y ybn+Ly i n order t o g e t some appropriate x,,+~ ,x,,+~. choice o f ,,b, ( i i ) f o l l o w s from (i). n

,,..

,...

,...

h

,...

,...

,...

Now we g i v e some necessary and s u f f i c i e n t conditions f o r the actual c o n s t r u c t i o n o f a separable system E ' such t h a t fx++ = frl

o

fz transforms

228

G. Pilz

(ao,O,O

,...) =aoz0 .i n t o

...,O,a 0 ,o,...)

(0,

=aoz

L

.

I n o r d e r t o s i m p l i f y t h e f o l l o w i n g expressions, we r e t u r n t o t h e Markov f o r a separable system w i t h maps a,p,y,6 ( D e f i n i t i o n 5). The should be meaning of [a,p;nl ($1 ,$,,-,) f o r appropriate maps @1 Remember t h a t we can take k = O i n D e f i n i t i o n 5, and we s h a l l do so. c l e a r , too. symbols [a,p;n]

)...

,...

P r o p o s i t i o n 10: L e t E = (Q,A,B,F,G) symnet ric

= (Q,A,B,a,p,y,6)

.

( i ) I f t h e r e e x i s t zero-symmetric maps a ' : Q

d:B

-, Q,

6':B -, A such t h a t , f o r some L E

+

be separable and zero-

Q, y ' : Q + A, and homomorphisms

No,

( 0 ) 6 ' 6 = 0 ( = zero map)

( 1 ) y[a',p';Ol6+

6'yB = 0

-

(*

( i y 1 [a I ,D I ;i 1 I ( 6,yp ,yap

.,yai -2

,..

( L ) y'[a',p';L-ll (6,yp,yap,...,ycrL-2p) then 1' : = (Q,B,A,a',p',y',6')

( ** )

fx,+x,(ao,O,O

+ 6 lyai

Ptroob: L e t

x

+

(1 5 i5 L 1

f u l f i l l s f o r each a o E A :

,...) = ( 0 ,... ,0,xL,xL+, ,... ) w i t h Q, 6':B

-

p=o

+ 6'yaL-lb= i d

( i i ) Conversely, i f zero-symmetric maps a ' : Q homomorphisms d:B

-

+

+

xL =ao.

Q, y ' : Q + A, and

A f u l f i l l (**) then they f u l f i l l (*).

.

s t a r t a t time 0 i n s t a t e qo = 0. The i n p u t sequence aoOO..

gives a sequence o f s t a t e s qo = 0, q1 = a ( O )+ p(ao) = p(ao), q2 = a p ( a o ) + p ( 0 ) =

,...,qi

( i 2 1 ) and produces t h e output sequence i-1 b o = v ( 0 ) + 6 ( a 0 ) = 6 ( a o ) , bl = v ( a ( a o ) ) + 6 ( 0 ) =uD(ao) bi = y a e(a,) (1 l i ) . Suppose t h e r e a r e maps a',p',y1,6' (as s p e c i f i e d i n t h e statement) then we g e t = ap(ao)

(ao,O,O,

fl,+l

= a i-lp(ao)

,...,

...I

=(xoyxl,

...) w i t h x o = y ' ( 0 ) + 6 ' ( b o ) = 6 ' 6 ( a 0 ) ,

xi = u ' [ a ' , B ' . i - l l (bo,b l , . . . ,bi-l) + 6 ' ( b i ) = ( u ' [ a ' , p ' , i - l I i-1 B)(ao). Hence (**I i s f u l f i l l e d i f f (*) holds.

+ 6'ya

and, f o r 1
(6,y0,yaB,

...,ya

0

Remark: I f L = O then (*) reduces t o 6 ' 6 = i d . This i s s o l v a b l e i f f 6 i s a monomorphism such t h a t I m 6 i s a semidirect summand ( r e t r a c t ) o f B y i.e.

if

N o f B such t h a t I m 6 + N = B y I m 6 n N {O). I n t h i s case, every b E B can uniquely be w r i t t e n as b = i + n w i t h i E I m 6, n E IN.

t h e r e e x i s t s a normal subgroup

This holds, because i f 6 has t h i s p r o p e r t y then 6 i s an isomorphism from A t o

I m 6 , having an inverse 6':Im 6 + A .

If B = I m 6+N,

I m 6 n N = { O ) , we can

extend 6 ' t o a homomorphism from 6 t o A by d e f i n i n g 6 ' = 6 ' ( i + n ) : = 6 ' ( i ) . Conversely, i f 6 ' 6 = i d holds f o r some homomorphism 6 ' : B

+

A then

p)+

229

New-rings and non-linear dynamical systems B = I m 6 t K e r 66' v i a b = 6 a 1 ( b ) t ( - 6 6 ' ( b ) + b ) , and I m 6 n K e r 6 6 ' = { 0 1 . L e t us now f i x t h e f o l l o w i n g

Notation: L e t z , ~ 'be as i n P r o p o s i t i o n 10, w i t h ( * ) holding. I f L 2 1 , l e t L lpl1 1 x1 : = tQ .B,A,a , ,y ,d ) w i t h 1 a (bl ,...,bL) : = (b2, bL,O) 1 13 (b) : = (0, 0,b) 1 Y (b, 9 . SbL) 1 6 (b) : = 6 ' ( b 1 Low l e t , f o r L 2 1, I f L = 0, take x : = ({O},B,A,O,O,O,6'). L- 1 L + 6'ya Then x2 : = (Q,A,A,a-B?&,-?,id : = y'[a',p',L3 (y,ya ya

...,

...,

--

.

,...,

1 1 : L e t x be as above. I f t s t a r t s i n s t a t e q, = ( 0 Proposition 1 i t assumes s t a t e s q1 = (0,. ,O,bo) .qL = (bo,.. qLtl = (bl,..

..

,...

.

under t h e i n p u t sequence (bo,bl,b2,...), c1 = y ' [ a ' , p ' , L - l ] t

6'(bLtl),

cLtl

(0

,...,O,bo) 1

PUJO~:We g e t q1 = a ( 0 1

t p (bl) = ( 0

t

=y'[a',B',L-lI

,... ,bo,bl),

c 0 = y ' [ a ' ,p' ,L-1 ] ( 0

(0

c, =y'[a',B',L-l]

,... ,O)

. , b L b L t l ,-.

and produces outputs co = 6 ' ( b o ) ,

6'(b,),

..., cL = y ' [ a ' , P ' , L - l l

(b

,bLtl)

+

,... ,0) then

1

13 (b,)

=

(0

+

(bo

,... ,bL)

+

6'(bL+2),...

,...,O,bO),

1

q2 = a ( 0

,...,O,bo)

+

and so on. The outputs a r e

,...,0) 6 ' ( bo) = 6 ' (b,) , ,...,O,bo) + 6 ' ( b l ) , etc. t

0

Before t h e n e x t p r o o f i t i s good t o have t h e f o l l o w i n g Lemma: I f 1 and 1' a r e as above, we g e t f o r a l l i €IN and bo, [a',~',L-ll

(0

,...O,bo, ...,b.1-1 )

= [a',p',i-I]

(bo

,...,b i - 1 )

... ,bi-l:

Pmob: This can e i t h e r be shown by i n d u c t i o n o r by r e c a l l i n g t h a t [a',p',L-l]

,...,O,bo,..,

(0

ybi-l)

i s t h e s t a t e o f 1' a f t e r s t a r t i n g a t "time"

...,bi-,.

L - i t 1 and i n p u t sequence b o y

Due t o time-invariance,

t h i s i s t h e same

as t h e s t a t e o f Z ' a f t e r s t a r t i n g a t time 0 and i n p u t sequence bo,...,bi-l. Theorem 6: L e t 1, 1' , x2 be as above. Then we have f o r a l l a o E A and L E IN L + ,f E l i t 12 (ao,O,Oy. ..I = (0 ,O,ao,O ) = aoz

,...

P4.006: We i n s e r t b o = 6 ( a o ) , bi = y a

,...

i-I

.

p(ao) (see P r o p o s i t i o n 10) i n t o t h e

formulas f o r ci i n P r o p o s i t i o n 11 and g e t

0

230

C.Pilz

c0 = 6 ' ( b0 ) = 6 ' 6 ( a o ) = 0 , and, f o r l < i < L ,

,... ,O,bo,...

ci =y'[a1,!3';L-11

(0

,b i - 1 ) t 6'(bi)

i-1 )+6'(bi)

( u s i n g t h e Lemma) i-2 i-1 !3(ao))+ 6 ' y a e(ao)

=y'[a',B',i-11

( bo,...,b

=v'[a',!3',i-11 o(ao)=O

(6(ao),v8(ao),ua~(ao,... ,ya

.....

i
I f , however, i > L, we g e t

,...,

ci = y ' [ a ' , B ' , L - l ] (biTL bi - 1 ) + 6 ' ( b . ) = y' [a' ,B' ,L-1 I (ya1 -L- 1e(ao) , ,yai'2fi(ao)) = (Y'[~',P',L-II

(ya

or, with i= L + j ( j > O )

i-L-1

,. .

+ 6'yai-lg(ao)

i-1 ~ , . . . , y a ~ - ~ ~ ) + b ' y ae)(ao),

,...,yaL+J-')+6'ya L+j-1 ) ( D ( a o ) ) = ,.y aj - 1 p(ao). 2 The s t a t e s o f X a r e given by qo = ... =qL = 3, qL+j = aj-'o(a0) f o r j 2 1. We c ~ = +( u ' ~[ a ' , B ' , L - l l

j-1 j (ra ,ya

show t h i s l a s t statement by i n d u c t i o n on j. For j = 1 we g e t qL+l = (a-R?)(qL)

+

!3(cL) = (a-@?)(O) + B(ao) = B(ao) =aog(ao). The i n d u c t i o n s t e p from

+ o ( c ~ + =~ (a-b<)aj-'P(a0) ) + which was t o be shown.

j t o j+li s g i v e n by qL+j+, +

B$-l 8(ao) = aJe(a,),

= (a-&)(qLcj)

,. .. o f E x1* x2 . We

F i n a l l y , we can now compute t h e outputs do,dl course, do = dl =

... = dL-l

j 2 1, dL+j = -;(qL+j)

+t

= 0, and then dL = -(;q).,

+ id(cL+j)

= -
get, o f

+ i d ( c L ) = 0 + cL = a,

and f o r

+ $aJ-'@(a0) = 0. Hence t h e r e s u l t .

Remark: ( i ) I f L = O then we g e t already fz,+E1

(ao,O,O

,... )

,...I= (ao,O,O

( i i ) I t i s easy t o show t h a t , i f L = 0,

fX*Xi,+Ez

(ao,al ,a2,... 1 = (ao,al ,a2,.. . I holds. Hence fEl+tx2 i s a l e f t i n v e r s e o f fE.

( i i i ) The same holds i f L = 1 and 6 = i d , f o r instance, b u t n o t i n general. ( i v ) Also, i t can be seen t h a t Theorem 6 i s n o t t r u e i n general i f zero-symmetri c. The c o n s t r u c t i o n i n Theorem 6 o f an i n v e r s e X1*X2

x

w i t h delay L o f

i s not

x

can be

used f o r decoupling procedures (see [161) i f A i s t h e d i r e c t product o f some o t h e r groups. I f o n l y one component i s e x c i t e d , u s u a l l y a l l components respond. Decoupling i s t h e procedure t o p l u g another system (z 1*x2, f o r instance) a f t e r X so t h a t o n l y one output channel responds t o t h e a c t i v a t i o n o f j u s t one i n p u t channel. For more on t h i s , see e.g. [161. I n v e r t i b i l i t y has t o do w i t h r e g u l a r i t y i n t h e n e a r - r i n g N(A). We c a l l a system z c S ( A ) keg&,

i f t h e r e i s some o t h e r system Y € S ( A ) such t h a t Z * Y * x

has t h e same i n p u t - o u t p u t behaviour ( = t r a n s f e r f u n c t i o n ) as 1 i t s e l f . S(A) i s

23 1

Nearrings and non-linear dynamical systems

r e g u l a r i f every E E S ( A ) i s regular. A system t E S ( A ) i s LdempoZent, i f t*t has t h e same behaviour as

z. We say t h a t N(A) i s A h n g &

hqgCLeah

i f f o r each

S(A) t h e r e i s a zero-symmetric YES(A) such t h a t

zero-symmetric system

t i t Z i + Y behaves l i k e 1. I n t h i s case, S(A) i s r e g u l a r and

\y

i s invertible with

delay L = O . Now we g e t from 9.155,

9.156,

9.158,

9.159 ( c ) , ( f ) and 9.62 o f [141:

Theorem 7: I f S(A), A f i n i t e , i s r e g u l a r then )

and W E (as defined above) a r e idempotent systems.

t*Y

The s e t o f a l l systems X*X

i)

( X E S ( A ) ) coincides w i t h t h e s e t X + E f o r

an idempotent system E. i i ) I f t h e zero system R ( w i t h f R = O ) and t h e i d e n t i t y system I ( w i t h fI = i d ) a r e t h e o n l y idempotent systems i n S(A) then every system i n S(A) i s i n v e r t i b l e with d e l a y

L = O (and conversely).

The s i t u a t i o n i n ( i i i ) happens i f f f o r each E E S ( A ) t h e r e i s no TES(A)

(iv)

such t h a t ttt T = 62.

I f S(A) i s even s t r o n g l y r e g u l a r then every t c S ( A ) i s t h e p a r a l l e l

(v)

connection o f a constant and f i n i t e l y many systems which are i n v e r t i b l e w i t h zero delay. The same statements hold, o f course, f o r subnear-rings o f S(A) i f they happen t o be r e g u l a r ( w i t h t h e obvious changes). F i n a l l y , we b r i e f l y mention t h e r e a l i z a t i o n problem. For a given f u n c t i o n f:L(A)

+

L(B) (A,B groups) we want t o f i n d a group Q and "separable" f u n c t i o n s

F:Qx A

+

Q,

G:Qx

A

-,8

such t h a t f o r 1 : = (Q,A,B,F,G),

f =f,.

This i s c e r t a i n l y

a n o n - t r i v i a l problem, and i t i s n o t always solvable, s i n c e we know from Example 1 t h a t n o t every f u n c t i o n f : L ( A ) separable systems

+

L(B) i s o f t h e form f = f xf o r some

x.

I f f i s l i n e a r and A,B a r e a b e l i a n then t h e usual methods and r e s u l t s o f

l i n e a r r e a l i z a t i o n theory, as developed by Kalman i n [81, apply. I n t h e o t h e r case, one might t h i n k about t h e Arbib-Manes c a t e g o r i c a l approach (see [21 and [13]). The obvious candidate i s t h e category C c o n s i s t i n g o f groups and maps between these groups. Since every s e t can be made i n t o a group (see e.g. [91), C has products ( = Cartesian products), coproducts ( = d i s j o i n t unions w i t h some group s t r u c t u r e s ) , and. f r e e o b j e c t s , hence a l s o f r e e dynamics i n t h e sense o f [2].

Hence we g e t a f r e e r e a l i z a t i o n ( l e f t a d j o i n t ) w i t h s t a t e group A* (any

group s t r u c t u r e on A*) i n the sense o f [21 (p. 680). b u t t h e maps

F,G a r e n o t

separable i n general. Also, s i n c e f ( A ) i s i n general n o t a group i f f i s n o t a

232

G. Pilz

homomorphism, we g e t d i f f i c u l t i e s w i t h c o e q u a l i z e r s and minimal r e a l i z a t i o n s ( c f . [131, p. 726). Also, t h e r e s t r i c t i o n s t o a f f i n e maps F,G ( i . e .

sums o f a

homomorphism and a c o n s t a n t map) and a b e l i a n A does n o t work: t h e c a t e g o r y C ' o f a b e l i a n groups and a f f i n e maps between these does n o t c o n t a i n (co)products.

So t h i s remains an open problem.

REFERENCES

[l] A d l e r , I.,Composition Rings, Duke Math. J. 29 (1962), 607-625. [21 Bobrow, L.S. and A r b i b , M.A. , D i s c r e t e Mathematics, Saunders, P h i l a d e l p h i a , 1974. [3] C a s t i , J.L., N o n l i n e a r System Theory, Academic Press, New York, 1985. [41 Gratzer, G., U n i v e r s a l Algebra, 2nd e d i t i o n , Springer-Verlag, New YorkH e i d e l b e r g - B e r l i n , 1979. [5] Hofer, G. and P i l z , G., Near-rings and automata, Proc. Conf. Univ. Algebra, K l a g e n f u r t , A u s t r i a , 1982, 153-162. The s y n t a c t i c n e a r - r i n g o f a l i n e a r s e q u e n t i a l machine, 161 Holcombe, W.M.L., Proc. Edinbg. Math. SOC. 26 (1983), 15-24. [7] Holcombe, W.M.L., A r a d i c a l f o r l i n e a r s e q u e n t i a l machines, Proc. Royal I r i s h Acad. 84A (1984), 27-35. [8] Kalman, R.E., Falb, P.L. and A r b i b , M.A. , Topics i n Mathematical System Theory, McGraw-Hill , New York, 1969. [91 Kertesz, A., E i n f u h r u n g i n d i e t r a n s f i n i t e Algebra, B i r k h a u s e r , Basel, 1975. [I01 Lausch, H. and Nobauer, W . , Algebra o f Polynomials, N o r t h H o l l a n d / American E l s e v i e r , Amsterdam, 1973. [Ill L i d l , R. and P i l z , G., A p p l i e d A b s t r a c t Algebra, S p r i n g e r - V e r l a g (Undergraduate Texts i n Mathematics), New Y o r k - H e i d e l b e r g - B e r l i n , 1984. [12] Menger, K., Algebra o f A n a l y s i s , N o t r e Dame Math. L e c t u r e s , No. 3, Notre Dame, I n d i a n a , 1944. [I31 Padulo, L. and A r b i b , F1.A. , System Theory, Saunders, P h i l a d e l p h i a , 1974. [14] P i l z , G., Near-Rings, 2nd e d i t i o n , N o r t h Holland/American E l s e v i e r , Amsterdam, 1983. [151 P i l z , G., S t r i c t l y connected group automata, submitted. 1161 Sain, M.K., I n t r o d u c t i o n t o A l g e b r a i c System Theory, Academic Press, New York, 1981.

Near-rings and Near-frlds, C. Betsch (editor) 0 Elsevier Science Publishen B.V.(North-Holland), 1987

233

REDUCED NEAR-RINGS

D. RAMAKOTAIAH and V. SAMBASIVARAO Department o f Mathematics, Nagarjuna U n i v e r s i t y , Nagarjunanagar-522 510, A.P. I n d i a ABSTRACT I n t h i s paper we introduce a p a r t i a l o r d e r r e l a t i o n i n a reduced nearr i n g and show t h a t t h e s e t o f a l l idempotents o f a reduced n e a r - r i n g w i t h i d e n t i t y forms a Boolean algebra under t h i s p a r t i a l ordering. F u r t h e r we introduce t h e n o t i o n s hyper atom and orthogonal subsets i n a reduced n e a r - r i n g w i t h i d e n t i t y and show t h a t a reduced n e a r - r i n g with i d e n t i t y i s isomorphic t o a d i r e c t product o f n e a r - f i e l d s i f and o n l y i f i t i s hyper atomic and o r t h o g o n a l l y complete. I n t r o d u c t i o n : Abian [ l ] introduced a r e l a t i o n 2 i n a reduced r i n g R by d e f i n i n g a 2 b y a,b E R i f and o n l y i f a2 = ab and showed t h a t t h i s r e l a t i o n i s indeed a p a r t i a l order r e l a t i o n i n R. I n t h i s paper, we examine t h e p o s s i b i l i t y o f extending t h e above mentioned p a r t i a l order i n a reduced r i n g t o a reduced near-ring and o b t a i n some o f t h e consequences which o f course a r e generalizat i o n s o f t h e r e s u l t s which were already obtained f o r reduced r i n g s , This paper i s d i v i d e d i n t o t h r e e sections. I n s e c t i o n

5

1, we show t h a t t h e

s e t o f a l l idempotents o f a reduced near-rings w i t h i d e n t i t y forms a Boolean algebra and o b t a i n some i n t e r e s t i n g consequences o f t h i s r e s u l t . I n s e c t i o n

5

2 , we introduce t h e n o t i o n o f orthogonal subsets i n a reduced n e a r - r i n g and

show t h a t t h e f o l l o w i n g c o n d i t i o n s f o r a reduced n e a r - r i n g w i t h i d e n t i t y w i t h p a r t i a l order 5 a r e equivalent: (i)

(N,2) s a t i s f i e s t h e ascending chain c o n d i t i o n

(ii)

( N & ) s a t i s f i e s t h e descending chain c o n d i t i o n and N i s o r t h o g o n a l l y complete

( i i i ) Any orthogonal subset o f N i s f i n i t e (iv)

N s a t i s f i e s t h e ascending (descending) chain c o n d i t i o n on a n n i h i l a t o r s .

I n section

5

3, we d e f i n e hyper atoms i n a reduced near-ring w i t h i d e n t i t y

and show t h a t a reduced near-ring w i t h i d e n t i t y i s hyper atomic and o r t h o g o n a l l y complete i f and o n l y i f i t i s isomorphic t o a d i r e c t product o f n e a r - f i e l d s .

D. Ramakotaiah and V. Sambasivarao

234

P r e l i m i n a r i e s : We r e c a l l t h a t an algebraic system N = (N,+,.,o) n m - h i n g i f and only i f ( i ) (N,+,o)

i s a ILegt)

i s a ( n o t necessarily a b e l i a n ) group, ( i i )

(N,.) i s a semi-group, ( i i i ) a(b+c) = a b + a c f o r a l l a,b,c E N and ( i v ) oa = o f o r a l l a E N. A near-ring w i t h o u t Ron-zero n i l p o t e n t elements i s c a l l e d a d u c e d near-ring. For any x E N, we denote t h e i d e a l generated by x by <x>.

For o t h e r terminology t h e reader i s r e f e r r e d t o [21. We note the f o l l o w i n g w e l l known r e s u l t s i n the theory o f near-rings. L e m 0.1:

Let

N be a reduced near-ring. Then ab

= 0 , a,b C:

N i m p l i e s ba =

0.

Further f o r any a E N, A(a) = {x E N I ax = 01 i s an i d e a l o f N. L e m 0.2:

L e t N be a reduced near-ring. Then

(i) N has IFP, t h a t i s , a,b E N, ab = o implies anb = o f o r a l l n E N. ( i i ) ne = ene f o r every idempotent e i n N and n ’ i n N. ( i i i ) IfN contains an i d e n t i t y , a l l idempotents i n N are c e n t r a l .

A c a r e f u l observation o f ( i i ) i n lemma 0.2 says t h a t a reduced n e a r - r i n g has a r i g h t i d e n t i t y i f and o n l y i f i t has an i d e n t i t y .

Lemma 1.1: L e t N be a reduced near-ring. I f a,b E N, ab = 0, then a+b = b+a. Further f o r a l l idempotents e,f E N, e f = o i m p l i e s ( e + f ) e = e and ( e + f ) f = f. Proof: Suppose a,b E N such t h a t ab = 0. Then by lemma 0.1, ba =

0.

This

f a c t and our supposition imply t h a t a(a+b-a-b) = o and b(a+b-a-b) = 0. Again by lemma 0.1 (a+b-a-b?

, we

have (a+b-a-b)a = o and (a+b-a-b)b = 0. Consequently

= o and hence a+b-a-b = 0. This shows t h a t a+b = b+a. The r e s t o f

the proof f o l l o w s from lemma 0.2 ( i i ) . Lemma 1.2: L e t N be a reduced near-ring w i t h

d e n t i t y . Then f o r a l l idem-

potents e,f E N, e f and e + f - f e a r e idempotents. Proof: By lemma 0.2 ( i i i ) , a l l idempotents i n N a r e c e n t r a l and the r e s t o f the p r o o f f o l l o w s as i n the case o f r i n g theory.

Reduced near-rings

235

We now introduce an order r e l a t i o n i n an a r b i t r a r y n e a r - r i n g N by s e t t i n g a 2 b f o r a,b E N i f and o n l y i f a2 = ab. I f a n e a r - r i n g i s a Boolean r i n g , than t h e r e l a t i o n 6 i s a p a r t i a l o r d e r i n N. What i s t h e c l a s s o f near-rings f o r which t h e above i n t r o d u c t e d r e l a t i o n i s a p a r t i a l o r d e r ? We answer t h i s question i n t h e f o l l o w i n g . Theorem 1.3:

L e t N be n e a r - r i n g and 5 a r e l a t i o n on N as defined above. Then

N i s a reduced n e a r - r i n g i f and o n l y if5 i s a p a r t i a l o r d e r r e l a t i o n on This i s t h e case, ( N , . , < )

N.

i s a p a r t i a l l y ordered semigroup.

Proof: Suppose N i s a reduced near-ring. C l e a r l y 5 i s r e f l e x i v e . Suppose E N such t h a t a 5 b and b S a. Then a2 = ab and b2 = ba, t h a t i s , a(a-b) = o

a,b

and b(b-a) = (a-b)'

0.

By lemma 0.1,

we have (a-b)a = o and (b-a)b = 0. Consequently

= 0 . Since N i s reduced, a-b = o and hence a = b. So i i s antisymmetric.

Suppose a 5 b and b 5 c. Then a2 = ab and b2 = bc, t h a t i s , a(a-b) = o and 3 3 2 0 . This i m p l i e s aca = acb, a c = a b and a -ac = a(b-c). From these 2 f a c t s , we have a2(a2-ac) = o and ac(a -ac) = 0 . By lemma 0.1, we have b(b-c) =

(a 2 -ac)a2 = o and (a 2 -ac)ac = 0 . Consequently (a2 -ac)' = 0. Since N i s reduced, 2 we have a -ac = o and hence a2 = ac. So i i s t r a n s i t i v e . Suppose x 5 y and a 5 b. Then x(x-y) = o and a(a-b) =

0.

This i m p l i e s xa(x-y) = o and ax(a-b) = o ,

t h a t i s , xax = xay and axa = axb. From these f a c t s , we have xa 5 yb. Therefore

(N,.,i)

i s a p a r t i a l l y ordered semigroup. Conversely,suppose Ii s a p a r t i a l

o r d e r r e l a t i o n on N. I t i s enough i f we show t h a t a2 = o i m p l i e s a = 0. It i s c l e a r t h a t o 2 a. Since a2 = o = oa, we have a 5

0.

Hence a = 0.

We introduce suprema and infima o f any subset o f a reduced n e a r - r i n g under t h e p a r t i a l o r d e r r e l a t i o n 2 i n t h e usual way. For any subset S o f N, sup S ( i n f S ) denotes t h e supremum (infimum) o f S. From now on,

N stands f o r a

reduced n e a r - r i n g with i d e n t i t y , unless s t a t e d otherwise, and 2 always stands f o r t h e p a r t i a l o r d e r defined above. Proposition 1.4: For any subset S of idempotents o f N, i f sup S ( i n f S ) e x i s t s , then i t must be an idempotent. Proof: Suppose sup S = x e x i s t s . Then s = sx f o r a l l s E S. This i m p l i e s s x 2 y n d s =< x2. Since sup S = x, we have x s x2 and hence x2 = x3. Now x(x-x2) = o and x 2 (x-x 2 = 0. Consequently (x-x 2 ) 2 = o and hence x = x 2 Suppose i n f S = x. Then x2 = xs f o r a l l s E S. Now x2 = xs = xs2 = x 2s = x 3

.

.

By an argument s i m i l a r t o the above, i t can be shown t h a t x i s an idempotent, too.

D. Ramakotaiah and V. Sambasivarao

236

I n p r o p o s i t i o n 1.4, i f x i s a l o w e r bound o f S, t h e n x i s a l s o

Remark 1.5: an idempotent.

F o r a l l idempotents e,f E N, e + f - f e = sup { e , f )

Theorem 1.6:

and

Thus any two idempotents commute w i t h each o t h e r a d d i t i v e l y .

e f = i n f [e,fl.

T h i s r e s u l t can be v e r i f i e d i n t h e usual way. The p a r t i a l o r d e r r e l a t i o n S i n N induces a p a r t i a l o r d e r r e l a t i o n which we a l s o denote by 6, i n B, t h e s e t o f a l l idempotents o f N. I n view o f theorem 1.6, we have

e

A

Theorem 1.7:

(B,<)

Theorem 1.8:

Define

i s a l a t t i c e w i t h o and 1. A,

v and

'

on B as f o l l o w s . L e t e,f

E B. D e f i n e

f = e f , e v f = e + f - f e and e ' = 1-e. Then (B,A,v,',o,~)

i s a Boolean

a1 gebra. L e t a E N. Then aN = A ( l - a ) i f and o n l y i f a i s an idempotent.

L e m 1.9:

T h i s i s t h e case, eN i s an i d e a l f o r each e E B and <e> = eN. Proof: By lemma 0.2 ( i i i ) , a l l idempotents i n N a r e c e n t r a l and i t i s a l s o c l e a r t h a t f o r each idempotent e, 1-e i s a l s o an idempotent i n N. Suppose a i s an idempotent i n N. L e t x E A(1-a).

Then ( 1 - a ) x = 0. By lemma 0.1,

we have

x = xa = ax E aN. So A(1-a) 5 aN. L e t x E aN. Then x = an f o r some n E N. T h i s i m p l i e s x ( 1 - a ) = 0. By lemma 0.1,

(1-a)x = o and hence x E A(1-a).

So

A(1-a) = aN. Conversely, suppose a E N such t h a t A(1-a) = aN. Since a E aN, we have (1-a)a = 0. By lemma 0.1,

a ( 1 - a ) = o and hence a i s an idempotent. By

lemma 0.1 and by t h e above argument, f o r each e E B y eN i s an i d e a l and <e>=eN. Lemma 1.10: Proof:

F o r any e,f E B, eN n f N = efN.

By lemma 0.2 ( i i ) , efN _c eN r l fN. L e t x E eN

n f N . Then x

= en and

x = f n ' f o r some n,nl E N. Now x = en = een = e f n ' E efN. So en n f N = efN. Lemma 1.11:

F o r any e,f

Proof: By lemma 1.2,

E B y e N + f N = (e+f-fe)N.

e + f - f e i s an idempotent and by lemma 1.9,

eN, f N and

( e + f - f e ) N a r e i d e a l s o f N. Since e,f E B y we have e E eN and f - f e E f N . So e + f - f e E eN+ f N and hence ( e t f - f e ) N _c eN+ f N . On t h e o t h e r hand, e = e + f e - f e = = e ( e + f - f e ) = ( e + f - f e ) e E (e+f-fe)N

show t h a t f N

and hence eN _c (e+f-fe)N.

5 (e+f-fe)N. So eN t f N _c (e+f-fe)N. Hence eN + f N

S i m i l a r l y we can = (e+f-fe)N.

231

Reduced near-rings Lemma 1.12: The s e t S = {eN

Ie

Proof f o l l o w s from lemmas 1.9, Theorem 1.13:

L e t S = {eN

Ie

E B1 i s a l a t t i c e under s e t i n c l u s i o n . 1.10 and 1.11.

E 6). Then the mapping a:B

a(e) = eN i s a l a t t i c e isomorphism and hence Proof: By lemmas 1.10 and 1.11,

(S,s)

+

S defined by

i s a Boolean algebra.

a i s a ,+and v-homomorphism. I t i s easy t o

v e r i f y t h a t a i s onto and f o r any e,f

E B, e 2 f i n B i f and o n l y i f a ( e )

5 a(f).

Suppose e,f E B such t h a t a(e) = a ( f ) . Then e = f n and f = e n ' f o r some n,n'

E N. Now f = e n ' = een' = e f = f n f

= e. So a i s one-one.

Therefore

a:B + S i s a l a t t i c e isomorphism and hence S i s a Boolean algebra. We r e c a l l t h a t a near-ring N i s b L t e g d a h i f and o n l y i f f o r each a E N, t h e r e e x i s t s a c e n t r a l idempotent e E N such t h a t = <e>. Theorem 1.14:

I f N i s a b i r e g u l a r reduced n e a r - r i n g w i t h i d e n t i t y and i f

S = CeN I e E B I and S ' i s the s e t o f a l l p r i n c i p a l i d e a l s o f N, then S = S '

and hence S ' i s a Boolean algebra. Further, the c a r d i n a l i t y o f t h e s e t o f a l l idempotents i n N i s the same as the c a r d i n a l i t y o f the s e t o f a l l p r i n c i p a l i d e a l s o f N. Proof: By lemma 1.9,

S 5 S ' . Since N i s b i r e g u l a r , f o r each a E N, t h e r e

e x i s t s an idempotent e E N such t h a t = <e> and hence by lemma 1.9,

=eN.

So S ' 5 S. Hence S = S ' and S ' i s a Boolean algebra by theorem 1.13. Since t h e c a r d i n a l i t y of B i s t h e same as t h e c a r d i n a l i t y o f S, by the above argument, t h e c a r d i n a l i t y o f B i s the same as the c a r d i n a l i t y o f 5 ' .

9 2 Throughout t h i s section,

N stands f o r a reduced n e a r - r i n g w i t h i d e n t i t y .

For any subset X o f N, A(X) denotes t h e s e t {n E N Proposition 2.1: c =

SUP

= o for all x

E XI.

L e t X be any subset o f N. Then t h e f o l l o w i n g are e q u i v a l e n t

f o r any c E N. (i)

I xn

X

( i i ) c i s an upper bound o f X and A(X) 5 A(c). I n f a c t , A(X) = A(c).

D. Ramakoraiah and V. Sambasivarao

238

Proof: ( i ) i m p l i e s ( i i ) : Suppose sup X = c. Then x S c f o r a l l x E X. L e t 2 f o r a l l x E X. This i m p l i e s c+d i s an

d E A(X). Then xd = o and x(c+d) = x

upper bound o f X. Since sup X = c, we have c 5 c+d. Consequently cd = o and 2 hence d E A(c). So A(X) c - A(c). L e t d E A(c). Then cd = o and x d = xcd = o f o r a l l x E X. Consequently xd =

So d E A(X) and hence A(X) = A(c).

0.

( i i ) i m p l i e s ( i ) : Suppose c i s an upper bound o f X and A(X)

5 A(c). L e t d be

any upper bound o f X. Then f o r a l l x E X, x2 = xc and x2 = xd. This i m p l i e s

5 A(c). Consequently c(c-d)

c-d E A(X)

C o r o l l a r y 2.2:

=

o and hence c 2 d. So sup X = c.

I f X i s a subset o f N such t h a t sup X = b e x i s t s , then f o r

a l l a E N, sup aX e x i s t s and equals ab. Proof: L e t a E N. Since x 5 b f o r a l l x E X, we have x ( x - b ) = 0. Then axa(x-b) = o and hence ax 2 ab f o r a l l x E X. So ab i s an upper bound o f ax. L e t d E A(aX). Then axd = o and hence xda = o f o r a l l x E X. This i m p l i e s da E A(X). Since sup X = b y by p r o p o s i t i o n 2.1, we have da E A(b). Consequently d E A(ab). So A(aX) _t A(ab). Hence by p r o p o s i t i o n 2 . 1 , sup aX = ab. D e f i n i t i o n 2.3:

A subset X o f non-zero elements o f N i s s a i d t o be o h t h o g o -

m d i f xy = o f o r a l l x,y E X w i t h x

$

y.

I

i E I}and Y = Cy. I j E J) be two orthogonal J subsets o f N such t h a t sup X = x and sup Y = y. L e t Z = { x . Y . I i E I, j E J I .

P r o p o s i t i o n 2.4:

L e t X = {xi

Then Z i s an orthogonal subset of N and sup Z = xy. Proof:

1 J

It i s c l e a r t h a t Z i s an orthogonal subset o f N. Since xi

5 x, we E I. This i m p l i e s x.y.(xi-x) = o and hence 1 J X . Y . X . = x.y.x f o r each i E I and f o r each j E J. Consequently x . y . 5 xyj f o r 1 3 1 1 J 1 J each i E I and f o r each j E J. Since y < y, we have y j ( y j - y ) = o f o r each j = j E J. Then xy.x(y.-y) = o and hence xy < xy f o r a l l j E J . Hence by t h e

have xi(xi-x)

= o f o r each i

J J j = above two arguments, x . y . 5 xy f o r each i E I and f o r each j E J . So xy i s an 1 J upper bound o f Z. L e t d E A(Z). Then x . y . = o f o r each i E I and j E J . This J J J J J f o r each j E J and dx E A(Y). Hence dx E A(y) and d E A(xy). So A(Z) 5 A(xy).

i m p l i e s y . d E A(X) and hence y . d E A(x) f o r each j E J. Consequently y.dx = o

Hence by p r o p o s i t i o n 2.1,

sup z = xy.

We note the f o l l o w i n g lemma Lemma 2.5: d E N.

I f f o r any x,y E N with xy = 0 , then (x+y)d = x d + y d f o r a l l

Reduced near-rings

239

Proof: L e t x,y E N be such t h a t xy = 0. Then by lemma 0.1, these f a c t s , we have x((x+y)d-yd-xd) lemma 0.1,

yx = 0. From

= 0 and y((x+y)d-yd-xd)

= 0. Again by

we have ((x+y)d-yd-xd)x = o and ((x+y)d-yd-xd)y = o and hence = 0. Consequently ((xty)d-yd-xd)'

((x+y)d-yd-xd)(x+y)

= 0. Since

N i s reduced,

(x+y)d-yd-xd = o and hence ( x t y ) d = xd+yd. As a consequence o f lemma 2.5,

we can deduce the f o l l o w i n g .

I f X i s an orthogonal subset o f N, then

P r o p o s i t i o n 2.6:

1 xN i s a d i r e c t XEX

sum o f N-subgroups o f N.

The proof o f t h i s p r o p o s i t i o n i s easy and w i l l be omitted.

I n order t o o b t a i n the main r e s u l t o f t h i s s e c t i o n , we need t h e f o l l o w i n g . Lemma 2.7:

I f {ail

i s a s t r i c t l y ascending chain o r a descending chain o f

elements i n N , then {ai-ai+ll Proof: L e t {ai) ai(aj-ajtl) = a.a 1

= aitlaj

i s an orthogonal subset o f N.

be a s t r i c t l y ascending chain. Suppose i j and i < j. Now = ai 2 - ai2 = o and ai+l( a . aj+l) = 2 2 = ai+l - aitl = 0. Consequently ( a . - aj+l ) a . = o and J 1

- aiajtl j

- ai+lajtl

-

= 0. So (a - a j + l ) ( a i -ai+l) = o and hence {ai -aitl} i s an ( a j ajtl )aitl j orthogonal subset o f N. By a s i m i l a r argument, t h e o t h e r case can be disposed

-

of. Lemma 2.8: sup {al,a2,

...,an]

I f {al,a2,

...,an}

i s an orthogonal subset o f N, then

e x i s t s and equals a +a +...+ a., 1 2

Proof: W r i t e a = a +a +...+an. Then aia = ai2 1 2 f o r a l l 1 5 i 5 n. Suppose ai 2 d f o r a l l 1 5 i 2 2.5, ad = a l d t a d t + and = al2 + a22 t +an. 2 orthogonal subset o f N, by lemma 2.5, we have a'

...

...

f o r a l l 1 5 i 5 n. So a . 5 a 1

5 n. Then a:

= aid.

,. ..,a ...+a$

Since {al ,a2 = a,2 t a;+

By lemma

1

i s an

So a'

= ad

and hence a = sup {al,a2,...,anl. Lemma 2.9: I f { a i l i s an orthogonal subset o f N, then {sup { a . J a s t r i c t l y descending chain (provided t h a t the suprema e x i s t ) and {SUP { a i 1 1 5 i 5

jll

i s a s t r i c t l y ascending chain.

1

j 2 ill i s

D. Ramakotaiah and V. Sambasivarao

240

Ij

Proof: W r i t e ci = sup {a.

J

2 i } . Then ci 2 a j f o r a l l j 2 i and ci+l

2 aj

f o r a l l j 5 i + l . This i m p l i e s ci 2 a . f o r a l l j 2 i + l . Therefore ci i s an

Ij

upper bound o f {a.

J

J

2 i + l I . So ci+l

descending chain. I f ci = ci+l aici+l

= ai

sup {a.

J

1j

5 ci f o r each i and hence {cil

f o r some i, then aici

:s

2 i+l) = sup {a.a.

orthogonal subset o f N. This shows tha:

Ij =

2

= ai.

is a

By c o r o l l a r y 2.2,

2 i + l ) = o since { a i l i s an

o and hence ai = 0 , which i s a

c o n t r a d i c t i o n . So {ci)

i s a s t r i c t l y descending chain. Write It i s clear t h a t d . = sup {a. I 1 5 i 5 j } . By lemma 2.8, d = a +a +...+a J 1 j 1 2 j' = 0 , which I d j } i s an ascending chain. Suppose d j = dj+l f o r some j . Then a j+

i s a c o n t r a d i c t i o n . So I d . ) i s a s t r i c t l y ascending chain.

J

The proofs o f t h e f o l l o w i n g lemmas 2.10,

2.11 and 2.12 a r e easy and w i l l be

omitted. Lemma 2.10:

The descending c h a i n and t h e ascending chain c o n d i t i o n s on

a n n i h i l a t o r s i n N are equivalent. Lemma 2.11:

I f {ail

i s an orthogonal subset o f N , then {A(al,a2,..

,ai

1I

i s a s t r i c t l y descending c h a i n o f a n n i h i l a t o r s . Lemma 2.12:

If {Ii) i s a s t r i c t l y ascending chain o f a n n i h i l a t o r s

n A(Ii) then Ii+,

*

( 0 ) and

i f o 9 ai E Ii+l n A(Ii),

then {ai}

n N, i s an orthogo-

nal subset o f N. D e f i n i t i o n 2.13:

A n e a r - r i n g N i s said t o be o h t h o g o m U y cornpLeAe i f every

orthogonal subset o f N has supremum. We now s t a t e t h e main r e s u l t o f t h i s section. Theorem 2.14: The f o l l o w i n g c o n d i t i o n s on N a r e e q u i v a l e n t : s a t i s f i e s the ascending chain c o n d i t i o n

(i)

(N,S)

(ii)

(N,5) s a t i s f i e s t h e descending chain c o n d i t i o n and complete

N i s orthogonally

( i i i ) Any orthogonal subset o f N i s f i n i t e (iv)

N s a t i s f i e s the ascending (descending) chain c o n d i t i o n on a n n i h i l a t o r s

The proof o f t h i s theorem i s a consequence o f the lemmas 2.7 t o 2.12.

24 1

Reduced near-rings 9 3

I n t h i s s e c t i o n we i n t r o d u c e t h e n o t i o n o f hyper-atoms and study some o f t h e consequences. Throughout t h i s s e c t i o n , N stands f o r a reduced n e a r - r i n g w i t h identity

.

D e f i n i t i o n 3.1: only i f x

j

A non-zero element a E N i s s a i d t o b e a hypm-atam i f and

a i m p l i e s e i t h e r x = o o r x = a and i f an

+

o f o r any n E N, t h e r e

e x i s t s s E N such t h a t ans = a. Example: L e t N ' be a n e a r - f i e l d . Then N x N ' i s a reduced n e a r - r i n g w i t h i d e n t i t y and S i s a p a r t i a l o r d e r i n g on N x

N'. Now f o r each non-zero element

a E N ' , (o,a) i s a hyper-atom i n N x N ' . Lemma 3.2:

L e t a be a hyper-atom i n

N. For any n E N,

i f an

+

0,

then an i s

a hyper-atom i n N. Proof: Since an

* o and a

i s a hyper-atom i n N, t h e r e e x i s t s s E N such t h a t

ans = a. Suppose x S an. Then xs 5 ans = a. Since a i s a hyper-atom, e i t h e r xs = o o r xs = a. Case ( i ) : I f xs = 0 , then xans = o and hence xa = 0. Since x 6 an, we have xL = xan = o and hence x = 0. Case ( i i ) : Suppose xs = a. x 5 an i m p l i e s x ( x - a n ) = 0. Then xs(x-an) = o and hence a ( x - a n ) = o and an(x-an) = 0. By lemma 0.1,

( x - a n ) x = o and (x-an)an = 0. Consequently

(x-an)2 = 0. Since N i s reduced, x-an = o and hence x = an. Suppose a n t t E N. Then a n t s = a f o r some s E

+

0,

N s i n c e a i s a hyper-atom. Now

( a n t ) s n = ( a n t s ) n = an. Hence an i s a hyper-atom. Lemma 3.3:

L e t a be a hyper-atom i n N. Then t h e r e e x i s t s an element s E N

such t h a t a = asa, t h a t i s , a i s r e g u l a r i n t h e sense o f Von Neumann.

*

o and t h e r e e x i s t s an element Proof: L e t a E N be a hyper-atom. Then a' s E N such t h a t a 2 s = a. T h i s i m p l i e s a(a-asa) = o and asa(a-asa) = 0. By lemma 0 . 1 , (a-asa)a = o and (a-asa)asa = o and hence (a-asa)2 = 0. Since N i s reduced, we have a = asa. Lemma 3.4:

L e t a be a hyper-atom i n N. Then t h e r e e x i s t s an element s E N

such t h a t as i s an idempotent hyper-atom i n N. Proof: Follows f r o m lemma 3.2 and lemma 3.3.

D. Ramakotaiah and V. Sambasivarao

242

The proofs o f the f o l l o w i n g lemmas 3.5 and 3.6 a r e easy and w i l l be omitted. Lemma 3.5:

The set o f a l l idempotent hyper-atoms i n N i s an orthogonal sub-

N.

set o f

Lemma 3.6:

I be the s e t o f a l l idempotent hyper-atoms i n N. Then n E N) i s a subnear-field o f N and

L e t IeiIi

f o r every i E I, Ni = b e i

Ni n N. = (0) i f i

J

D e f i n i t i o n 3.7:

9

I

j.

N i s s a i d t o be hgpe,t-uLotnic provided f o r each non-zero

element n E N, t h e r e e x i s t s a hyper-atom 3. E N such t h a t a 5 n. Lemma 3.8:

L e t N be a hyper-atomic near-ring and l e t IeiIiEI

be the set o f

a l l idempotent hyper-atoms i n N. Then f o r each non-zero element q E N, there e x i s t s an idempotent hyper-atom ek such t h a t qek the sup {nei

I i E I1 exists

* 0. Moreover,

f o r each n E

N,

and equals n.

Proof: Since N i s hyper-atomic, i f q a E N such t h a t a 5 q. Then a2 = aq

* 0.

* 0 , then

t h e r e e x i s t s a hyper-atom

By lemma 3.4,

there e x i s t s s E N such

*

t h a t as i s an idempotent hyper-atom. Now we w i l l show t h a t qas 0. Suppose qas = 0. Then aqas = 0. This i m p l i e s a2 as = 0 and hence as = 0, a contradiction. So qas

qek

* 0.

* 0.

Thus there e x i s t s an idempotent hyper-atom as = ek i n

Since (nei)*

= nein

N such t h a t

f o r a l l i E I,n i s an upper bound o f h e i

I

i E I}.

Then nnei = nuei f o r each i E I . Suppose u i s an upper bound o f h e i I i E I}. 2 I f n $ u, then n -nu 0. By the above argument, t h e r e e x i s t s an idempotent hyper-atom ei i n N such t h a t (n2-nulei 0 , which i s a c o n t r a d i c t i o n . Hence

*

sup b e i

I i E I1

Lemma 3.9:

= n.

L e t N be hyper-atomic and l e t Ieili

I be the s e t o f a l l idemN onto a Ni o f n e a r - f i e l d s Ni.

potent hyper-atoms i n N. Then a : n + (neiIi subnear-ring o f the d i r e c t product

n

€ I i s an isomorphism o f

iEI Proof: C l e a r l y a i s a homomorphism. Suppose a ( n ) = 0. Then (nei)i This implies sup h e i [ i E I ) = 0. By lemma 3.8,

I =

n = 0. Hence a i s a mono-

morph ism. We now s t a t e and prove the main r e s u l t o f t h i s section.

0.

243

Reduced near-rings Theorem 3.10: N i s isomorphic t o a d i r e c t p r o d u c t o f n e a r - f i e l d s i f and o n l y i f N i s hyper-atomic and o r t h o g o n a l l y complete. Fi, where each Fi i s a n e a r - f i e l d . L e t o Proof: Suppose N = T'I iE1 I and t h e r e e x i s t s j E I such t h a t n . 0. Now Then n = (ni)i x = (o,o

J

,...,o,n.,o,....o) J

+nE

N.

*

i s a hyper-atom i n N and x 5 n. L e t X by any

orthogonal subset o f N. L e t ki:N

+

Fi be t h e c a n o n i c a l p r o j e c t i o n . W r i t e

*

o f o r some x E X and ai = o o t h e r a = {aili wise. Since X i s orthogonal, i t f o l l o w s t h a t a = sup X. Conversely, suppose

I , where ai

= ki(x)

i f ki(x)

N i s hyper-atomic and o r t h o g o n a l l y complete. By lemma 3.9, i t i s enough i f we Ni o f lemma 3.9 i s onto. L e t (niei) E n Ni. + n i€1 i€1 By lemma 3.5, t h e s e t hiei I i E I1 i s a n orthogonal subset o f N. Since N i s o r t h o g o n a l l y complete, sup h i e i I i E I 1 = h e x i s t s . Now by c o r o l l a r y 2.2 and show t h a t t h e mapping a:N

lemma 3.5, f o r each j E I , we have he = e.h = e . sup {n.e. J J J 1 1 Now a ( h ) = (heiIi So a i s onto. I = (nieili

I

i E I1 = n.e J j*

REFERENCES

[I] Abian, A .

, Direct

product decomposition o f commutative semisimple r i n g s ,

Proc. Amer. Math. SOC. 24 (1970).

€21 P i l z , G., Near-rings, North-Holland, Amsterdam 1977. [31 Raphael, R. and Stephenson, W . , Orthogonally complete r i n g s , Can. Math. B u l l , Vol. 20, No. 3 (1977).

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Near-ringsand Near-fields, C. Betsch (editor) 0Elsevier Science PubMers B.V. (North-HOW), 1987

245

NON-COMMUTATIVE SPACES AND NEAR-RINGS INCLUDING PBIRD's PLANAR NEAR-RINGS AND NON-COMMUTATIVE GEOMETRY Maic SASSO-SANT

INTRODUCTIOY T h i s work shows some r e l a t i o n s between n e a r - r i n g s , non-commutet i v e g e o n e t r x and c o m b i n a t o r i c s (PBIBD's and RIBD's). Many i n t e r e s t i n g problems f o r f r u i t f u l f u t u r e r e s e a r c h i n t h i s a r e s may be found i n [ 2 ] o r [4] We s t a r t w i t h some p r e r e q u i s i t e s i n Eon-commutative geometry.

.

I NOX-COMMUTATIVE GEOMETRY 1.1.DEFINITION: A space i s a s t r u c t u r e ( X , 0 ) where X i s a ncnempty s e t ( e l e m e r t s a r e c a l l e d p o i n t s ) and b i s a s i l t s e t o f t h e power s e t X. 0 7 :

1.2.DEFINITION: An L-space i s a s t r u c t u r e (X,) non-empty s e t X and a nnspping:

c o c s i s t i n g of

3

possessing t h e following p r o p e r t i e s : (LO)

x u x

=

I4

t X U Y (12) z ~(x-y)\(x)

(L1)

X,Y

II)

f o r all x g X f o r a l l x,y 4 X x u y = x-z

(incidence condition) for a l l x , y , z t X , x 9 y (exchang;e s o n d i t i c n )

The ? o i n t x i s c a l l e d b a s e p o i n t (Aufpunkt) of xuy d e n o t e d by x f x u y . A p o i n t s e t o f t;he f o m xu;.' with x y is called l i n e (Linie).

+

-

I.~.DEFINITION: An LG-space i s a n L-space w i t h (L3)

xuy

= y u x =

xuc 4

x u 2

=

zux

(X,Y,Z €XI

M. Sasso-Sant

246

L i n e s w i t h (L3) a r e c a l l e d e t r a i g h t l i n e e (Geraden). 1.4.DEPINITIOX: L e t (X,,) An e q u i v a l e n c e r e l a t i o n

11

s

be an L-space and := { x U y / x , y c X ) on is called parallelism i f

3

1

xux YUY f o r a l l x,yd X I f L e 9 and x S X t h e n t h e r e e x i s t s e x a c t l y one L ' such

(PO)

(PI)

t h a t x 6 L ' and 1,' 11 L ( E u c l i d e a n p a r e l l e l i s m c o n d i t i o n ) (P2) I f xuy xIuy' then y u x uy",~' (x,y,x',y'e X) (symmetry c o n d i t i o n )

I

1.5.DEPINITIOIT:

i s a t r i p l e (X,,,

An LF--pace

1)

with t h e

p1'3 p *?rt i a s

(X,-)

(a)

1

(b1

i s an b s p a c e is 3 paralle~sm

1.G.DEFZNITION: An LP-space i s c a l l e d s e l f a d j o i n t ( s e l b s t a d j u n gie7.t) i f (P2) x u y yux f o r a l l x,y E X ( s t r o n g synrnetry c o n d i t i o n ) A 3 p q u i - a l s n c . ' c l a s s xu? is c a l l s d direction.

u

1

I.?.DEFIiITTION: subspace, i f :

(U1) (u2)

X,

r,y

eu 3 X U Y SU ~ 1 ,3L o u

XLU,

{XI

A s u b s e t Us X o f a n LP-space

(X

(X,,,

1)

is c:alied

UL)SU

( x a x ) , 0 3ce c a l l e d t r i v i a l subspRce3 o f ( X , U ,

1).

An LP-spsce i s c a l l e d p-irnitive, i f c n l y t h e t r i v i a l subspaces e x i s t ; o t h e r v i s e i t i s c a l l e d imprirnitive.

I.E.DEFIRITION: A s k e w a f f i n e space i s a n LP-space where t h e l'amaschke-condition h c l c l s : ?am: 1: (x,y,z} +, x u y I I x ' u y ' ( x ' , y ' € X) t h e n t h e r e s x i s t z ' C X such t h a t xuz U x l u z ' an11 yue y1-z'

(x,,l

)

The f o l l o w i n g - l o s u r e c o n d i t i o n i s c a l l e d p a r a l l e l o F r a m n c l s s u r e condition :

-

PGM: If x , y , z a r e p a i r w i s e d i f f e r e n t p o i n t s o f a n LP-spacs and (X,U, ther. t h e r e ? x i s t s w r X such t h a t x ~ y zl u w

xL1z

I

a)

jruw

.

247

Non-commutative spaces and near-rings

An LP-space i s c e l l e d Desarguesien i f t h e f o l l o w i n g D e s a r g u e s i e n c l o s u r e cor,dition holds:

&

a r e p a i r w i s e d i f f e r e n t p o i n t s o f t h e s e t X end

I f u,x,y,z

i f u-x = u-x' (XI( X) then t h e r e e x i s t d i f f e r e n t p o i n t s y ' , z ' ( X such t h a t u 4 y = u-y' , u u z = I i u z ' , xUy)x'-y'

xuz

1 x ~ u z ' , yuz[

y ~ u z l .

A s u b s e t FCX o f an L-space

I.9.DEFINITION: if

(X,)

is called

flat,

(FI) (F2)

x , y ( L e t , x,g6F, x x,yCF, x

+

jr;

+

y II) L G F x C L € g , y 4 M r 9 ; L,M$F

+ lLnM[

6 I

An i m p r i m i t i v e FkewJffine space i s c e l l e d s e m i e f f i n e , i f e v p r y subspace i s f l a t .

I1 GEOMETRIC AND PLANAXR NEAR-RINGS 11.1 .DEFINITION: A n e a r - r i n g ( N , + , - ) w i t h t h e d i s t r i b u t i v e l a w (s+b)c = a c + bc i s c a l l e d g e o m e t r i c 5 6 i f : (")N i s zerosym-

L HI

m e t r i c , i . e . O R =ao = o f o r 311 EI E N. ( 2 ) F o r all a € N t h e r e e x i s t s 1 E N such t h a t a'! = a . ( j ) I f o = a = bc t h e n aN = bN ( a , b , ( . € N ) .

.

F o r t h e d e f i c i t i o r , o f 3 p l a n a r n e s r - r i n g , s ? e [7] The f o l l o w i c g theorem:; 3 r e t r u e . Some o f them were proved independectly by R.Scapellato. THEOREM 1 : I F N i s t h e n Na THEOREM 2 : ~f m i s Na i s a THEOREK 3 : I f N i s then N l I THEOREM 4 : (And&):

3 geometric n e a r - r i n g 2 n d Pa + ( 0 ) ( a E N ) , i s a l s o a geometric near-ring. a plans? n e o r - r i n g and 2 3 (s E N ) , t h e f i P planar near-rlng, too. 6 g e o m e t r i c neRr-ring, I 8 p r o p e r i d e a l of N , i s a geometric near-ring. Every f i n i t e p l a n a r n e a r - r i n g i s g e o n e t r i c .

IN./-[

THEOREM 5: Every i n t e g r a l p l a n a r n e a r - r i n g

i s geoaetric:.

The l a s t theorem g l v e s 3 g c s n e t r i c i n t e r p r e t a t i o n of thi. B a t s i n t h e s t r u c t u r e t h e o r e n of 3 p l a n a r n e a r - r i n g :

-

x u y = By-x

-

+ x , where By-x

= By-x~[o)

M. Sasso-Sant

248

I11 PBIBD’s, PLANAR NEAR-RINGS AND NON-COMMUTATIVE GEOMEZRY

.

F o r t h e d e f i n i t i o n o f a t a c t i c a l c o n f i g u r a t i o n s e e [3’J III.1.DEFINITION: S t a r t with 3 t a c t i c s 1 c o n f i g u r a t i o n ( v , 8 , E ). Let 7 = { A c V / I A l = 2 ) and l e t eC = {A,,, A2, Am} bs a p a r t i t i o n o f T. Then & i s a n s s s o c i a t i o n schame on V if, g i v e n ( x , y j € A h , the number z C V such t h a t { x , z ) € A i and { ; J , z ) ~ A depends ocly 31: h , i 3r,d j and not on x and/or y. That i s , t h e r e h h i s a number p i j such t h a t f o r f x , g ) € A h t h e r e a r e e x a c t l y p L.l d i s t i n c t e l e m e n t s z € V , such t h a t { x , z J c A i and { y , x ) eAj. A s s o c i g t i o n schemes w i t h m = 1 o r m = v(v-1)/2, v = I V l 3re declared ”uninteresting”.

......,

III.2.DEFINITION: Suppose ( V , @ , E , K ) i s a t a c t i c a l c o n f i g u r a t i o n w i t h a s s o c i a t i o n schem?. T h i s s t r u c t u r e i s a PBIBD ( p a r t i n l l y b a l a n c e d i n c o m p l e t e block d e s i g n ) , i f : a . ) f a r eech A i € & , t h e r e e x i s t s a number Ai such t h a t { x , y } € Ai i m p l i e s x , y belong Yo e x a c t l y blocks of t h e r e e x i s t s a numbe? ni, s u c h t k z f f o r b.) for each Aie e s c h x E V , t h e r e a r e e x s - t l y ni d i s t i n c t e l e m e n t s y E V , such t h e t { x , y ) 6 Ai. SO a PSIBD h3.5 p a r a m e t e r s p hi j , l j ,ni i n a d d i t i o n t o th o s e 3f a ts c t i c 3 1 con fip r a t i on

a ,

L

x.

.

Now t h e q u e s t i o n i s : “Which p l a n a r n e a r - r i n g s y i z l C PBIED’s ? ” *

a

III.3.LEN;uA 1 : Let N be a f i n i t e p l s n a r n e a r - r i n g , a € N ( N = N+u f i x e d , b Q N. Then [art + S b a N ) g i v e s a t a c t i c a l configuration

1

ITI.4.LEMMA 2 : Let N be 3 f i n i . t e p l a n a r n e a r - r i n g . Then { (&aN) + b b 6 N } f o r A C N: g i v e s a t a c t i c a l c o n f i g u r a t i o n , if ~t l e a s t one a # C , ( N ) , where C1(N):= ( x 6 N / xN = -xN + x i

1

III.5.DEFINITION:

.

A n e a r - r i n g N i s c a l l e d symmetric, i f f XN = -xN for a l l x E N

1II.G.TEEOREM: Suppose (N, +, . ) i s a f i n i t e , symmetric, p l a n a r n e a r - r i n g and @ i s t h e s e t o f a l l b l o c k s [ aN + b a 6 N: b 6 N ) Then t h e f o l l o w i n g i s t r u e :

I

I.>( N , m )

i s a PBIBD

+

b ) i s a PBIBD

2.)

(N, aN

3.)

(N, ( g A a N )

a

C1(N))

+

(a0U* fixed)

b ) f o r an A c N ” i s a PBIBD ( a t l e a s t one

.

249

Non-commutative spaces and neawings

REMARK: If C,(P!)

i n 6.1.),

= to)

t h e n t h e PBTBD is a BTBD and

( N , (8) i s a Desarguesian, s k c w a f f i n e space w i t h p a r a l l e l c l o s u r e c o n d i t i o n (PGM). If C,,(N) f { o ) t h e n ( N , @ ) i s a Desarguesian, s e r n i a f f i n e s p a c e w i t h PGM. I f cne c o r l s i d e r s 6.2.) t h e n t h e b l o c k s R N + b a r e l i n e s o f one d i r e c t i 3 n o f t h i s space an2 i n 6 . 3 . ) t h e b l o c k s a r e t h e unior? o f l i l i e s wi%h d i f f e r e n t d i r e c t i o n s , b u t f i x e d b a s e p o i n t . F o r p a r a m e t e r s ni, we have :

Ai,

a.)

ni = l x y y l

b.)

1.)

2.)

p 9 j o i a PBIBD o v e r a p l a n a r n e a r - r i n g

-

-

1 = k

ph 3 1 + p i i ;j pii = 0 , pg

63"

=

1

,......,m

7 and p1 z I hj = 'I, where k = 3

4.)

pg = k - 2, i f k ag pig = 0, i g

5.)

p6. =

6.:

p Jh h = k - 3 ,

5.)

for a11 i

1

3

+

JJ

0,

j

+

g,

pEj

I

j + g + h ,

(semiaffine space) j f h ,

k a 3

The i n d e x g d e n o t e s 3 s t r a i g h l ; l i n e d i r e c t i o n , t h a t incans i f 5 = x-y , then x u y = yux.

EXAMPLES: 1 . ) Consider

(z,-, +)

with t h e u s u a l modulo a d d i t i o n a n 3 t h e f i x e d p o i n t f r e e automorphism group = {id,'f) with ' f ( x ) = -x. T h i s i s a v e r y i n t e r e s t i n g example, because t h e b l o c k s a r e c i r c l e s o f a MBbius p l a n e , t h e y form a B I B D , t h e y a r e Lines o f a Desarguesian s k e w a f f i n e space w i t h PGM, and a l l t h e l i n e s form a PEIBD. 2.) We c c n s i d e r +> a g a i n with t h e u s u a l a c d u l c 3 d d i t i 3 r snd t h e f i x e d p o i n t f r e e autonorphlsm group = [ i d , ' f ) , with y(x)= - x . The b l o c k s form 6 PBIED snd t h e y a r e l i n e s o f a Desargues'Lan s a i a f f i n e space w i t h PGM. See a l s o [2]

(z9,

4

9

.

S t a r t i n q with Leinma 1 one may a s k t h e q u e s t i o n , which g l a n e r n e a r - r i n g s l-ead t o symmstric b l o c k d e s i g n s ? The snswer i s s t i l l unknown. The r e s d e r may f i n d examples of s p e c i s l symmetric b l o c k d e s i g n s , Hsdamard d e s i g n s and p r o j e c t i v e p l a n e s i n L4J

.

M. Sosso-Sant

250

IV NON-COMMUTATIVE GM)METRY OVER

(r,M(r))

We b e g i n w i t h a n s d d i t i v e , n o t n e c e s s a r i l y a b e l i a n group ( L e t M( ) d e n o t e a s u b s e t o f t h e s e t o f a l l f u n c t i o n s from

r to r.

--

r , +).

We make t h e f o l l o w i n g d e f i n i t i o n : u :

rxr (x,Y)

yr) xuY

:=

~ ( r ) ( ~ - x+ )x,

and we a s k t h e q u e s t i o n , under which’ c o n d i t i o n s t h e s p a c e ( , M( )), , becomes a non-commutativa space? I t ’ s a g e n e r a l i z a t i o n of t h e c o n c e p t of Theobald r53 [6] I f one c o n s i d e z s t h e s p e c i a l f u n c t i o n s M( ) : = {yseW( ) / y a ( n ) = m } where ( , +, i s a n e a r - r i n g and n , a € , t h e n we g e t t h e d e f i n i t i o n : xuy = ( y - x ) r t x. Among m s n y o t h e r r e s u l t s t h e f o l l o w i n g i s i m p o r t a n t :

r

r

r

r

a >

r

r

.

IV.’!.fHEOREM: I f (N, i s 3 f i n i t e , zero-symmptric, quasii n t e g r a l n e a r - r i n g , t h e n (E, + , * ) i s a g e o m e t r i c n e a r - r i n g , where q u a s i - i n t e g r a l m e a n s a - b = o =? a = o v b = o . + , a >

T h i s s e c t i o n m o t i v a t e s t h e f o l l o w i n g s e c t i o r , V.

V STElONGLY GEOMETRIC PJEAR-RINGS FROM AUTOMORPHISM GROIJPS V.1 . D E F I N I , T I O N : A g e o m e t r i c n e a r - r i n g N i s callecl. s t r o n p l g geom e t r i c , if N c o n t a i n s a r i g h t u n i t 1 ( i . s . n . 1 = n f o r a l l n C N ) .

9

V.2.DEFINITION: C o n s i d e r a n automorphism group on a group , +). A n o r b i t ( y > , Y E % i s c a l l e d p r i o c i p a l , i f l + ( y o t h e r w i s e i t i s c o l l e 3 secondar:r o r b i t . We denote wLth H t h e f a m i l y o f a l l p r i n c i p a l o r b i t s and with N t h e f a a i l y o f a l l sec o n d a q o k b t t s . From e v e r y p r i c c i p a l o r b i t we choose r e p r e s e n t a t i v e s . N o w we d e f j n e a p r o d u c t on +) i n t h e f o l l o w i n g way:

(r

4

r

(r,

]=l+l,

25 1

No n-commu ta live spaces and near-rings

(r ,

With t h e l e s t d e f i n i t i o n +, ) becomes 8 s t r o n g l y g e o m e t r i c n e a r - r i n g , with d i s t r i b u t i v e l a w ( a t b ) * c = a'c + b-c.

V.3.THECREM:

V.4.THEOREM:

?he n e a r - r i n g

(r , +,

0 )

i s quasi-integml.

The p r o o f s o f a l l theorems i n t h i s s e c t i o n may be found i n (4)

.

REFERENCES

"73 [2]

And& J.

B e t s c h G.-Clay

Z.ur Geometrie d e r Frobeniusgruppen. Mathem. Z e i t s c h r i f t 154 (1977) S.159

-

15R

Block d e s i g z s from Frobefiius g r o u 2 s J.R. and planar iear-rings.

Proc. Conf. f i n i t e g r o u p s (Psr.1: C i t y , Utzh) Acad. Press (1976), pp. 475 - 502

[3]

P i l z G.

Near-Rings The Theory and i t s A p p l i c a t i o n North Holland (1983)

[4]

Sasso-Sant M.

Nichtkoqmutstive R&me und Fastringi:. Diplomarbeit U n i v r r s i t a t Saarbriickpn (1986)

[5] Theobsld E.

Nich$kommutstive Geometrien iiber F a s t r i n g e n . D i p l o a a r b e i t U n i v e r s i t a t S a a r b r u c k e n (198'1 )

[6] Theobald E.

IJear-.-ings and non-commutative geometry. I n : Proceed. Conf. ' o n N a a r - r i n g s and N e a r - f i e l d ? , ed. by G. F e r r e r o and C. F e r r r r o C o t t i , S a n Senedetko d e l Trorito, pp. ?',,I - 218 (1981).

Author's address Maic Sasso-Sant I V . Gartenreihe I 8 663 S a a r l o u i s West

-

Germeny

This Page Intentionally Left Blank

Near-rings and Near-fields, G.Betsch (editor) 0 Elsevier Science Publishers B.V. (North-Holland),1987

253

ON GEOMETRIC NEAR-RINGS

Raffaele SCAPELLATO Dipartimento d i Matematica, Universith d i P a r m a , Parma, I t a l y ”

In t h i s p a p e r we g i v e a short account of o u r r e s u l t s on geometric near-rings.

A zero-symmetric

near-ring

is s a i d geometric if i t s a t i s f i e s the fol-

lowing conditions:

(G1) for e v e r y a € N t h e r e i s a n e E N such t h a t e a = a ; ( G 2 ) for every a , b e N, if ab#O then Nab=Nb.

An L-space

( s e e ( 1 1 ) is a p a i r ( S , U ) , where S is a set a n d U is a

map from S x S to P ( S ) , such t h a t , i f U ( x , y ) i s denoted by xUy, the following conditions hold for every x , y , z

t

S:

(LO) x U x = ( x ) ;

(L1) x , y

(L2) i f z

E

t

xuy; xUy \ i x ) t h e n xUz=xUy.

Theobald [ 6 ] proved t h a t , i f N is a near-ring

a n d U i s defined b y

xUy=N(y-x)+x, then the p a i r (N,U) is a L-space iff N is geometric. This is a link between near-rings

a n d non-commutative geometry i n the sense

of Andrk ( s e e [ 11 ) . For geometric a p p l i c a t i o n s of n e a r - r i n g s see a l s o ,

e. g.,

[21, l31,[41.

The following r e s u l t s (except Th.3) have been proved i n [ 51.

L E M M A 1 . I f I i s a proper i d e a l of a geometric near-ring

N , then

1N=O and N/I is geometric. THEOREM 2. A f i n i t e non-constant near-ring

is geometric i f f i t is

strongly monogenic.

I n the next statement we w i l l denote by F the set of a l l non-zero maps x * a x .

THEOREM 3. A near-ring

N is both geometric a n d s t r o n g l y monogenic

iff F i s a g r o u p . *Work supported by t h e i t a l i a n M P I .

R. Scapellato

254

PROOF. Suppose t h a t N is both geometric a n d s t r o n g l y rnonogenic. Let a

E

N with aN=N, let e

E

N with e a = a . If b

E

N we get b=ac for some c , so

eb=eac=ac=b, a n d e i s a left i d e n t i t y . By (G2) we h a v e e t Ne=Nae, so e=dae for some d . Since d a i s c l e a r l y a left i d e n t i t y , i t i s now e a s y to conclude t h a t F i s a group. Conversely, i f F is a g r o u p , let a

6

N such t h a t a N # O .

There is b

t

N

such t h a t a b is a left i d e n t i t y : now N=abN c aN a n d N is s t r o n g l y rnonogenic. Finally, let a , b

t

N a n d ab#O. For c

E

N , i f cb=O it is c l e a r t h a t

c b E Nab; otherwise the map x c c x belongs to F a n d i t i s e q u a l to the map x - d a x

for a s u i t a b l e d . We get c b = d a b , t h u s c b t Nab, a n d ( G 2 )

follows. This completes t h e proof.

REFERENCES

[ 11 Andre, J . , Nichtkomrnutative Geornetrie und verallgerneinerte HughesEbenen. Math. Z . 177 (1981), 449-462. 121 Ferrero, G . , Sterns p l a n a r i e BIB-disegni, Riv. Mat. Univ. Parrna, ( 2 ) 11 (1970), 79-96. 131 Ney, H . , P l a n a r n e a r - r i n g s a n d t h e i r r e l a t i o n s t o some non-commutative s p a c e s , i n Proc. Conf. on Near-rings a n d Near-fields, Harris o n s b u r g , 1983. [ 41 Sasso-Sant, M., Nichtkommutative Raume und F a s t r i n g e , to a p p e a r . [ 51 Scapellato, R . , On geometric n e a r - r i n g s , Boll. Un. Mat. I t . , ( 6 ) 2-A ( 1983) , 389-393. [ 61 Theobald, E., Near-rings a n d non-commutative geometry, i n Proc. Conf. on Near-rings a n d Near-fields, 5. Benedetto del Tronto, 1981.

Near-rings and Near-fslds, G.Betsch (editor) 0 Elsevier Science PublishersB.V. (North-Holland), 1987

A

TERNARY

255

INTERPRETATION @F THE

TNFRA-NEAR

RINGS

Mlrela Stefhescu 1.

INTRODUCTION

In studying some g e n e r a l i z a t i o n s of r i n g s a s infra-near rings (see S t e f h e s c u

[lll ),weak

r i n g s (see Climescu [3,41

,

Cupona [5J ) , p r e r i n g s (see Janln [6,7] ),we have noted t h a t cert a i n facts can be b e t t e r explained by means of a ternary i n t e r p r e t a t i o n o f t h e a d d i t i v e composition law. T h i s i n t e r p r e t a t i o n might be of considerable use i n studying a f f i n e infra-near r i n g s ( Stefgnescu LlO3 ) a s well a s i n studying i d e a l s o f those alge-

b r a i c systems. Given t h e development of t h e theory of t e r n a r y groups and t e r n a r y rings,such an i n t e r p r e t a t i o n using a t e r n a r y operation Instead o f t h e binary a d d i t i o n may be i n t e r e s t i n g I n itself. F i r s t we r e c a l l some d e f i n i t i o n s and p r o p e r t i e s o f generali-

z a t i o n s of r i n g s involved I n our considerations. (1.1) Definition. A l e f t h f r a - n e a r r i n g is a triple (I,+,.),

where I i s a nonempty s e t

, + and

(addition and m u l t l p l i c a t i o n )

.

a r e binary operations on I

, such t h a t

t h e following condi-

tions are fulfilled: (1) (I,+) ie a group (generally,noncommutative) i

(il) (I,.)

I s a semigroup;

( i 1 i ) t h e m u l t i p l i c a t i o n is l e f t i n f r a - d i s t r i b u t i v e w i t h r e s p e c t t o t h e a d d i t i o n , 1.e.

x

. (y + z ) = x.y

- x.0

+

X.Z

, for

a l l x,y,z t I .

The concent o f r i g h t infra-near ring is analogous. If x.0 = 0 f o r a l l x ~ 1 , t h e nwe o b t a i n a l e f t near-ring.

If

(I,+,.) i s a l e f t and a r i g h t infra-near ring w i t h a commutative

M. Stefanescu

256

addltion,we obtain an equivalent d e f i n i t i o n o f a weak rhg, t h e general case. We mention t h a t t h e weak rings were first Introduced by A1.Cllmescu 131 in 1961 i n a p a r t i c u l a r case,when x.0

= 0.x = x f o r a l l

X E

I

In 1964,Climescu [41 has noted a ge-

n e r a l possibil.%ltyQf o b t a i n i n g weak r i n g s s t a r t i n g from a g i v e n ring. The p a r t i c u l a r case was rediscovered by G.Cupona

(53 I n

1971, who c a l l e d it "quasi-ring".

Let ( I , + , . ) be a l e f t infra-near ring. One can e a s i l y check that f o r each

- x.0

, is

X G I t h e mapping f x r

I--+I,given by fx(y)

an endomorphism of the a d d i t i v e group (I,+)

I=

.

-

x.y

This a l -

most obvious remark may help us t o study t h e l e f t infra-distribut i v e a s s o c i a t i v e multiplications over an a d d i t i v e group (see Stefgnescu (123 ) , t h e discussion being unexpectedly i n t e r e s t i n g .

One of t h e first examples o f r i g h t near-rings was the s e t of a l l a f f i n e transformations over a vector space or over an Abelian a d d i t i v e group endowed w i t h the pointwise addition and the mapping composition. This is a l s o a l e f t infra-near ring and a

few p r o p e r t i e s of t h i s s e t come from i t s infra-near r i n g struc-

ture

. It is s u r p r i s i n g how much algebra can be obtained only by

considering t h i s weaker s t r u c t u r e (see ClO'J

,a

?aper i n which we

have studied and generalized the infra-near ring of a f f l n e trans-

formations from t h i s point o f view ). If a l e f t infra-near ring s a t i s f i e s the condition I 0.x = 0

, for

all xcI,

then we c a l l it a l e f t C-infra-near 0.x = x

, for

all x

rinp;

, while

in the case

I

61,

we c a l l it a l e f t Z-infra-near

ring.

we give two examples o f f i n i t e l e f t C-infra-near rings: .'1

The c y c l i c group (&,,+),with

1.x = 2.x = x , f o r a l l x e Z 3

.

t h e multiplication

I 0.x

= 0,

A ternary interpretation of the infra-near rings

.'2

The symmetric group S3

3:

where 3a = 2x = 0 and x+a = b

257

{O,a,b = 2a,x,y = x+a,z

+

, with

x

a

x+b],

t h e multiplicatlon given

by one of the following t h re e p o e s l b i l l t i e s r

(I) 0.8 = 0,a.s =

x.8

= y.s =

z.8

= a

, b.8

=

(11) 0.8 = 0,a.s = x . 6 = y.6 = a, b.s = z . s =

(111) 0.8 = 0,a.s = x . 8 = a , b.8

P

y.6

P

z.8 =

B

, for a l l

BC

%'

8

,for a l l s e S 3 1

8

,for a l l B E + .

Many o t h e r eaamples of l e f t infra-near rings and a general s t u d y of such algebraic systems can be found in [llJ

mark that

, If

. Let u s re-

(I,+) is a group,then (I,+,+)I s a l e f t and a

right

lnfra-near ring. I n a l e f t lnfra-near r i n g we do not obtain 0.0 = 0. For examp l e , i f we consider I = 25 and the binary operations I x

+ y = (x1+91,72+92,X?)+Y3

,XI++Yl+,x5+x2 .Y3+Xl+.Yl+Y51#

x.y = ( X I .Y1,'2*92,a~ ,a&, w i t h a3,a4,a5 fi x e d elements in Z,then we obtain a l e f t infra-near ring for which

0.0 = (0,0,a3,a4,a5),whlle 0 = (0,0,0,0,0).

The r e a l difference between t h e general case (see L83

in

which th e multiplication is an "independenttt operation on an ad-

d i t i v e group,

- that means the multlplication

cond!tions w i t h respect t o t h e addition,

does not f u l f i l any

- and our case

( I n which

t h e multiplication I s l e f t l n fra -d i st r i b u t i v e over t h e addition) is perceptibly I n the theory of congruences,hence i n defining homomorphisms and Ideals of such st ru ct u res . (1.2) Definition. A fno-Slded Ideal of a l e f t infra-near ring (I,+,.)I s a nonempty subset J o f I s a t l s f y l n g the conditions: (I(J,+) ) is a normal subgroup of t h e group

- x.0 e J

(ii) x . j (ill ) ( j

+

x).y

, for

a l l x E I and j E J

- x.yeJ , f o r

(I,+)i

i

a l l x , y E I and

j C J

.

If J o n l y satisfies t h e conditions (Iand ) (ii) (reSpeCtiVdg, (Iand ) (ill )) , J I s c a l l e d a l e f t Ideal (respectlvely,a right

I d e al) of

I.

M. StefGnescu

258

If (I,+,.)is a l e f t near-ring,hence x.0 = 0 for a l l x e 1 , t h e n

we obtain t h e d e f i n i t i o n s o f the two-sided,left and r i g h t idealB of a l e f t near-ring.

The similitude is a m a s i n g , B u t i n f a c t the

s i t u a t i o n 1s more complicated

, since

even t h e two sided i d e a l s

of a l e f t infra-near ring a r e not closed under the multiplloa-

tion,hence generally they a r e not subinfra-near r i n g s o f I. For l e f t near-rings a s i m i l a r assertion is t r u e only f o r the r i g h t ideal8 (see P i l z L9l

, 1.28

and 1.33).

O f c o u r s e , i f J is a two-sided i d e a l of the l e f t Fnfra-near

ring I

, then

I/J = f x + J / x

E

I) i s a l e f t infra-near r i n g with

respect t o t h e usilal operations beween cosets.

(1.3) Definition. Let (I,+,.)and (It,+,.) be t w o l e f t infra-near r i n g s . A mapping y t 141’ i s c a l l e d a homomorphism of l e f t infra-near rings i f t h e following two conditions a r e f u l filled : (1)

‘P(x + Y) = \ p ( x ) + \ P ( Y )

,

.

fix). y(y) , f o r a l l x , y c I , k e r ‘4 = jx E I / yJ (x) = 0’5 , i s a two-sided

(ii) .p(x.y) =

The kernel of ‘f

i d e a l of 1,while t h e image o f p

, Im y

=

(p(x> / x E I

a subinfra-near ring of 1’. Moreover,the %-sided

f , is

i d e a l s of I

a r e exactly t h e kernels of the homomorphisms from I t o another l e f t infra-near ring

.

(1.4) Definition. Let I be a l e f t C-bfra-near be a group. If t h e r e e x i s t s a mapping

= m.x

, such

(I) m.(x

MxI4M

that t h e following axioms a r e f u l f i l l e d

+ y) = m.x

(ii) (m.x).y

= m.(xy)

- m.0

+ m.y

,p ( m , x )

(M,+,p) a right

, f o r a l l m h M and x,y 0 I 4

, I-group.

Obviously I is i t s e l f a r i g h t I-group and each right i d e a l of I i s a r i g h t I-subgroup

, while

=

I

, f o r a l l m cM and x,ye I ;

(111) 0.x = C , f o r a l l x C I

then we c a l l

,441

r i n g and (M,+)

a r i g h t I-subgroup of I i s

259

A ternary interpretation of the infra-near rings

n o t a r i g h t ideal of 1,even in th6 cam when it i e a normal oubgroup of (Il+).

Denote by D = i d G I / d.0 = 0 5 and W

wrI / w.0 =

o

w),

t h e s u b s e t of a l l l e f t d i s t r i b u t i v e elements of I and t h e s u b s e t of

a l l weakly l e f t d i s t r i b u t i v e elements of I , r e s p e c t i v e l y . I f I

i s a l e f t C-infra-near

x,yr:I

, hence

+

ring and (x

y).O

e

x.0

+

y.0 f o r

0 i s a r i g h t d i s t r i b u t i v e element of I

all

, then

D is

a l e f t subinfra-near r i n g of I which i s a normal eubgroup of (Il+)

.

and W i s a r i g h t I-subgroup of I

r e c t decomposition I = D + W x = (x

- x.0)

2. TERNARY

+

, where

x.0

x

;

Moreover w e have t h e semidi-

indeed,for any x E I ,we have

- x.OED

.

and x . O E W

GROWS

Let u s note t h a t t h e l e f t i n f r a - d i s t r i b u t i v i t y of t h e m u l t i p l i c a t i o n w i t h r e s p e c t t o t h e a d d i t i o n reminds one of a l e f t distri-

b u t i v i t y of the binary multiplication over a ternary composition defined by

8

( y , t , z ) := y

-t+z

(y,@,z) = y

Indeed,we have then for a l l x l y l z 6 1

.

for all x , y , t , z 6 1

for a l l y , t l z GI.

+

Moreover we have x.(y

.

= (x.y1x.Olx.~)

z and x.(y,O,z)

-t

+ z)

= xdy-x.t+x.z,

B u t such t r i p l e s were obtained by Dijrnte

and by Baer i n t h e t w e n t i e s i n connection w i t h some g e o m e t r i c a l f a c t s

( t h e Erlangen Program on groups) ; In

[11

,R.Baer had n i c e l y ex-

plained the geometrical and algebraic reasons to take a s Fnva-

riants f o r a group of transformations t h e expressions of t h e form x

.

-y+z

S t a r t i n g f r o m t h i s point

Certaine

was

guided t o consider a ternary operation on a group (G,+) defined by

(x,y,z)

I=

x

-y+z

f o r a l l x,y,z E G (we t r a n s l a t e i n t o an

a d d i t i v e notation t h e paper written by Certaine)

.

By considering

t h e a b s t r a c t properties of such an o p e r a t i o n , C e r t a h e has defined ternary groups.

M.Stefdnescu

260

(2.1) Definition. Let I! be a nonemptg eet endowed with a tar-

nary composition

,.) : T

(.?.

%

T x T 4 T and With a f i x e d element

O d T such that t h e following conditions a r e f u l f i l l e d : (1) (x,O,(y,Z,u)).= ((x,O,y),z,u)

, for a l l x t T , for a l l x e T

(2) (x,O,O) = x

(3)

(x,x,O) = 0

, for

a l l x,y,z,u e T ;

;

.

Such an a lg ebra i c system i s called a ternary group of first type

(Certaine 121

, Definition

3

By defining a binary operation on T

(4) x

+ y r = (x,O,y)

, for a l l

x,yeT

,

we obtain a group (T,+) w i t h 0 a s i t s n eu t ral element ( 123

Theorem 4 )

.

,

?%en t h e binary and t h e ternary operations a r e r e l a t e d by t h e

eq u ali ty :

(5)

(x,y,z) = x

- y + z , f o r a l l x,y,z

E T

,

then we say t h a t t h e ternary group i s reg u l ar

, the

In a binary group

n e u t ra l element is unique

statement does n o t hold f o r a ternary group satisfy (11,

( 2 ) and ( 3 ) i n D e f i n i t i o n ( 2 . 1 1 ,

, but

.

A s i mi l ar

if 0 and O1 both

then t h e cor-

responding binary groups by (4) a r e isomorphic a s it i s proved in

(21

( t h e isomorphism i s given by x ~ ( x , O , O 1 )

, for

xe!&

Moreover it i s possible t h a t d i s t i n c t ternary groups correspond t o the same binary group,aa it is shown

in c2’J f o r t h e

c y c li c group w i t h two elements. To i l l u s t r a t e this,we consider th e cy c li c group o f order 3 3a = 0

,

(T = {O,a,b) ,+) w i t h b = 2a and

. We f i n d three p o s s i b i l i t i e s t o define t ern ary group8

o f f i r s t type,namely

t

(I) the r egu l a r ternary gmup (x,y,z) = x X,Y,Z

-y+

, f o r my

6T 3

(11) the s p e c i a l products a r e (a , a , a ) = (O,b,b) = b

, (b,b,b)

=I

26 1

A ternary interpretation of the infra-near rings

P

(O,a,a)

a

L

, (b,a,a)

(a,b,b)

o

, the

0

other ternary pro-

ducts being r e g u l a r ;

(a,a,b) = (b,b,a) = 0

(111) the s p e c i a l products a r e

, (O,a,b)

(O,b,a) = (b,a,b) = a

= (a,b,a) = b

, the

,

other pro-

ducts being regular, ~e obtain anotkz concept of ternary group, which we call

of

seccpld

type,

type, while a ternary group of Certain's type is called of f & s t type.

(2.2) Definition. A couple ( T , ( . , * , . ) )

.,

set and (., .)

I

Tx Tx T

,where T i s a nonempty

, if

called a t e r n a r y group of second type

it s a t i s f i e s the

following axioms f o r a l l x , y , z E T and a fixed element O € f

=

( 6 ) ((X,Y,O),Z,O)

(7)

(O,X,O)

(8)

((O,O,x),x,o>

= x

(X,(YtZ,O>,O~

, then

I

,

, = 0

(2.3) Proposition. If ( T , ( . , . , . ) ) cond type

, is

i s a t e r n a r y operation on T

3T

(T,+)

, where

i s a t e r n a r y group o f set

i s t h e binary composition

defined by (9)

+

x

y

I=

(x,y,O)

, for

a l l x,yC:T

,

is a group.

Proof. The a s s o c i a t i v i t y

comes from (6)

, by

using the defi-

n i t i o n o f t h e binary operation. The other two axioms t r a n s l a t e d into the addition complete the definition of a group. We a l s o have the equalities:

,for

(7')

(x,O,O)

(8')

(x,(O,C?,x),C) = 0

= x

all x%T

, for

,

all xbT ,

(2.4) Definition. A t e r n a r y group of second type i s c a l l e d

regular

,

i f the ternary operation is connected w i t h t h e binary

composition (9) by the following equality!

(lo)

(x,y,z) = x

+y

-z

, for

all x , y , z E T

.

As in the case of t e r n a r y groups of first type, t h e r e

are

M. Stefanescu

262

many t e r n a r y groups of second t y p e a s s o c i a t e d t o t h e same group by t h e e q u a l i t y ( 9 ) . For example, for t h e c y c l i c group of o r d e r 2 , (T = { O , a ) , + ) , we may d e f i n e a t e r n a r y composition: ( O r O r O ) = = (a,O,a) = (a,a,O) = ( a , a , a ) = 0, (O,a,O) = (O,O,a) = ( a t O , O )

=

= ( O , a , a ) = a , which p r o v i d e s T w i t h t h e s t r u c t u r e of a t e r n a r y g r o u p of second t y p e w i t h (T,+) a s t h e a s s o c i a t e d group. T h i s t e r n a r y group is n o t r e g u l a r , s i n c e ( a , a , a ) = 0 # a = a + a a.

-

The following ternary compositions on T = \O,a)

,namely (7), (8)

only two of the three axioms I n Definition (2.2) for (tl),

(61,031 for (t,) and (6),(7) for (t,)

satisfg

I

o

{

(O,O,O) = (a,O,O> = (O,a,a) = (a,a,o) =

,

(O,o,a) = (a,O,a) = (O,a,O) = ( a , a , a ) = a ; (0,0,0)= ( O , O , a )

= (o,a,o) = (O,a,a) = 0

,

= (a,O,a> = (a,a,O) = ( a , a , a ) = a ;

lo ,o,o)

= (a,O,a) = (O,a,a) = 0

(t3) (O,O,a) = (a,O,O) = (o,a,o)

b

= (a,a,O) = ( a , a , a ) = a

.

These uodels prove t h e following

(2.5) Proposition. The system of axioms in Definition (2.2) is independent. (2.6) Definiti0n.A ternary group

(of first or of

(I,(.,.,.))

second type) together w i t h an a s s o c i a t i v e binary m u l t i p l i c a t i o n which i s l e f t d i s t r i b u t i v e w i t h respect t o the t e r n a r y composi-

t i o n , i . e . which s a t i s f i e s the e q u a l i t y (11) x.(y,z,w)

= (x.~,x.z,x.w)

, for

a l l x , y , z , w ~ I,

i s c a l l e d a l e f t t e r n a r y near-ring (of first o r of second type).

(2.7) Fropositlon.(l) If ( I , ( . , . , . ) , . ) is a l e f t t e r n a r y near-ring of first type

, then

(I,+,*), where + is defined by (4),

.

., ,

is a l e f t infra-near r i n g , (ii)If (I,(., .)

i s a repular

l e f t ternary near-ring of second type, t h e n (Il+, defined by (9)

, is

a l e f t infra-near ring.

,where + is

263

A ternary interpretation o f the infra-near rings

Proof. (i) Since (I,+) I s a group (Certalne 127, Theorem 4) , and (I,.) is a semlgroup by assumption, we have to check the left

i n f r a - d i s t r l b u t i v i t y of t h e m u l t i p l i c a t l m w i t h r e s p e c t t o t h e addition. We have

I

= ((x.y,0,0>,x.0,x.z) But x.0

+

(O,X.O,x.Z)

= (x.O,x.O,x.z) - - x.0 + X . Z y,zGI

,

+ z ) = x.(y,O,z) = (x.y,x.O,x.z)

x.(y

= (x.y,0,(0,x.0,x.z~)

= x.y + (O,X.O,X.Z).

= (x.O,O, (O,X.O,X.Z))

= x.(O,O,z) hence x.(y

=

+

=

= ((X.O,O,O) ,x.O,X.Z)

, therefore

X.Z

z ) = x.y

-

(O,x.O,x.z) f x.0 + xlz , f o r all x,

.

(ii)Here we obtain immediately

= (x.y,x.z,x.O)

= x.y

+

- x.0

X.Z

x.(y

t z)

, from

= x.(y,z,O)

=

t h e r e g u l a r i t y of t h e

t e r n a r y group. B u t taking y = 0 we obtain (12)

therefore

x.z

+

x.(y

x.C = x.0

+ z) = x.y

+

X.Z

- x.0

,for a l l x , z e I

+

X.Z

The o t h e r exioms i n the d e f i n i t i o n o f

,

.

, for

a l l x,y,z € 1

left

infra-near r i n g s

come from t h e hypotheses and t h e Proposition (2.3)

.

Ye note t h a t i n the g e n e r a l case t h i s c o m u t a t i v i t y g h e n

, as

by (12) does n o t h o l d f o r l e f t infra-near rings

one can

see from the exemple I = E5 i n Section 1.

T h i s way we obtain a s a p a r t i c u l a r case of t h e t e r n a r y near-

-rings the ttpreringsttdefined by Janin [6,73

.

Indeed, a p r e r h g

is a ternary near-ring of first type or of second type which satisfies for all x,y,zqI a commutativity condition of the ternary com-

position

, namely

(x,y,z) =(y,x,z)

(x,y,z) = (z,y,x)

- for

- f o r the

first type and

t h e sec-nd e T e ,which has a n idempotent

m u l t i p l i c a t i o n being l e f t and r i g h t d i s t r i b u t i v e w i t h r e s p e c t t o t h e ternary composition.

We t r a n s l a t e the d e f i n i t i o n of a two-sided i d e a l of a l e f t infra-near ring into ternary language. In a left ternary near-ring,

we say t h a t a nonempty subset J i s a n idea1,lf it t h e following conditionsr

satisfies

M.Stefdnescu

264

, ((x,O,a),x,O)e

J

(i) (a,b,O)(

J

for a l l a , b € J and ~ € 1 1

, for a l l a d J and x G I ; (ill)((x,O,a).y,x.y,O) € J , f o r all a E J and x , y e I (ii) (x.a,x.O,O)

E J

.

Let I be a l e f t ternary near-ring of first type i n which x.0 f 0 b u t 0.x = 0 f o r a l l x G 1 . In addition, we assume t h a t f o r any x,y( I

, (x,O,y).O

, hence

= (x.O,O,y.O)

i s r i g h t d i s t r i b u t i v e with respect t o the

t h e element 0

t r i p l e s o f t h e given

, a s defined above, which a r e semigroups under multiplj.catlon , a r e a l s o closed under t h e t e r n a r y operation. We note t h a t x.wEW , for any X E Iand w e W

form. Then t h e subsets D and W

In f a c t , t h e following proposition holdsi

Proposition. Let I be a l e f t ternary near-ring of f i r s t

(2.8)

type which s a t i s f i e s t h e conditions

= x

(I) 0.x

, for

(ii)(x,y,z).O

I

a l l XO ; I

, for

= (x.O,y.O,z.O)

a l l x,y,zeI ;

, for a l l X Q ; I (w.x,O,d.x) , f o r a l l d c D , w e W and

(iii) (O,x.O,x) = (x,x.O,O)

( i v ) (w,O,d).x =

/ deD,weW

Then : (i) I = (D,G,W) = i ( d , O , w ) x

f

xaI.

and foa? each

e I , t h e elements d t D and w c g W such that x = (d,O,w)

a r e uni-

quely determined; (ii)Using ternary expressions f o r two a r b i t r a r y elements

of I

,

the multiplication is given by t h e formula8

= (d.d',O,(w,O,d.w'))

(13) ( d , O , w ) . ( d ' , O , w ' > Proof. (i) For each -

x C I

belongs t o D and w = x.0 EW

, we

.

determine d = (x,x.O,O) which w i t h d and w j u s t

B u t x=(d,O,w),

determined above (one can verify this relation by straightforward calculations). w,wl t w

, we

have

If (d,O,w)

= (d',O,w')

, with

d , d l E D and

=

(((O,d',O),O,(d,O,wR)),O,(O,w,O))

= (((0,d',0~,0,(d',0,w~),0,~0,w,0~~ and after some skillful calcu-

l a t i o n s we obtain = (w',w,O)

(0,d:d) = ( w ' , w , O ) C W n D

= 0 , B u t then d1 = ( d ' , O , O )

, hence

(O,d',d)=

= (dl,O,(O,dl,d)) =

A ternary interpretation of the infra-near rings

= ((d',OiO),d',d)

(d',d',d)

p

(d',d',(O,O,d))

p

= (O,O,d) = d

t((d',d',O),O,d)

265 0

I n t h e same manner, we prove

that w' = w

(it) The formula (13) is obtained by using t h e condition (iv) the e q u a l i t y w.x = w f o r a l l x 6 1 and w t w .

t a k i n g i n t o account

It might be Fnteresting t o study ternary near-rings In an Independent context

. Here we

have used t h e t e r n a r y i n t e r p r e t a -

t i o n f o r showing t h e r e l a t i o n between infra-near rings and pre-

rings and f o r explaining t h e l e f t l n f r a - d l s t r i b u t i v i t y a s a ref l e c t i o n o f a l e f t d i s t r i b u t i v i t y of t h e m u l t i p l i c a t i o n with r e s p e c t t o a t e r n a r g composiflon. REFEFf ENCES

C11. Baer,R.

- Z u r EinfUhrung des Scharbegriffs, J.reine Math. ,160 (1929)

C23. Certaine,J.-

, 199-207.

The t e r n a r g operation (abc) = ab-lc of a

group, Bull.Amer.Math.Soc.,49

c3]. Climescu,Al.-

Anneaux f a i b l e s

(11) (1961)

ClFmescu,Al.

[4],

angew.

9

(1943)

, 869-877.

, Bul.Inst.Polit.Ia$i

,7

1-6

- A new c l a s s of weak rings , (Romanian) ,

ibidem, 1 0 ( 1 4 ) ( 1 9 6 4 ) , 1 - 4 .

[5]. Cupona,G.

- On

quasirings

Phys,ldaC$doine

, (Macedonian

, 20

, Eull.Soc.lyfath.

)

(1969) ,19-22 (1971) 4 MR

44 #I703

c63. J a n h , P . - Une g h 6 r a l l s a t i o n de l a notion d'anneau.Pr6-

anneaux.C.R.Acad.Scl.Paris,Sec.A

, 269

(1969)

,

62-64

[7].

Janln,P.

- Une g h 6 r a l i s a t i o n de l a notion Prealghbres. C.R.Acad.Sci,mris

(81

.

(1969)

, 120-122.

Murdoch,D.C. ,Ore,O.

d'algbbre.

, Sec.A , 269

- On generalized rings, Amer. J.Math. ,

M.StejZnescu

266

63 (1941) 973-78

0

[91. P i l e ,G.- Near-rings .The theory and its applications

North-Holland Mathematics Studies 23

, North-Holland

Pub1 .Comp. ,Amsterdam, 1977.

[lo].

Stefhescu,MFrela

- Infra-near

An.St.Univ.Al.I.Cuza

1111.

Stefhescu,Mirela

-A

r i n g s o f a f f i n e type,

I a s i , 2 4 (1978) ,5-14

generalization o f the concept of

n e a r r i n g rInfra-near r i n g s

45 [12].

- 56

,

ibidem, 25 ( 1 9 7 9 ) ,

9

StefZinescu ,Mirela

- Multiplications i n f r a - d i s t r i b u t i v e 6

s u r un groupe,Publ .hth.Debrecen, 255

.

- 262.

27 (198C3,

Near-rings and Near-fiilds,C. Bersch (editor) 0 Elsevier Science Publishen B.V. (North-Holland), 1987

267

ON TWO-SIDED IDEALS IN MATRIX NEAR-RINGS

ANDRIES PJ VAN DER WALT Department of Mathematics, University of Stellenbosch, 7600 Stellenbosch, Republic of South Africa

Matrix near-rings over arbitrary near-rings were introduced in [ 2 ] ,

where

some results about the correspondence between the two-sided ideals in the base near-ring

R

and those in the matrix near-ring

Mn(R)

were given. The pur-

pose of this paper is to give a f u l l e r account of this correspondence which turns out to be quite a bit more complex than in the ring case, although the overall picture is pleasantly similar. (R,+,-) be a right near-ring with identity 1.

Let

direct sum of

n

copies of

The elements of Rn

Rn

will denote the

(R,+), and similarly for subgroups of

(R,+).

are thought of as column vectors and written in transposed

form with pointed brackets, eg

.

The symbols

1.

1

and n ,

will

1

denote the jth coordinate injection and projection functions respectively, The elementary

-

n

X

r E R

s

R and

rs, all

s E R.

For typographical reasons f r .

[r;i,j] . The subnear-ring of

written

are the functions fr.: Rn -P Rn, I' i(r): R -P R is the left multiplica-

1

11

tion

n matrices over

fr. = liX(r)n.. Here

where

is sometimes

11

M(Rn)

generated by the

fr , 1.1

is the

near-ring of n X n matrices over R, denoted Mn(R). Now l e t I be a (two-sided) ideal o f R. There are two obvious ways in which one can let an ideal in Mn(R) correspond to I . The first is to define

I*

:= {A

E Mn(R) IAp E I n , vp E Rn}

as was done in [ 2 ] , where it was shown that the mapping I I+ I* is an injection from the set of all ideals of R into that of Mn(R) (see 4 . 3 of [2])

.

I+ where

The second is to define :=

Id({fa. ( a E I , 1

i,j

1.1

< n)),

Id(T) denotes the ideal generated by the set T .

for all

a E I , 14 i,j

PROPOSITION 1

7' 5 I*

PROPOSITION 2

The mapping

PROOF Suppose

1,

and

Since

fyj E I*

n, we immediately have the following for any ideal

7,

7

1 of R.

0

I+ is an injection. are ideals of

R

such that

1, # 1 2 ;

say

A.P.J. van der Walt

268

Il \ I,.

a E

E 1;.

Then l : f

Next, if J

J,

:=

{x E

E I;,

If l : f

is any ideai of

R-I

3

by Proposition 1.

< a , ~ ,..., O> f

=

r;.

(R) we define

for some A E

R ~ X = 71.~0

It was proved in [2] that J,

fI1 E 1;

then

..., O>

fIl <1,0,

But this is not the case because

J,

p

E R ~ 1, G j

is an ideal of

R.

nl.

The following is an important

result in this connection:

PROOF J iff

c_ (J,)* was shown in 4.7 of [2]. By 4.4 of the same paper fyl E J

a E J,.

This shows that

5 J.

(J,)'

0

In view of this result it is of interest to know whether the inclusion

I+ 5 I* of Proposition 1 can be proper. That this is indeed the case is shown in the next

EXAMPLE 4

Let

,

R = Z,[x]

the zerosymmetric polynomial near-ring in one

indeterminate over the integers (see Pilz [3] p 220 onwards). Consider the

7

ideal

:=

(2)[XI.

We proceed to show that in

2 end we prove that A = [x ;l,l] (f;l + f;2)

l2 for

Clearly, A = <2pq,0> E

A

$ I+

all

-

I+

M2(R)

2 [x ;l,l]

E R

2

I*,

and to this

2

-

,

f

[x ;1,2] E I* so A E I*.

\

I+.

TO see

that

it suffices to carry out the following steps:

(In the following discussion we let X,Y denote arbitrary matrices in MZ(R); B, B1

and

B2 will be elements of

arbitrary elements of (a)

R

2

;

I+;

, , are

a.,b.,c, are appropriate integers; and 1

1

1

Show that niX = a.p + b.q

+

2c.pq +

i E {1,21.)

. ..

(b) Show that n.(X( + ) - X) = (a. + 2 c . s )+~ (b. + 2 c . r ) ~ 2c.uv

1

1

+ ...

(c) Note that T.B = 2f.(p,q) for some polynomial f. in 1

(This is because (d)

I+

c -

2 variables.

I:.)

Prove that Tr,B = 2a.p + sb.q + 4cipq

+

.. .

for any

B E I+.

This is done recursively by showing the assertion (i) is true for

B = [2axt;i,jl;

(iii) if true for B,

is also true for

B1 + B2' BX (by (a) and (c));

(iv) if true for B,

is also true for

X(B

(ii) if true for B1,BZ, is also true for

For any ideal J

in

Mn(R)

+

Y)

-

XY

(by (b) and (c)1 .O

we therefore have the following diagram:

On two-sided ideals in matrix nearrings

Here

269

is the inclusion injection and, in general, not the identity function.

I

This relationship reminds one of the phenomenon of enclosing i d e a l s that one has in polynomial near-rings over commutative rings (see [ l ] or 1 3 1 ) .

However,

in that case one deals only with so-called full i d e a l s of the polynomial nearring, i.e. those ideals that are also ideals of the polynomial ring, whereas the situation we are concerned with here applies to any ideal

J of Mn(R).

The following proposition provides more information about the mappings and

(

PROPOSITION 5 (i) A

5B

(ii) If A

For ideals

+8

then

A+

(vi) PROOF

(i) This is clear.

(iv) (v)

A

and

implies A+ c - B+

(A n B)* = A* n B*. (A n B)+ c - A+ n B+ (A + 8 ) * 3 A* + 8* ( A + 8 ) + = A+ + 8+

(iii)

( )*

1'. 8

of

and A*

R we have:

5 B*.

8*. (In fact, this holds for arbitrary intersections.)

(ii) This was actually shown in the proof of Proposition 2. fiii) This is a standard result about noetherian quotients.

See [31, 1-44.

(iv) Obvious fran (I).

+

E An + 8" = ( A + 8)" for all p E Rn. (vi) It is clear that A+ + B+ 5 (A + 8)+. TO prove the reverse inclusion, a+b note that (A + 8)' is generated by the set if.. la E A, b E 8, 11 a+b - a 1 < i , j < n). Since f. , - fij + fyj E A+ + '8 we have our result. o (v) If A E A*

8*, then

Ap

11

Since O* = 0 we conclude from (iii) above and Propositim 3 .

COROLLARY 6 R

is subdirectly irreducible iff

Wn(R) is.

The next result gives some information on the behaviour of products: PROPOSITION 7

If BC

5A

for ideals A ,

8 and c of R, then 8+C* c_ A*.

270

A.PJ. van der Walt

c*,

PROOF We have to show that if B E 8+, C E p E Rn.

Since Cp E

all y E

en.

cn

An

then BCp E

for all

we need only show that By E b B is of the form f ,, , b E B ,

p E Rn

for all

This is easy enough when

17

An

for

and this

observation provides the starting point for a recursive proof based on the way b is generated by the set if. ,Ib E 8 , 1 G i,j G nl. 0 in which 8' 11

The ideal

COROLLARY 8

A

of

is nilpotent iff A+

R

The following result takes 4.11 of [ Z ]

P

PROPOSITION 9

P*.

So suppose

A+B*

$ P*.

P*

is prime, and let A,B

A8 $ P,

BY 7 this implies

treated similarly.

0

R

iff

P*

is a prime

M,(R).

P is prime (semiprime), then so is

In 4.11 of [ 2 ] it was proved that if

PROOF

Mn(R).

further:

is a prime (semiprime) ideal of

(semiprime) ideal of

is nilpotent in

so

$ P. P

Then

A+$*

is prime.

$ P*

by 5(ii), so

The semiprime case is

0

If we denote by rad S the prime radical of the near-ring

then we have

S,

the

5

COROLLARY 10 rad M n( R )

(rad R) * .

PROOF By S(iii) (rad R ) * =

n{P*[P

is prime in

R}

2

rad Mn(R).

0

It would be interesting to know whether the inclusion in this result can be proper, especially since it was proved in [41 that the J -radical of 2 is of the form (J2(R))*. We close with a result about

R.

generated near-ring

where

A*,

Mn(R)

is an ideal in a distributively

A

First we establish a purely group theoretic result

which is probably well known, and so we provide only a sketch of the proof: Let

LEMMA 1 1

that a = al

... +

a

m

+

+ bm

=

PROOF We have (a2 + a

m

-

... +

bm-l

be a group with

(G,+)

... + am, b a

+

b

+

b

1

+

c = a - a + c - b

... + am )

a

- (a3 +

- ...

- b + b

m

=

d, where

1

PROPOSITION 12 Let R

A 2 J1(R), where A

a,al,

... +

b = a

+ b2 +

-

(a

... +

2

+

... + am) + bl +

(amdl + am) - a

m

from which the result follows.

be a

dg

f G such

Then c := al + bl + a2 + b2 + b , . E J1(G), the commutator subgroup of G.

d

+

..., am, b,bl, ..., bm

+

bm-l

+

0

near-ring with identity, and suppose

is an ideal of

R

and

61(R) is the commutator sugbroup

27 1

On two-sided ideals in matrix near-rings of

(R,t). Then

t

... +

PROOF

A*

in

M2(R)

[al;l,l] + [b1;1,21+

the form

[c ;2,1] + [d ; 2 , 2 ] , 4

consists of all and only those matrices of

... +

where

q

In [2] it was noted that if

[am;l,l]

+

[bm;1,2] + [c1;2,1] t [d1 ;2,2]

Cai, Chi, Zci, Idi E A .

R

is dg, then so is

Mn(R). By applying

matrices in A*

to the vectors

matrices in A*

are of the given form. On the other hand, it follows from

<1,0> and

it is easy to see that all

Lemma 1 1 that every matrix of the given form is in

A*.

0

It will be obvious that an extension of this result to possible.

M,(R),

n > 2,

is

However, the formulation of the extension does not provide additional

insight, so we shall forego stating it.

ACKNOWLEDGEMENT Work on this paper was financially supported by the Council for Scientific and Industrial Research.

REFERENCES [l] [2] I31 [4]

Lausch, H and W Nobauer, Algebra of polynomials, North-Holland, 1973. Meldrum, JDP and APJ van der Walt, Matrix near-rings, Archiv der Math (Hasel). To appear. Pilz, G, Near-rings, Revised Edition, North-Holland, 1983. Van der Walt, APJ, Primitivity in matrix near-rings, Quaest. Math. To appear.

This Page Intentionally Left Blank

Near-rings and Near-fields, G. Betsch (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1987

273

SOME PATHOLOGY FOR RADICALS I N NON-ASSOCIATIVE NEAR-RINGS Stefan Veldsman Department o f Mathematics, U n i v e r s i t y o f P o r t E l i z a b e t h , P o r t E l i z a b e t h Republic o f South A f r i c a ABSTRACT. I t i s well-known t h a t every semisimple c l a s s i n t h e c l a s s m s s o c i a t i v e r i n g s o r a l t e r n a t i v e r i n g s i s h e r e d i t a r y , t h i s bei n g an easy consequence o f t h e Anderson-Divinsky-Sulifiski p r o p e r t y ( c f . [ 11). Dropping t h e a s s o c i a t i v i t y , one g e t degenerate r a d i c a l s i n t h e sense t h a t a r a d i c a l c l a s s has a h e r e d i t a r y semisimple c l a s s i f and o n l y i f i t i s an A - r a d i c a l , i . e . t h e r a d i c a l o n l y depends on t h e I n the s t r u c t u r e o f t h e u n d e r l y i n g a b e l i a n groups ( c f . [ 4 ] and [ 5 ] ) . class o f a l l near-rings ( o r abelian near-rings) t h e s i t u a t i o n i s not as bad, here we have b o t h h e r e d i t a r y and n o n - h e r e d i t a r y semisimple classes, see, f o r example [ 3 ] . Dropping t h e a s s o c i a t i v i t y i n t h i s case, t h e r e s u l t i s even worse than i n t h a t o f n o n - a s s o c i a t i v e r i n g s . We show t h a t i n t h e c l a s s o f a l l a b e l i a n , n o t - n e c e s s a t i l y a s s o c i a t i v e zero-symmetric n e a r - r i n g s t h e o n l y r a d i c a l s w i t h h e r e d i t a r y semisimple classes a r e t h e two t r i v i a l r a d i c a l c l a s s e s .

1.

PRELIMINARIES L e t W be t h e c l a s s o f a l l a b e l i a n n o t - n e c e s s a r i l y a s s o c i a t i v e zero-symmetric

near-rings.

Near-rings w i l l be r i g h t n e a r - r i n g s and we use t h e n o t a t i o n and

n o t i o n s o f P i l z [6].

W , being a u n i v e r s a l c l a s s o f a-groups i s hence s u i t a b l e

f o r t h e development o f a Kurosh-Amitsur r a d i c a l t h e o r y , basics o f which we r e f e r to

171. A c l a s s R 5 W i s a radical class (in t h e sense o f Kurosh-Amitsur) i f ii)

R i s homomorphically closed For every N E W , R(N) := c ( I I I an i d e a l i n N w i t h I E R) E R

ii)

R(N/R(N)) = 0 .

(i)

I R(N)

w i l l denote t h e semisimple cZuss o f R. SR Using standa r d techniques, i t can be proved, f o r any r a d i c a l c l a s s R : AS u s u a l , SR = {N E W

is

= 0)

h e r e d i t a r y i f from N E SR and I an i d e a l i n N, I E SR f o l l o w s .

(i) (ii)

I f K i s an i d e a l o f N and N / K E SR, then N’N)’I c _ R ( K). 5 R(N)’ f o r every i d e a l I o f N.

SR i s h e r e d i t a r y i f and o n l y i f R ( I )

( i i i ) R(N1 8N1) = R(NI) BR(N2). For N E W , N+ w i l l denote t h e u n d e r l y i n g a d d i t i v e gymup o f N and No w i l l be t h e zero n e a r - r i n g b u i l t on N+ by t h e m u l t i p l i c a t i o n ab = 0 f o r a l l a,b E N. The c l a s s o f a l l zero n e a r - r i n g s i n W w i l l be denoted by 2 .

N E W,

No E W holds.

I f I i s an i d e a l o f N, i t w i l l be denoted by I A N.

Obviously, f o r an)

214

S. Veldman

I n t h e sequel, a l l n e a r - r i n g s considered, w i l l be from W. 2.

RESULTS Betsch and K a a r l i [ Z ] have proved t h e n e x t theorem i n t h e c l a s s o f a l l near-

As i t makes no use o f t h e a s s o c i a t i v i t y , i t remains v a l i d i n W , and we

rings. have

5.1 Theorem L e t R be a r a d i c a l c l a s s w i t h h e r e d i t a r y semisimple c l a s s SR and assume SR c o n t a i n s a zero n e a r - r i n g N ( = No) w i t h N # 0.

Then 2

c_ SR.

Our main r e s u l t depends on t h e n e x t two c o n s t r u c t i o n s : For N

E W , we d e f i n i e two n e a r - r i n g s r(N) and h(N) by:

r(N)'

= NtON+4Nt

= A(N)+

w i t h m u l t i p l i c a t i o n d e f i n e d by (a,b,c)(x,y,z)

=

(ax

{

,az , 0 ) , az , c z )

(axtcy

in

r(A)

i n A(A).

Both T(N) and A(N) a r e a b e l i a n , r i g h t d i s t r i b u t i v e , zero-symmetric, not-necess a r i l y a s s o c i a t i v e (even i f N i s a s s o c i a t i v e ) and hence i n W. L e t B = {(a,b,O) B

N 4 No.

B

I a,b E

Then B i s an i d e a l i n b o t h T(N) and A(N) and

N}.

Furthermore,

r (B N ) "= No

and A(N) B

N.

2.2 L e t R be a r a d i c a l c l a s s w i t h h e r e d i t a r y semisimple c l a s s SR. No

E SR, then N

If

N E R and

= 0.

Proof Because SR i s h e r e d i t a r y , we have N = R(NBNo) = R(B)

zR( ( N ) )

A r(N).

On t h e o t h e r hand, R ( m ) = R(No) = 0 , N@N0

hence

R ( T ( N ) ) C_ R(NBNO) = N. By t h e above, we have N = R ( r ( N ) ) A

L e t a,b E N. (O,ab,O)

Because (a,O,O) E (N,O,O)

= (a,O,O)(O,O,b)

E N

Hence ab = 0 f o r a l l a,b E N. 2.3

r(N).

E

(N,O,O)

=

N A T(N), follows.

Thus N = No E R 11SR = 0, i . e . N = 0.

Lemma

L e t R be a r a d i c a l c l a s s w i t h h e r e d i t a r y semisimple c l a s s SR. No i

R, t h e n N

= 0.

If N E SR and

275

Some pathology for radicals in non-associative nearrings

Proof As i n t h e previous lemna, i t can be shown t h a t NO

= R ( A ( N ) ) A A").

For a,b E N, (O,b,O) (ab,O,O)

E (O,N,O)

= (O,U,a)[(a,O,O)

No A A(N).

t

(O,b,0)1

Hence

- (O,O,a)(a,O,O)

E

No

z (O,N,O).

Thus ab = 0 f o r a l l a,b E N.

E R I I SR

Hence N = No

= I ) , i.e.

N = 0.

We can now prove o u r main r e s u l t : 2.4

Theorem

L e t R be a r a d i c a l c l a s s i n W.

Then SR i s h e r e d i t a r y i f and o n l y i f R = 0 or

R = W. Proof Obviously o n l y t h e s u f f i c i e n c y needs v e r i f i c a t i o n . Indeed, i f Z

R, l e t N = No E Z w i t h N f R.

N/R(N) i s a zero n e a r - r i n g , Z 0 # B E R. Z c - R.

Then Bo t Z

For any N

5 SR

5 SR

I f R # I ) , then Z

5 R.

Hence 0 # N/R(N) E SR and because

f o l l o w s from 2.1.

Because R #

I),

let

and by 2.2 the c o n t r a d i c t i o n B = 0 f o l l o w s .

E W , we have N/R(N) E SR and (N/R(N)I0 E R.

€ Z

c_ R.

Hence

By 2.3,

N/R(N) = 0, thus N = R(N) REFERENCES

[I] Anderson, T., D i v i n s k y , A. and S u l i r l s k i , A., H e r e d i t a r y r a d i c a l i n a s s o c i a t i v e and a l t e r n a t i v e r i n g s , Canad. J . Math., 17, 1965, 594-603. [ 2 ] Betsch, G. and K a a r l i , K., S u p e r n i l p o t e n t r a d i c a l s and h e r e d i t a r i n e s s o f semisimple classes o f n e a r - r i n g s , C o l l . Math. SOC. J . B o l y a i , 38, Radical Theory, North H o l l and, 1.985, 47-58. [ 3 ] Setsch, G. and Wiegandt, R., Non-hereditary semisimple classes o f nearr i n g s , S t u d i a Sci. Math. Hungar., 17, 1982, 69-75. [ 4 ] Gardner, B.J., Some degeneracy and pathology i n n o n - a s s o c i a t i v e r a d i c a l theory, Annales Univ. Sci. Budapest, 22-23, 1979/80, 65-74. [ 5 ] Gardner, B.J., Some degeneracy and pathology i n non-associative r a d i c a l t h e o r y 11, B u l l . A u s t r a l . Math. SOC., 23, 1981. 423-428. [ 6 ] P i l z , G., Near-rings, (North-Holland, 1977). [ 7 ] Wiegandt, R., Radical and semisimple classes o f r i n g s , Queen's papers i n pure and a p p l i e d mathematics, No.37, Kingston, O n t a r i o , 1974.

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Newrings and Near-felds, G.Betsch(editor) 0 Elsevier Science Publishers B.V.(North-Holland), 1987

217

PARTIALLY AND FULLY ORDERED SEMINEARRINGS AND NEARRINGS Hanns Joachim WEINERT 1. INTRODUCTION

According to [ 1 1 I and [121, a (in this paper always right distributive) seminearring, is defined to be an algebra S = (S,+,*) such that (S,+) and (S,-) are semigroups and (a+b)c=ac+bc holds for all a,b,c E S. Seminearrings occur very naturally as transformations of semigroups ( r , + ) (cf. Thm. 3.1). They are also the common generalization of nearrings and semirings, and we refer to [I31 and [ I 4 1 for corresponding investigations of statements shared by both, nearrings and semirings. In a similar way we deal in this paper with seminearrings established with a partial order on S. These considerations will also provide new results on partially ordered (p. 0 . ) nearrings (cf. 151, [81 - I 1 1 1 1 and p. 0. semirings (cf. [21, [151), in particular in the fully ordered ( f . 0.) case. which is likewise In the papers just cited, a nearring ( S , + , * ) a p. 0. group (S,+,<) is called a p. 0. nearring iff the set P(S) of all positive elements of ( S , + , S ) is multiplicatively closed (hence a subseminearring). The latter turns out to be equivalent with P (S)S M r (S) (called IIIr in the f o l l o w i n g ) , where Mr ( S ) [Mi(S)1 denotes the set of all right [left] monotone elements of ( S , - , < ) (cf. (2.1)). By results of [51 , there are (proper) f. 0. nearfields S satisfying also P(S) CMk(S) (called 111%), as well as those for which IIIQ fails to be true. Taking also the sets N ( S ) for all negative elements of (S,+,<) and Wr(S) [W,(S)l of all right [left] antitone elements of (S,-,S) into consideration (cf. (2.2)), we introduce IVr by N ( S ) C Wr(S) and IVL by N(S) S Wi(S). For each ring (S,+,') which is also a p. 0. group ( S , + , S ) , we shall see that any of these four properties IIIr - IVi implies the other three (Remark 2.5). In the case of nearrings, merely IIIi c) IVa holds in general, IIIQ*IIIr and IIIQ 9 IVr for zero-symmetric ones, and 111, o IVr if S satisfies at-b) =-(ab) (cf. Thm. 2.4). More general, let (S,+,-) be a seminearring which is also a p. 0. semigroup ( S , + , S ) . We call such a structure ( S , + , - , S ) a w e a k l y partia l l y o r d e r e d I w . p. OJ s e m i n e a r r i n g and introduce the sets P(S), and the properties 111,' already used above for w.p.0. seminear-

...

...

218

H.J. Weinert

rings in 5 2.

The c o r r e l a t i o n s between t h e s e f o u r p r o p e r t i e s , t h e

main s u b j e c t o f t h i s s e c t i o n , t u r n o u t t o s t a g g e r from " t h r e e do n o t imply the f o r t h " t o " p a i r w i s e e q u i v a l e n t " , depending on f u r t h e r a s s u m p t i o n s on S. Some o f t h e s e r e s u l t s are mentioned above. They prove it meaningful t o d e f i n e nearring

-

a p.

0.

-

g e n e r a l i z i n g t h e c o n c e p t of a p.

s e m i n e a r r i n g a s a w. p .

0.

seminearring s a t i s f y -

0.

i n g I I I r y which i s now i n d e p e n d e n t o f and m o s t l y s t r o n g e r t h a n Those p .

P ( S ) ' P ( S ) C P(S).

t r a n s f o r m a t i o n s of p. Thm. 3 . 2 ) .

0.

0.

seminearrings occur n a t u r a l l y as

semigroups (r,+,S),

H e r e 111, i s i m p l i e d by

say

Mr(S) = S ,

I I I Q and IVR i s o n l y t r u e f o r a s p e c i a l c h o i c e of

C o n v e r s e l y , e a c h p.

0.

S =

(rry+,-,s) ( c f .

whereas e a c h of IV,,

seminearring s a t i s f y i n g

(r,+,S)

(Cor. 3 . 3 ) .

M (S) = S r

c a n be re-

p r e s e n t e d i n t h i s way (Thm. 3 . 4 ) . For s t a t e m e n t s on f .

n e a r r i n g s S, t h e c o n s t a n t s u b n e a r r i n g of

0.

S and t h e zero-symmetric

o n e , s a y S1 and S 2 , are of some i m p o r t a n c e

( c f . [ill). W e i n t r o d u c e i n 5 4 c o r r e s p o n d i n g s u b s e m i n e a r r i n g s S 1 and S 2 f o r s e m i n e a r r i n g s S w i t h a z e r o . If ( S , + ; , 6 ) i s a w. p . 0 . seminearring,

t h e same h o l d s f o r ( S 1 , + , ~ , S 1 and ) ( S 2 y + y - , < 2 )w i t h

t h e induced p a r t i a l o r d e r s S ( c f . Thm. 2 . 1 0 ) . For a f . 0. s e m i n e a r i r i n g S, t h e o r d e r o f S 1 + S 2 i s u n i q u e l y d e t e r m i n e d l e x i c o g r a p h i c a l l y by ( S 1 , S 1 ) and ( S 2 , S 2 ) , and, a p a r t from some e x t r e m e cases, a f . s e m i n e a r r i n g S such t h a t

Sl${o}+S2

0.

h o l d s i s non-archimedean

o r d e r e d (Thm. 4 . 2 ) . The a p p l i c a t i o n s t o f .

0.

n e a r r i n g s i n Thm. 4 . 3

p r o v i d e w e l l known a s w e l l a s new r e s z t s , i n p a r t i c u l a r t h a t Thm. 9 . 1 4 6 of

[ l l ] i s o n l y t r u e f o r zero-symmetric

nearrings. Since we

need v a r i o u s examples t o prove r e s u l t s i n 5 2 and § 4 and to show t h a t t h e y a r e f a i r l y s t r o n g , w e have c o l l e c t e d them i n

5.

There i s one more p o i n t c e n t r a l t o t h e d i s p o s i t i o n of t h i s p a p e r . To o u r own s u r p r i s e , a l l r e s u l t s on w. p . 0 . do n o t depend on t h e a s s o c i a t i v i t y o f

s e m i n e a r r i n g s (S,+,',S)

( S , + ) , which a l s o d o e s n o t

s i m p l i f y t h e i r p r o o f s . S i n c e one-sided d i s t r i b u t i v e a l g e b r a s ( S , + , * ) w i t h n o t n e c c e s s a r i l y a s s o c i a t i v e a d d i t i o n a r e of i n t e r e s t i n o t h e r contexts (cf. [4]

,

[6])

, we

f e e l t h a t w e should e x p r e s s t h i s gener-

a l i t y i n o u r p r e s e n t a t i o n . So, s i m i l a r l y a s i n [ 1 3 1 , w e g e n e r a l i z e t h e c o n c e p t of a s e m i n e a r r i n g a s f o l l o w s : DEFINITION 1 . 1 . An a l g e b r a ( S , + , ' )

i s c a l l e d a seminearring i f f ( S , + ) i s any g r o u p o i d , ( S ; ) a semigroup and ( a + b ) c= a c + b c h o l d s f o r a l l a,b,cES. T o a v o i d c o n f u s i o n , w e u s e t h e term n e a r r i n g a s u s u a l , a l t h o u g h a s e m i n e a r r i n g f o r which ( S , + ) i s a q u a s i g r o u p would now be t h e adequate concept.

279

Partially and fully ordered seminearrings and near-rings

Various statements on those seminearrings have been given in [I31 and [141. However, anyone not interested in this generalization, can read this paper without taking notice of Def. 1.1. In particular, all examples of seminearrings used to prove statements are of course additively associative. Finally, we need a few conventions. For subsets A,B of a seminearring (S,+,') we introduce A+B={a+b I a E A , ~ E B ]and AB={ab I a E A , b ~ B 1 , and use e. g. aB for {ajB. If a groupoid or a seminearring S has an additive neutral, this unique element is called the z e r o of S and denoted by 0. In the latter case, o is said to be l e f t [ r i g h t ] a b s o r b i n g iff o S = { o ) [ S o = { o } l holds. A seminearring with a (two-sided) absorbing zero is also called zero-symmetric.

2. BASIC STATEMENTS ON (WEAKLY) PARTIALLY ORDERED SEMINEARRINGS

According to 5 1 we do not assume additions to be associative (if not explicitely stated) and define the more general concept of a weakZy p a r t i a l l y o r d e r e d I w . p . 0 . ) s e m i n e a r r i n g S = ( S , + , * , = < ) by

I

(S,+,')

I1

a < b implies a + c < b + c and

is a seminearring, (S,<) a p.

0.

set, and

c+a6c+b for all

a,b,cES.

In particular, S is called a w e a k l y f u l l y o r d e r e d Iw. f . 0 . 1 s e m i is a f. 0. set, and trivially p. 0. iff the partial order equals equality. Note that a w. p. 0. seminearring S is just a seminearring (S,+,.) and a p. 0. groupoid (S,+,<) (cf. [I], Chap. X). By 11, a < b implies a+c c b+c if c is right cancellable in IS,+). So if (S,+) is cancellative, the monotony law I1 yields its strict version n e a r r i n g iff (S,<)

IISt a < b

implies a + c
I

a < a+p for all a E

we introduce the set seminearring S = (S,+,.,<), which are in fact the right positive elements of the p. 0. groupoid (S,+,S). Due to 11, p < s for P E Pry s E S implies s E Pry hence Pr(S) is an upper set of ( S , S ) and thus a subgroupoid of (S,+) if Pr$@. The same holds for the sets PQ(S) and P(S) =Pr(S) rl PR(S) of all l e f t p o s i t i v e or p o s i t i v e elements of S, respectively. If S has a zero 0, one has P =Pa = P = { p S~ 1 0 s p}. Dually with respect r to the relation <, we have the corresponding definitions and stateBy

Pr = Pr(S) = {p E

S

of all r i g h t p o s i t i v e elements of a w. p.

S]

0.

280

H.J. Weinert

ments f o r t h e sets N r ( S ) , NII(S) and N(S) of a l l [ r i g k t , l e f t ] n e g a t i v e e l e m e n t s of S.

briefly

Note t h a t

r* An element m of a w . p . 0 .

(2.1)

n < p holds for a l l

ncN

II

and

pepr,

NI16P

a cb

implies

seminearring

amsbm

for a l l

S = (S,+,',<) such t h a t

a,bES

i s c a l l e d r i g h t m o n o t o n e , and s t r i c t l y r i g h t m o n o t o n e i f f ( 2 . 1 ) h o l d s w i t h am
t h e s e sets, i f n o t empty, are subsemigroups of (S,.), and ML(S) i s a l s o a d d i t i v e l y c l o s e d . An element W E S such t h a t (2.2)

a
implies

awtbw

for a l l

a,bE S

i s c a l l e d r i g h t a n t i t o n e . W e d e n o t e t h e set of a l l t h e s e e l e m e n t s

by Wr(S). Likewise w e i n t r o d u c e t h e s e t W ( S ) of a l l l e f t a n t i t o n e 11 elements o f S and W ( S ) = W I I ( S ) n Wr(S). R u l e s l i k e W r W r E M r or W M CWr

are obvious. Note i n t h i s c o n t e x t t h a t (S,+,-,6) remains r r a w . p . 0 . s e m i n e a r r i n g r e p l a c i n g 5 by i t s c o n v e r s e r e l a t i o n S d . But whereas P ( S ) and N ( S ) change t h e i r r o l e s under t h i s p r o c e d u r e , Mr(S), M Q ( S ) , W r (S),and W II ( S ) remain i n v a r i a n t .

DEFINITION 2 . 1 .

L e t S be a w . p . 0. s e m i n e a r r i n g . W e d e n o t e , i n t h e

f o l l o w i n g o r d e r of s u c c e s s i o n , t h e p r o p e r t i e s (2.3)

P(S) C Mr(S),

N(S) C Wr(S),

P(S) C MII(S), N ( S ) C WR(S)

by 111,' I V r , I I I Q and I V R , and use a sequence a s e . g. ( l , l , O , l ) t o i n d i c a t e by "1" which of t h e s e p r o p e r t i e s h o l d s f o r S and by " 0 " which n o t . The c o r r e s p o n d i n g sequence i s c a l l e d t h e t y p e of S, and 1 S a p a r t i a l l y o r d e r e d ( p . 0 . ) s e m i n e a r r i n g i f f 111, h o l d s . Hencef o r t h w e w r i t e e . g . ( 1 1 0 1 ) = ( 1 3 ) f o r t y p e s i n an obvious and conv e n i e n t i n t e r p r e t a t i o n . As shown by examples c o l l e c t e d i n § 5 ( a l l with a s s o c i a t i v e a d d i t i o n ) w e s t a t e a t f i r s t : REMARK 2 . 2 .

The p r o p e r t i e s ( 2 . 3 ) a r e i n d e p e n d e n t , and t h e r e a r e

even f . 0. n e a r r i n g s and f . 0. s e m i r i n g s of t y p e ( l o l l ) = ( I l ) , f . 0. s e m i r i n g s o f t y p e (1101) = ( 1 3 ) and o f t y p e ( 1 1 1 0 ) = (14), and w. f . 0 . s e m i r i n g s o f t y p e (0117) = ( 7 ) (Expls. 5.3 and 5.91. Moreover, t h e r e a r e w . p . 0 . s e m i n e a r r i n g s S = ( S , + , * , < ) such t h a t a l l p o s s i b l e t y p e s from ( 0 ) t o ( 1 5 ) r e a l l y o c c u r .

Partially and fully ordered seminearrings and nearrings

28 1

On the other hand, additional assumptions on S cause implications between the properties ( 2 . 3 ) . For instance, there are no f.0. nearrings of type ( 1 3 ) or ( 1 4 ) , in fact not even p. 0. ones, and also no w. f.0. nearrings of type (7) (cf. Thm. 2 . 4 ) . For the corresponding investigations we need one more notion and call a w.p. 0. seminearring (S,+,-,6)with a zero p a r t i a Z l y o r d e r e d b y X=P(S) f r o m t h e l e f t iff the following holds for all a,bE S : (2.4)

a Sb

there is some X E X satisfying x+a = b.

Q

Likewise one defines S to be p . o . by X=P(S) f r o m t h e r i g h t with respect to a + x = b , and both concepts apply in fact to ( S , + , < ) . 1 ) Turning now to implications between the properties ( 2 . 3 ) for particular w. p. 0. seminearrings S = ( S , + , - , < ) , of course in the meaning of Def. 1 . 1 , we use the following abbreviations for additional assumptions on S: S has a left absorbing zero 0. (r) S has a right absorbing zero 0. (p) S is p.0. by its set P(S) of positive elements from one side. (n) S is a nearring (i. e.(S,+) a group), which yields ( 8 ) and (P). S is a nearring satisfying also the sign-rule a(-b) = -(ab). (s) (9.)

L W A 2 . 3 . Let

S = ( S , + , * , < ) be a w.p.0. seminearring and write P(S) = P etc. Then we have the following implications depending on the assumption on S fixed at each arrow:

-__ -__ P

111,

PP C P

NP C N

111,.

R

P

IVr

PNEN

IIIQ

NNEP

PP G P r

r

PN E N

S

-

IVr

IIIR

r

S

NP C N

IVQ

R

n

R

NN E P

IVg r

Proof. From n < o S p for each n E N , P E P and ( a ) we obtain NMrEN and PMrSP, hence IIIr implies N P E N and PPEP. All other implications indicated by ( R ) or (r) follow similarly. Now we assume P P E P [PNEN] and (p). Then ( 2 . 4 ) yields 111, [IV,] by right distributivity and the definition of P [N]. The implications for which

282

H.J. Weinert

( n ) o r ( s ) i s assumed a r e clear. The n e x t theorem y i e l d s i n p a r t i c u l a r s t a t e m e n t s on w . p .

0.

near-

o n e s ( ( n ) and ( r ) ) , and s u b n e a r r i n g s

r i n g s S ( ( n ) ) , zero-symmetric

of a n e a r f i e l d ( ( s ) and ( r ) ) ,which w e do n o t f o r m u l a t e e x p l i c i t e l y . I n t h i s c o n t e x t one a l s o c h e c k s t h e e q u i v a l e n c e s

7p E P 7n c N

(2.7)

IIIR

and

b-a€ P

implies

pb-pa

E

P

(2.8)

IVL

and

b-a

implies

nb-na

E

N,

clearly for a l l

a , b , n , p E S.

E

P

The r i g h t s i d e of

t o t h e supplementary axiom A 4 i n [ 5 ] f o r f . THEOREM 2 . 4 .

Let

S =

0.

(S,+,-,<) be a w . p . 0 .

and

( 2 . 7 ) corresponds

nearfields.

s e m i n e a r r i n g . Then,

u s i n g t h e above c o n v e n t i o n s , we have t h e f o l l o w i n g i m p l i c a t i o n s :

(2.9)

IVR

7I11

(2.10)

IVr

‘sc

== > I I I r

and

rYP

111,.

Moreover, a w. f . 0 .

n e a r r i n g s a t i s f y i n g I V r i s zero-symmetric. ( 2 . 9 ) and t h e e q u i v a l e n c e ( 2 . 1 0 )

Proof. The second i m p l i c a t i o n of

f o l l o w from ( 2 . 5 ) and ( 2 . 6 ) . To show r i n g S, a t f i r s t w e assume (-p) E N

IVQ

I V Q y P E P and

e,

I I I Qf o r a w . p . o . n e a r -

a < b , i . e . b-aE P. Then

and ( 2 . 8 ) imply ( - p ) b - ( - p ) a = - ( p b ) + p a E N , hence p a = p b + n

for some shows

IVr

nEN,

IVR

*

which y i e l d s

pa
by t h e d e f i n i t i o n of N . T h i s

I I I R , and s i m i l a r l y w e g e t

I I I Q=) I V R

using

(2.7).

For t h e l a s t s t a t e m e n t w e r e f e r t o Lemma 4 . 1 . From t h e above w e o b t a i n w i t h o b v i o u s p r o o f s : REMARK 2.5.

For a w . p . 0 .

r i n g S, from ( 2 . 9 ) and i t s l e f t - r i g h t

d u a l v e r s i o n it f o l l o w s t h a t each of t h e p r o p e r t i e s ( 2 . 3 ) i m p l i e s t h e o t h e r t h r e e and c h a r a c t e r i z e s S a s a p a r t i a l l y o r d e r e d r i n g . REMARK 2 . 6 . Def.

2.1

The c o n c e p t of a p . 0 . n e a r r i n g S i n t r o d u c e d by I I I r i n ( 2 . 5 1 , e q u i v a l e n t w i t h t h e u s u a l one which d e f i n e s

i s , by

a p a r t i a l o r d e r on ( S , + , - )a c c o r d i n g t o ( 2 . 4 ) by a s u b s e m i n e a r r i n g X=P,

assumed t o s a t i s f y t h e p r o p e r t i e s g i v e n i n f o o t n o t e 1 ) .

REMARK 2 . 7 .

Each p . 0 .

(1100)= (12),

n e a r r i n g S h a s one of t h e t y p e s

( l o l l ) =( 1 1 )

p o s s i b l e f o r p.

0.

and

(1111) = (15),

(1000) = (8). Only ( 1 5 ) and ( 1 2 ) are

n e a r f i e l d s , and t h e r e a r e even f .

0.

n e a r f i e l d s of

t h i s k i n d a s shown i n [ 5 ] . Moreover, f . 0 . n e a r r i n g s o f t h e t y p e s ( 1 5 ) and ( 1 2 ) are zero-symmetric, t h o s e of t y p e ( 1 1 ) n o t ( b y ( 2 . 9 ) ) , whereas b o t h c a s e s o c c u r f o r 5.1 and 5 . 5 ) .

f.0.

n e a r r i n g s of t y p e ( 8 )

( c f . Expls.

Partially and fully ordered seminearrings and near-rings REMARK 2.8.

The n e x t c l a s s a r e w . p . 0 . (0111) = ( 7 ) ,

s a t i s f y I I I r y hence w i t h and ( 0 0 0 0 ) = ( 0 )

283

n e a r r i n g s S which d o n o t (0011) = (3)

(0100) = ( 4 ) ,

a s p o s s i b l e t y p e s . The l a t t e r t h r e e o c c u r even a s

w. f . 0 . n e a r r i n g s ( E x p l s . 5.2 and 5 . 7 ) . By t h e l a s t s t a t e m e n t of Thm. 2 . 4 and ( 2 . 9 ) , t h e r e are no w. f . 0 . n e a r r i n g s of t y p e ( 7 ) . W e d o n o t know w h e t h e r t h e r e are w . p . 0 . There a r e w . f . 0 .

REMARK 2 . 9 .

(0111) = ( 7 ) ,

(1110) = ( 1 4 )

o n e s ( b u t c f . Expl. 5 . 3 ) .

s e m i r i n g s of t h e t y p e s

and

(1101) = ( 1 3 ) ,

( 1 0 1 1 ) = ( 1 1 ) ( c f . E x p l s . 5.3 and 5.9).

T h i s shows t h a t IIIr and I I I Q a s w e l l a s I V r and I V L a r e indepens e m i r i n g s . However, s i n c e s t a t e m e n t s on s e m i -

dent a l s o for w.p.0.

r i n g s are m u l t i p l i c a t i v e l y l e f t - r i g h t d u a l , one s h o u l d speak a b o u t

a p-

0.

semiring

S

o n l y i f I I I r and IIIR h o l d . T h i s s t r o n g e r con-

c e p t w a s i n t r o d u c e d i n [151 by

P ( S ) C M(S) = M r ( S ) fl M Q ( S ) .

W e close t h i s s e c t i o n d e a l i n g w i t h q u e s t i o n s on s u b s t r u c t u r e s :

U = (U,+,.) be a s u b s e m i n e a r r i n g of a w . p . 0 . (S,t,-,4). Then (U,+,.,<) i s a w . p . 0 . seminear-

THEOREM 2 . 1 0 .

a) Let

seminearring

S =

r i n g w i t h r e s p e c t t o t h e i n d u c e d p a r t i a l o r d e r , and one h a s (2.11)

C ( S ) n U E C(U)

for

C = P, N, Mry

Wry

M R and W Q .

b ) However, even i f S i s f u l l y o r d e r e d , a s s o c i a t i v e and commutative w i t h r e s p e c t t o b o t h o p e r a t i o n s and s a t i s f i e s I I I r , IV,,

I I I R and

I V k y none of t h e s e p r o p e r t i e s need t o be t r u e f o r U.

c ) S u f f i c i e n t c o n d i t i o n s such t h a t 111,. t r a n s f e r s from S t o U a r e Mr(S) = S ( c f . Thm. 3 . 2 ) o r P ( U ) C P ( S ) , which i s P(U) = P ( S ) t l U by ( 2 . 1 1 ) . The l a t t e r i s a l s o s u f f i c i e n t t o t r a n s f e r I I I Q . I n t h e same way work

Wr ( S ) = S

f o r IVr

and N ( U ) E N ( S ) f o r I V r and I V Q .

d ) A s s u m e t h a t U and S have a common zero, which i s a l w a y s t h e c a s e i f b o t h a r e n e a r r i n g s . Then

P(U) = P(S) fl U and N ( U ) = N ( S ) I l U hold, IIIQor IVQy the

and i f S s a t i s f i e s one of t h e p r o p e r t i e s I I I r , IV,,

same p r o p e r t y h o l d s f o r

U.

P r o o f . A l l s t a t e m e n t s of a ) are checked s t r a i g h t f o r w a r d , and c ) follows from ( 2 . 1 1 ) Example 2.11.

and i m p l i e s d ) . For b ) w e g i v e t h e f o l l o w i n g

a ) A semiring ( S , + ; ,

l i s t e d i n Thm. 2.10.

S = {o,a,b}

o < a < b

b ) i s g i v e n by

2)

s a t i s f y i n g a l l assumptions

+e b

b

.

H.J. Weinert

284

One also checks P(S) =M(S) = S and N(S) = W(S) = {o}. For the subsemiring U = {a,b} we have P(U) =M(U) = U , however, "U) = {a} and W(U) = {b}. So U satisfies 111,. and 111I?' but not IVr and IVI?' Another semiring ( S , + , - , 6 ) of this kind is given by

S = {a,b,c,d) a
b

b

b

b

b

b

b

b

d l b

b

c

c

d l b

b

a

a

.

Here one obtains P(S) = @ , M(S) = {a,b}, N ( S ) = {a} and W(S) = S . The subsemiring U = {a,b,c} satisfies P ( U ) = {c}, M(U) = {a,b}, but not 111, and I11 N(U) = {a) and w ( U ) = U, hence IVr and IV 2' R' 3 . TRANSFORMATIONS ON PARTIALLY ORDERED GROUPOIDS

The following characterization may be considered as known by its specializations bo additively associative seminearrings (cf. [33 , [ 1 2 ]) and nearrings. THEOREM 3.1. Let ( r , + ) be a groupoid and S = rr the set of all transformations f: r + r. For f,gES, define (3.1)

(f+g)( 5 ) =f(5)+9(5) and (f-g)( 5 ) = f(g(S)) for all

S E r .

Then S = (S,+,') is a seminearring such that (S,+) is associative, commutative, left [right] cancellative or a quasi-group, respectively, iff ( r , + ) has this property. Iff ( r , + ) has a zero w, S has a zero o given by o ( 5 ) = w, and of = o holds for all f E S. In this case, the sets S, of all constant transformations of r and S 2 = {f E S I f ( w ) = w) are subseminearrings of S (cf. 5 4 ) . can be represented in this Conversely, each seminearring (S,+,.) way. Take any groupoid ( I ' , + ) containing (S,+) properly as a subgroupoid, and define for all a € S

Then

a + fa

provides a monomorphism of (S,+,.)

into

(rr,+;).

Note in this context that there always exists such a groupoid (I',+)which has the same properties as (S,+) concerning associati-

vity, commutativity, left [right] cancellativity, or being a quasigroup. Apart from the fact that a quasi-group need not be embeddable into a loop, ( r , + ) may be assumed to have a zero in all cases under consideration. On the other hand, (I',+) = (S,+) will do the j o b

285

Partially and fully ordered seminear-rings and neur-rings ax = bx f o r a l l is r i g h t reductive. iff

x

a = b y i. e. i f f (S,.)

E S always i m p l i e s

t o a p . 0 . groupoid ( r , + , < )a s sumed t o be n o n - t r i v i a l l y p. o., and d e f i n e f o r f , g E S = r r

THEOREM 3.2. Now we a p p l y Thm. 3.1

(3.3)

f

s

g

f (5) 6 g(5)

w

Then we o b t a i n a p . 0 .

for all

seminearring

r

0. by

But (S,6)

which a l w a y s st I1 or is

S = (S,+,-,<)

s a t i s f i e s M ( S ) = S and Wr(S) = @ . I f p.

S E T .

(r,+,<) satisfies

P ( r ) from one s i d e , t h e same h o l d s f o r (S,+,<)

is never f . o . ,

sI

even i f

-

(I',S)

p ( r ) E p ( r )1,

and P ( S ) .

is. W e f u r t h e r s t a t e = { n E s,

I

n(r) 5 N(r)),

(3.4)

P(S) = tpE

(3.5)

MQ(S)= { h E S

h(a) 6 h ( B )

for a l l

a , @ r~} ,

(3.6)

WR(S)= { h ~ S l a < =,@h ( a ) 5 h ( P )

for all

a,PET}.

I

a
N(S)

and

r

h a s a z e r o and hence a l s o S, t h e zero-symmetric subseminearr i n g S 2 = (S2,+,*,S) i s a p . 0. one which s a t i s f i e s Mr (S2)= S 2 and If

Wr(S2) = (01,

and (3.4)

-

(3.6) remain v a l i d f o r S

2'

P r o o f . The g e n e r a l s t a t e m e n t s on S, a l s o M (S) = S (which y i e l d s IIIr) r and (3.4) are checked s t r a i g h t f o r w a r d . They imply t h e c o r r e s p o n d i n g ones on S2 by Thm.2.10. c o n t r a d i c t i o n , and

For Wr(S) = @ , assume

w(q) = 5

f o r some f i x e d

wEWr(S)

n

E

I-.

by way of

Define

f,gc S

by f ( 5 ) = a < p = g ( c ) f o r any p a i r a < B i n r , and f ( c ) = c , g ( c ) = c f o r a l l 5 $; 5 Then f < g and (fw) ( q ) < (gw) ( r l ) c o n t r a d i c t f w 5 gw

.

W,(S). I n t h e same way one o b t a i n s Wr(S2) = {o), s i n c e f o r 09w E S 2 t h e r e i s some rl E i- such t h a t w ( n ) = 5 w holds. T o show ( 3 . 5 ) f o r S, assume a t f i r s t h E M L ( S ) . For any p a i r a < @ of r , d e f i n e f , g e S by f ( 5 ) = a and g ( 5 ) = I3 f o r a l l 6 E r Then f < g i m p l i e s hf S h g , hence h ( a ) 6 h ( B ) by (3.3) and ( 3 . 1 ) . So M ( S ) i s c o n t a i n e d i n t h e s e t of a 1 1 i s o t o n e t r a n s f o r m a t i o n s of L! ( r , + , 5 ) d e f i n e d on t h e r i g h t s i d e of (3.5), and t h e o t h e r i n c l u s i o n i s c l e a r . The proof of (3.5) f o r S 2 i s s i m i l a r , b u t it n e e d s some c a r e f o r t h e p a i r s w < @ and a < w i n r , i f t h e r e a r e t h o s e . Likew i s e one shows t h a t W Q ( S ) [W, ( S 2 ) ] c o i n c i d e s w i t h t h e s e t of a l l a n t i t o n e t r a n s f o r m a t i o n s c o n t a i n e d i n S [ i n S21y i. e. (3.6). a s c l a i m e d by

W E

.

COROLLARY 3 . 3 .

Concerning I V r y I I I Q and IV,

f o r t h e p . 0. seminear-

r i n g s S and S2 c o n s i d e r e d i n Thm. 3.2 w e s t a t e :

NU) = @

(3.7)

N ( S ) C Wr(S)

o

(3.8)

P(S) E M ~ ( s )

C)

Ip(r)I

s

(3.9)

N(S) E WQ(S)

*

1N(r)l

5 1

I

H.J. Weinert

286 (3.10)

N ( S 2 ) C- Wr(S2)

w

N ( T ) ={&I}.

(3.11)

P ( S 2 ) E MQ(S2)

w

P ( r ) = {wf

or

P ( r ) = {w,n},

the l a t t e r

o n l y i n t h e c a s e t h a t t h e r e a r e no d i f f e r e n t

( r ,+ ,5 )

e l e m e n t s comparable i n (3.12) Proof.

N ( S 2 ) E WQ(S2) w

w < TZ.

except

N(r) ={w}.

( 3 . 7 ) and ( 3 . 1 0 ) a r e o b v i o u s . For ( 3 . 8 ) w e s t a t e t h a t t h e r e

i s some h E P ( S ) \ M Q ( S ) i f f a p a i r

i n P ( r ) e x i s t s . The l a t t e r n + n ' i n P ( r ) i m p l i e s nSn+n' and n ' Sn+n', hence n < n + n ' o r n ' < n + n ' f o r n + n ' E P ( r ) . I n t h e same way one o b t a i n s ( 3 . 9 ) . For ( 3 . 1 1 ) , P ( S z ) E M R ( S 2 ) f a i l s t o be t r u e i f f t h e r e i s some h E S2 s a t i s f y i n g h ( r ) C P ( r ) and h ( a ) >h(@) f o r some a < @ i n r . The l a t t e r i s i m p o s s i b l e f o r P ( r ) = {u}, and c l e a r l y t h e case i f I P ( r ) I 2 3. For P ( r ) = {w,n} such a mapping h h a s t o s a t i s f y h ( a ) = n > w = h ( B ) f o r some (y. < 8 , and h e S 2 f o r c e s a S w . C o n v e r s e l y , t h e r e i s such an h E S2 i f w + a < B e x i s t s i n r . E x a c t l y t h i s i s e x c l u d e d i f f w < n i s t h e o n l y n o n - t r i v i a l rel a t i o n . Following t h e same p a t t e r n , one shows ( 3 . 1 2 ) where t h e case

i s e q u i v a l e n t with IP(S) I L 2 ,

IN(r)

-

I

=2

IVQ

-

makes no d i f f i c u l t i e s .

A s a consequence of Cor. 3 . 3 ,

IVr

S < q

since

IVr

h o l d s f o r S and even

IVQ

f o r S 2 . So t h e r e remain t h e f o l l o w i n g p o s s i b l e t y p e s (1101) = (13),

( 1 1 1 1 ) = (15),

and t h e same l i s t i n c l u d i n g

( 1 0 1 0 ) = ( 1 0 ) and (1011) = ( 1 1 ) and

( 1 0 0 0 ) = ( 8 ) f o r S2,

(1001) = ( 9 )

f o r S.

A l l t e n c a s e s r e a l l y occur f o r a d d i t i v e l y a s s o c i a t i v e s e m i n e a r r i n g s ,

s i n c e t h e r e are p .

semigroups

0.

( r y + , 6 ) s a t i s f y i n g t h e correspon-

d i n g p r o p e r t i e s on t h e r i g h t s i d e s of

(3.10)

t h o s e w i t h a z e r o w s e l e c t e d by ( 3 . 7 )

-

r

semigroup

= (IN , + , S 0

-

(3.121, a s w e l l as

( 3 . 9 ) . E. g . ,

for the f .

0.

of " p o s i t i v e " i n t e g e r s i n c l u d i n g 0 one ob-

)

t a i n s a n S of t y p e ( 9 ) , b u t an S 2 of t y p e ( 1 3 ) whereas f o r e a c h p . 0 . group

( r , + ,2 )

t h e p.

0.

n e a r r i n g s S and S2 have t y p e ( 8 ) . W e c l o s e

w i t h t h e f o l l o w i n g r e p r e s e n t a t i o n theorem: THEOREM 3.4.

For each p.

( S , + , - , S ) which s a t i s f i e s M ( S ) = S t h e r e i s an order-isomorphism i n t o a p . 0 . s e m i r n e a r r i n g [ n e a r r i n g ] (rr,+,-,s) of a l l t r a n s f o r m a t i o n s of a p. 0. g r o u p o i d [ g r o u p ] ( r ,+ ,I) 0.

seminearring [nearring]

.

P r o o f . C l e a r l y , each p .

0.

groupoid

subgroupoid of a p . 0 .

proper p . 0 .

( r , + , < ) . So

a S b

c)

fa 5 f b

a +fa

for a l l a,bES.

(S,+,I) i s a

[semigroup, group]

w e can a p p l y t h e c o n v e r s e p a r t of Thm. 3.1

mains t o show t h a t t h e monomorphism satisfies

[semigroup, groupl groupoid

of If

and it re-

(S,+,* ) i n t o aSb

( r r ,+,

holds, we

.)

287

Partially and fully ordered seminear-rings and near-rings

o b t a i n by ( 3 . 2 ) f f

thus

a a

b5 = f b ( 5 )

(5)

= a[

S

(5)

= a

I b

= fb(S)

for a l l

[ E S = M ~ ( S )and

for a l l

5

by ( 3 . 3 ) . The c o n v e r s e f o l l o w s from t h e l a s t f o r m u l a .

f a s fb

4 . CONSTANT AND ZERO-SYMMETRIC

SUBSEMINEARRINGS with a zero o we d e f i n e

For any s e m i n e a r r i n g S = ( S , + ,

(4.1)

E h S ,

S1 = { s l e S I

s 1o = s l }

S2 = { s ~ E S s I 20=o}.

and

I t i s e a s i l y checked t h a t e a c h of t h e s e s u b s e t s i s e i t h e r empty o r

-

-

a s u b s e m i n e a r r i n g o f S. One a l w a y s h a s

s2+g

(4.2)

02=o

s1 n s 2

-

SSICS1

+0

and

sln s2

=

to},

s o = o * o = ( s +o)o= s o+02 = o2 shows t h e o n l y n o n - t r i v i a l 2 2 2 i m p l i c a t i o n . R e c a l l t h a t each n e a r r i n g S i s a d d i t i v e l y a s e m i d i r e c t

since

sum of i t s s u b n e a r r i n g s S1 and S 2 , e q u i v a l e n t l y , e a c h S E S h a s a s = s + s E S +S2. S e m i n e a r r i n g s , however, a r e

unique decomposition

1 2 1 f a r away from t h i s n i c e s i t u a t i o n . N e v e r t h e l e s s , e a c h s e m i n e a r r i n g S with a zero s a t i s f i e s

(4.3)

S1 =

(4.4)

o2 = o

(03

*

S

2

= S

0

s 1 s = s1

and

S o = to} for a l l

and s1 E S1,

S E S

0

o S = {o}.

0 s = o * s s = ( s 0 ) s = s ( 0 s ) = s l y and ( 4 . 3 ) 1 1 1 For a s e m i n e a r r i n g S w i t h a l e f t a b s o r b i n g z e r o , w e

The l a t t e r f o l l o w s from from

SSIES1.

call S

1

t h e c o n s t a n t and S2 t h e z e r o - s y m m e t r i c s u b s e m i n e a r r i n g of S

( t h e i r o c c u r r e n c e i n Thm. 3.1 may j u s t i f y t h e f i r s t term). So w e

s t =s 1 1 1 groupoid (S,+)

c a l l a s e m i n e a r r i n g S, a c o n s t a n t one i f f i t h a s a z e r o and holds

s l S t lE

for a l l

S1.

Note t h a t e a c h ( p .

w i t h a z e r o p r o v i d e s such a ( w . P . Now l e t (S,+,*SS)be a w . p . 0 . b i n g z e r o . Then, by Thm.2.10,

w. p . that

0.

0.)

0.)

c o n s t a n t seminearring.

seminearring w i t h a l e f t absor-

and (S2,+,.,S2) a r e seminearrings w i t h r e s p e c t t o t h e induced r e l a t i o n s s i s u c h

P ( S i ) = P(S) n Si

p r o p e r t i e s 111,'

and

(S1,+,*,S1)

N(Si) = N(S) fl Si

h o l d and e a c h of t h e

I I I R and I V R t r a n s f e r s from S t o S1 and t o S2.

IVr,

I n t h i s c o n t e x t w e o b v i o u s l y have: LEMMA 4 . 1 .

Each w . p .

0.

c o n s t a n t seminearring ( S l , + , - , S l )

M r ( S 1 ) = M R ( S 1 )=W,(S1) = S I S whereas

is not t r i v i a l l y p. (1011) =(11)

0.

Wr(S1) = @

holds i f f

satisfies (S1,S1)

I n t h e l a t t e r c a s e , t h e t y p e of S1 i s always

s i n c e O E N ( S , ) . Hence, i f a w. p.

0.

(S,+S*,S)w i t h a l e f t a b s o r b i n g z e r o s a t i s f i e s I V r ,

seminearring

i t s subseminear-

H.J. Weinert

288

s1

ring

satisfies

s1 = t o }

( c f . (4.3)) or i s t r i v i a l l y p.

For t h e n e x t theorem r e c a l l t h a t a

f.0.

0.

semigroup (S,+,<)

z e r o o i s c a l l e d a r c h i m e d e a n i f f f o r each

a E

with

S t h e subsemigroup

of S g e n e r a t e d by a , s a y [a1 , i s n o t p r o p e r l y bounded from above i f o c a and n o t p r o p e r l y bounded from below i f a < o ( c f . [ l ] , Chap. X I ) . Obviously, t h i s c o n c e p t a p p l i e s a l s o t o a (S,+,<)

f.0.

w i t h a z e r o and i t s s u b g r o u p o i d s [ a ] g e n e r a t e d by

THEOREM 4.2.

L e t (S,+,*,<)

be a f . 0 .

groupoid a c S.

seminearring w i t h a l e f t a b -

S1=f { o l s S 2 h o l d s f o r t h e s u b s e m i n e a r r i n g s i n t r o d u c e d above. Then t h e induced o r d e r on t h e s u b s e t S1+S2 of S

s o r b i n g z e r o such t h a t

-

i s u n i q u e l y d e t e r m i n e d by (S, (4.5)

s1+ s 2 < t l + t 2

s 1 + s 2 $ t l + t 2of

for a l l

(4.6)

,S 1 )

according t o

and ( S 2 , S 2 )

s1 <1 t ,

or

s1 = t l ,

n1 < S 2 < p 1 f o r a l l n1 e N ( S 1 ) \ {o} and p1 E P ( S ~ ) \ { ~ } .

Moreover, a p a r t from t h e extreme c a s e s t h a t

S1 5

h o l d , t h e o r d e r of

(S,+,',2) i s non-archimedean.

Proof. M u l t i p l y i n g

s1+ s 2 < t l + t 2 by

yields

s 2 <2 t 2

S1+S2. T h i s y i e l d s i n p a r t i c u l a r

s1 S t l .

If

o

E

P ( S ) G Mr ( S )

s1 = t l h o l d s , s 2 2 t 2

s 1+ s 2 2 t l + t 2 by 11, hence w e o b t a i n

o2

S2

or

S2< 0 5 s

1

from t h e r i g h t

implies t h e contradiction For t h e c o n v e r s e i m -

s2 < t,.

(4.5) , assume a t f i r s t s1 < t l . Then w e have s1+s2< 1t +t2 f o r a l l s 2 , t 2E S 2 , s i n c e s1+s22 t l + t 2 y i e l d s a s above s1 2 t l . I f s1 = t l and s2 < t 2are assumed, a g a i n by I1 w e g e t s 1+ s 2 <= t +t2.

p l i c a t i o n of

of (4.5) which i s o n l y c l a i m e d f o r s 1 + s 2 ft l + t 2 .Now (4.6) f o l l o w s by n l + o < o+s2 < p 1 +o f o r a l l s 2 c S 2 .

This completes t h e procf

I f t h e extreme c a s e s a r e e x c l u d e d , t h e r e are

tl

E

S1 and a 2 E S 2

s a t i s f y i n g o < t , and o < a 2 , o r t l < o and a 2 < o . The f i r s t i m p l i e s S2 < t l by (4.6) and hence [a,]

tl

c

[a,].

SO

< t,

,

the l a t t e r likewise

(S,+,<) i s n o t archimedean o r d e r e d . I n t h e e x t r e m e t h e o r d e r of S may be

c a s e s , which r e a l l y o c c u r ( c f . E x p 1 . 5.61, archimedean or n o t .

Note t h a t (4.5) c o r r e s p o n d s t o t h e u s u a l l e x i c o g r a p h i c o r d e r of S1xs2 i f f

s 1 + s 2 = s l + t 2 always i m p l i e s

s2 = t 2 . Moreover,

t h e r e are

v a r i o u s seminearrings S s a t i s f y i n g S = S1+S2, i n p a r t i c u l a r t h o s e f o r which (S,+) i s a s s o c i a t i v e and c a n c e l l a t i v e . The l a t t e r are s e m i d i r e c t sums of

( S l y + ) and ( S 2 , + ) ,

and a c o n s t r u c t i o n method

s t a r t i n g from g i v e n n e a r r i n g s S1 and S 2 due t o [ l o 1 c a n b e g e n e r a l i z e d c o r r e s p o n d i n g l y , a l s o i n c l u d i n g p a r t i a l o r f u l l o r d e r relat i o n s . The most s i m p l e case ( c f . P r o p . S . 4 ) i s t o d e f i n e t h e opera-

289

Pariially and fully ordered seminear-rings and neaerings

t i o n s on

S1xS2

by

( a , . a 2 ) + ( b l , b 2 ) = ( a l + b l , a 2 + b 2 ), ( a l , a 2 ) ( b l y b 2 )= ( a l b l , a 2 b 2 ) .

(4.7)

A s u s u a l f o r r i n g s and n e a r r i n g s ( c f . [ill, 1 . 5 5 ) , we t h e n c a l l

t h e ( e x t e r n a l ) d i r e c t sum of t h e s e m i n e a r r i n g s S1

S = (S,xS2,+,*)

and S 2 . I f b o t h have a m u l t i p l i c a t i v e l y i d e m p o t e n t z e r o (as assumed i n our context) ,

al

(a, ,02)

-+

and a 2 -+ (ol , a 2 ) p r o v i d e monomorphisms

o f Si i n t o S. So one can change from ( a l , a 2 ) t o t h e n o t i o n a l + a 2 and t o t h e c o r r e s p o n d i n g i n t e r n a l d i r e c t sum. W e need t h i s f o r t h e n e x t theorem and d e a l w i t h r e f e r e n c e s i n i t s p r o o f . THEOREM 4 . 3 .

a)

(S,+,.,6) b e

Let

a f.

0.

nearrinq

and assume

S l = f { o l = / = S 2 f o r t h e c o n s t a n t and zero-symmetric s u b n e a r r i n g S1 and S of S. Then t h e o r d e r of S i s non-archimedean and u n i q u e l y d e t e r 2 mined by t h e i n d u c e d o r d e r s on Si a s t h e l e x i c o g r a p h i c o r d e r on

.

.

a c c o r d i n g t o ( 4 . 5 ) I n p a r t i c u l a r , w e have ( 4 . 6 ) 1 2 b ) There a r e f . 0 . n e a r r i n g s S which a r e t h e d i r e c t sum of t h e i r S = S +S

subnearrings

Sl$.{o}$S2

such t h a t b o t h have no z e r o d i v i s o r s .

c) L e t (S,+,*,S) be a f . 0 . d e t e r m i n e s t h e o r d e r of More p r e c i s e l y , a f .

0.

nearring.

(S,+,-),

Then t h e p o s i t i v e cone P = P ( S )

b u t by no means ( S , + , - , < )

seminearring

(P,+,',S)

cone P ( S ) = P ( T ) of non-isomorphic f . 0 . Proof. P a r t a ) i s a c o r o l l a r y are i m p o s s i b l e f o r a f . 0 .

itself.

may be t h e p o s i t i v e

n e a r r i n g s S and T .

of Thm. 4 . 2 ,

s i n c e t h e extreme c a s e s

n e a r r i n g . I t c o r r e s p o n d s t o [ill, 9.141

and g e n e r a l i z e s Lemma 1 of [91. P a r t b ) i s shown by Expl. 5 . 5 . c o n t r a d i c t s Thm. 1 of S S

1 2

[a]

( [ 11I

, 9.146) , where

It

it i s used t h a t

= S S = to) h o l d s f o r a d i r e c t sum of n e a r r i n g s Siy which i s 2 1

n o t i m p l i e d by t h e above d e f i n i t i o n . For c ) c f . Expl. 5.1 a ) . THEOREM 4 . 4 .

Let (S,+;,<)

s o r b i n g z e r o . Then, f o r a l l

be a f . 0 . s2 E S 2

s e m i n e a r r i n g w i t h a l e f t aband p l c P ( S 1 ) , t h e p r o d u c t

s2p1 i s a d d i t i v e l y i d e m p o t e n t . Hence s 2 p 1 = o h o l d s i f no i d e m p o t e n t s e x c e p t o., i n p a r t i c u l a r i f (S,,+)

( S l y + )h a s is l e f t [ o r r i g h t ]

cancellative. P r o o f . A t f i r s t w e assume

s2s1c s1

s2 = p2 f P ( S 2 ) .

Then I I I r , ( 2 . 5 ) and

imply p2p1 E P ( S ) n S1 = P ( S 1 ), hence

p2p1 = o o r o < p 2 p l .

I n t h e l a t t e r case w e can a p p l y ( 4 . 6 ) t o g e t p 2 + p2 < P2P1' BY 111, and p l p l = p l y t h i s y i e l d s p p + p 2 p 1

H. J. Weinert

290

t a i n e d i n S2, from p, = e R p l = o it f o l l o w s t h a t a w i t h a l e f t i d e n t i t y i s zero-symmetric

f.

0.

nearring

( c f . [ l l l , 9.136 and 9 . 1 3 7 ) .

5. EXAMPLES The p u r p o s e of t h i s s e c t i o n i s t o c o m p l e t e s t a t e m e n t s and p r o o f s in

2 and § 4 ,

n o t t o g i v e i m p r e s s i v e examples. I n p a r t i c u l a r , w e

show t h a t a l l t y p e s o f w . p . 0 .

s e m i n e a r r i n g s r e a l l y o c c u r and t h a t

o u r r e s u l t s c o n c e r n i n g t h e p r o p e r t i e s (2.3) a r e f a i r l y c o m p l e t e . So

we p r e s e n t e a c h t y p e by a w . p . 0 . s e m i n e a r r i n g S chosen a s s t r o n g a s w e c o u l d do, e. g , a s a n e a r r i n g , w i t h a f u l l o r d e r , o r w i t h a ( l e f t ) a b s o r b i n g z e r o . For t h e same r e a s o n , a l l s e m i n e a r r i n g s c o n sidered i n t h i s section are ~ l s a cdditively associative. W e c l e a r l y u s e t h e f . 0 . n e a r f i e l d s of t h e t y p e s ( 1 1 1 1 ) = ( 1 5 ) and t h a t e a c h c o n s t a n t w. f . 0 . s e m i n e a r r i n g h a s The f o l l o w i n 9 examples a r e , a s f a r a s p o s s i -

and (1100) = ( 1 2 ) , t y p e (1011) = ( 1 1 ) .

b l e , arranged according t o § 2. 5 . 5 , 5.6 and 5.7,

Expls.

The i n s e r t e d Prop. 5.4

prepares

where t h e f i r s t b o t h c o n c e r n

4 . A l l proofs

have been o m i t t e d . They a r e s t r a i g h t f o r w a r d even i f sometimes a

l i t t l e b i t tedious. Example 5.1.

a ) L e t (2,+,-,6) be t h e f . 0 . r i n g of i n t e g e r s i n t h e

u s u a l meaning. Define a new m u l t i p l i c a t i o n by S= (X,+,o,S)

is a f.

0.

zero-symmetric

and W ( S ) = M R ( S ) = Wk(S) = { O } ,

r

aob = a . I b l .

Then

n e a r r i n g s a t i s f y i n g Mr(S) = S

hence of t y p e

(1000) = ( 8 ) . N o t e

t h a t t h e r i n g Z a s w e l l as S have t h e same f . 0 .

seminearring

( N o , + , - , < ) a s p o s i t i v e cone ( c f . Thm. 4.3 c ) ) .

b ) The zero-symmetric satisfies of t y p e

f.

0.

Mr(U) = M k ( U ) = U

subseminearring U = (-No,+ and

Wr(U)

=W,(U)

={03.

,o

,I )

of S

Hence it i s

(1010) = ( 1 0 ) .

Example 5.2.

be t h e f . 0 . g r o u p o b t a i n e d as t h e d i r e c t (Z,+,<),o r d e r e d l e x i c o g r a p h i c a l l y : a l < b l o r a l = bl , a 2 5 b 2 . D e f i n e a p r o d u c t by

L e t (S,+,<)

sum of two c o p i e s of (al ,a2) S (bl,b2)

**

Then ( S , + , - , S ) i s a zero-symmetric w. f . b u t n o t 111,. Example 5.3.

This implies type a ) A zero-symmetric

0.

nearring satisfying I V

(0100) = ( 4 ) w.f.0.

by Thm. 2 . 4 .

s e m i r i n g (S,+,-,S) which

Mr(S) = W r ( S ) = N ( S ) = {o} and MR(S)= Wk(S) = P ( S ) = S g i v e n on S = { o , a , b , c , d } by satisfies

is

r’

29 1

Partially and fully ordered seminearrings and nearrings

o < a < b < c < d

Hence S i s of t y p e

b

b

b

d

d

o

b

b

b

d

d

d

d

d

o

a

a

a

(0111) = (7).

R e c a l l t h a t t h e r e a r e no

n e a r r i n g s of t h i s t y p e , and t h a t S c a n n o t b e p . 0 . by s i d e . By t h e way,

is a w.f.0.

U = (a,b,d)

b

a

.

w. f .

0.

P ( S ) from one

s u b s e m i r i n g of S w i t h

t h e element a a s l e f t absorbing z e r o , p . 0 .

by

P ( U ) and of t y p e

(0011 ) = ( 3 ) .

d i s t r i b u t i v e , w e can s w i t c h t o i t s m u l t i p l i -

b ) S i n c e S i s two-sided

c a t i v e l y l e f t - r i g h t d u a l v e r s i o n , s a y ( S ' , + , - , S ) , a zero-symmetric f.0.

s e m i r i n g of t y p e

a f.0.

(1101) = (13).

n e a r f i e l d of type

which i s a l s o of t y p e

(1100)

Note t h a t t h e p o s i t i v e c o n e of

is a f . 0 .

s e m i n e a r f i e l d ( c f . [131)

(1101) = ( 1 3 ) .

PROPOSITION 5.4. L e t ( S 1 , + , ' , < ) be a c o n s t a n t and (S2,+,',S2) a 1 zero-symmetric w . p . 0 . s e m i n e a r r i n g , b o t h n o n - t r i v i a l l y P . o . , and S = S 1 x S 2 t h e d i r e c t sum o f ( S l , + ) and ( S 2 , + ) .

W e assume b o t h t o be

c a n c e l l a t i v e semigroups. D e f i n i n g a l s o t h e m u l t i p l i c a t i o n a s i n ( 4 . 7 ) ,

i s t h e d i r e c t sum of

(S,+,')

a) (S,+,*,S)

is a w . p . 0 .

graphic order

( S l , + , - ) and ( S 2 , + , * ) .

seminearring with r e s p e c t t o t h e l e x i c o -

( a l , a 2 ) I (bl , b 2 )

iff

P ( S ) c o n s i s t s of a l l ( p , , s 2 ) s u c h t h a t that

a l <1 b l 0,

or

< p1

a l = b l , a 2 I2 b 2 ,

a n d a l l (ol , p 2 ) such

and N ( S ) i s d e s c r i b e d s i m i l a r l y . I f

02Sp2,

P ( S 1 ) = (o.,},

p(S1) 9 t o l l , IIIr f o r S f a i l s t o be t r u e . I I I Q f o r S

I I I r f o r S and I I I r f o r S2 a r e e q u i v a l e n t . I f

f o r S holds i f f holds i f f

M (S2) = S 2 .

r

v a l i d , and I V Q i f f b ) (S,+,',<)

r

P ( S 1 ) = { o , } and

S2S2 = {021 or

is also a w.p.0.

antilexicographic order alIlbl.

IV

S 2 S 2 = {02) o r b o t h ,

P ( S 2 ) S 2 ={ 0 2 ] , are

N ( S 1 ) = { o l } and

N ( S 2 ) S 2 = {02}.

seminearring with r e s p e c t t o t h e

( a , , a 2 ) 5 ( bl , b 2 )

iff

a2

c2

b2 o r

For e a c h (S,+,-,S) o b t a i n e d i n t h i s way, I11

r

a2=

and I V

b2' al-

ways f a i l t o be t r u e , whereas I I I R h o l d s f o r S i f f it h o l d s f o r S2, and l i k e w i s e w i t h I V Q . T o g e t h e r w i t h o u r examples f o r S1 and S2, v a r i o u s examples, i n p a r t i c u l a r of w. f . 0 .

Prop. 5 . 4 p r o v i d e s

s e m i n e a r r i n g s S of t h e

t y p e s ( 1 1 ) t o (8) and ( 3 ) t o ( 0 ) . Moreover, one c a n u s e o t h e r m u l t i plications for

S = SlxS2.

A r a t h e r s i m p l e one i s g i v e n by

H.J. Weinert

292

1 y a2b2) ( a l b l y a2b2) = (a ( a , , a 2 )( b l , b 2 ) = { ( a l b l . 02) = ( a l y02)

But w e u s e h e r e o n l y Prop. 5.4 Example 5.5. F o r any f .

for for

=ol 1 b l $;ol.

b

t o g i v e some needed examples:

c o n s t a n t n e a r r i n g S l and any f . 0 .

0.

zero-

symmetric n e a r r i n g S2 s a t i s f y i n g Mr(S2) = S2 and S2S2$. { 0 2 } , t h e d i r e c t sum S a c c o r d i n g t o Prop. 5.4 a ) i s a f . 0. n e a r r i n g of t y p e In p a r t i c u l a r , S

(1000) = ( 8 ) .

chosen so ( c f , Expl. 5 . 1 ) . Example 5.6. A

1

h a s no z e r o d i v i s o r s and S2 may be

T h i s p r o v e s Thm. 4.3 b )

seminearring

f.0.

S such t h a t

.

S 1 5 0 2 S2

hold ( c f .

Thm. 4 . 2 ) i s o b t a i n e d by Prop. 5.4 a ) e . g . for S1 = ( - R o y + ,w
From Prop. 5.4 b ) w e o b t a i n w. f . 0 .

(0011) = ( 3 )

types

r i n g S1 and a f . 0 .

and

zero-symmetric

(1100) = ( 1 2 ) .

(1111) = (15) o r

(0010) = (2)

of t h e t y p e s

and

n e a r r i n g s S of t h e

u s i n g any c o n s t a n t f . o . n e a r -

(OOOO)=(O)

n e a r r i n g S 2 e i t h e r of t y p e

L i k e w i s e , w. f . o , (0001) = ( 1 )

seminearrings S

( t h e r e a r e no n e a r r i n g s

of t h i s k i n d ) a r e o b t a i n e d u s i n g zero-symmetric s e m i n e a r r i n g s S 2 with types ( . . l o )

e . g . t h o s e of E x p l s . 5.1 b ) and 5.8.

o r (..Ol),

,F

Example 5.8. W e d e f i n e two s e m i n e a r r i n g s S and T on t h e same set {n,o,a,p} with the f u l l order

0

n

n

a

p

n

o

a

p

P

P

P

P

n < o < a < p by t h e t a b l e s o

a

p

n

n o

na

n p

0

a P

n

0

0

0

a

n

a

a

p

P

n

P

P

P

c l e a r l y b o t h zero-symmetric. The f i r s t one,

(S,+,*,<)

0

0

0

0

,

is a f . 0 .

s e m i n e a r r i n g such t h a t M ( S ) = WL(S) = S, b u t Wr(S) = S x { n } and r MQ(S) = Sx{p} h o l d , t h u s o f t y p e ( l o o t ) = ( 9 ) . Similarly, the other one i s a w. f . 0 . s e m i n e a r r i n g ( T , + , - , < ) of t y p e ( 0 1 1 0 ) = ( 6 ) . Example 5 . 9 .

Each p.

multiplication

0.

semigroup

s t = s a p.

0.

(S,+,<)

p r o v i d e s by t h e t r i v i a l

s e m i n e a r r i n g ( S , + , - , S ) such t h a t

M ( S ) = M (S) = W ( S ) = S and W r ( S ) = @ h o l d .

r

II ( l o l l ) = (11)

R

iff

N(S)$. @

( c f . Lemma 4 . 1 ) .

Thus S h a s t h e t y p e C l e a r l y , one o b t a i n s

(S,+,*,S) is a i s i d e m p o t e n t . S i n c e t h e r e a r e v a r i o u s even semigroups (S,+,I) o f t h i s k i n d s a t i s f y i n g N ( S ) +$, e . g - w i t h

f . 0 . n e a r r i n g s S i n t h i s way. On t h e o t h e r hand, p. o s e m i r i n g i f f.

0.

(S,+)

Partially and fully ordered seminearvings and nearrings

293

a zero, we get f.0. semirings S with a left absorbing zero of type (11). Now we can turn to the multiplicatively left-right dual f.0. semiring S ' which has the type (1110) = (14). Example 5.10. For the f. 0. group ( S , + , S ) of Expl. 5.2 define a product by (al,a2) (bl,b2) = (alb2+ a2bl,a2b2). Then (S,+, * , S ) is a commutative w. f.0. ring of type (0000), whereas its positive cone U=P(S) is a zero-symmetric w. f.0. semiring of type (0101) = (5). The latter was the last type needed to prove Remark 2.2. NOTES AND REFERENCES Footnote 1) The importance of these concepts is that one can introduce partial orders on groupoids (S,+) by (2.4) using suitable subsets X E S . For semigroups, this was extensively investigated in [151, and we note here only the following special result. Let (S,+) be a semigroup with a zero o and X E S . Then (2.4) defines a relation on S such that ( S , + , < ) is a p. 0. semigroup iff x+y = o for x,y E X implies x = y = o and c+X S X+c holds for all C E S . In this case, X coincides with the set of positive elements of ( S , + , < ) . Bibliography Fuchs, L., Partially ordered algebraic systems, Pergamon Press, Oxford 1963, Vandenhoeck & Ruprecht, Gottingen 1966. Hanumanthachari, J., K. Venu Raju and H. J. Weinert, Some results on partially ordered semirings and semigroups, Proc. First Intern. Symposium on Ordered Algebraic Structures, Marseilles, June 1984, Heldermann-Verlag, Berlin 1986. Hogewijs, H., Semi-Nearrings-Embedding, Med. Konink. Acad. Wetensch. Lett. Schone Kunst. Belgie K1. Wetensch. 32 (1970) 3 - 11. Karzel, H., Zusammenhange zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und 2-Strukturen mit Rechtecksaxiom, Abh. Math. Sem. Univ. Hamburg 32 (1968), 191 - 206. Kerby, W., Angeordnete Fastkorper, Abh. Math. Sem. Univ. Hamburg 32 (1968), 135 - 146. Kerby. W. and H. Wefelscheid, Bemerkungen iiber Fastbereiche und scharf zweifach transitive Gruppen, Abh. Math. Sem. Univ. Hamburg 37 (1972), 20 29.

-

Pilz, G., Geordnete Fastringe, Abh. Math. Sem. Univ. Hamburg 35 (19701, 83 - 88. Pilz, G., Direct Sums of nrdered Near-Rings, J. Algebra 10 (1971), 340 - 342.

H.J. Weinert

294

[9] Pilz, G. , Zur Charakterisierung der Ordnungen in Fastringen, 253. Monatsh. Math. 76 (1972), 250

-

[lo] Pilz, G., A construction method for near-rings, Acta Math. Acad. Sci. Hungar. 24 (1973), 97 - 105. [ll] Pilz, G., Near-Rings, North-Holland, Amsterdam 1977. [12] Weinert, H. J., Related representation theorems for rings, semirings, near-rings and semi-near-rings by partial transformations and partial endomorphisms, Proc. Edinburqh Math. SOC. 20 (1976 77), 307 315.

-

-

1131 Weinert, H. J., Seminearrings, seminearfields and their semigroup-theoretical background, Semigroup Form 24 (1982), 231 - 254. 1141 Weinert, H. J., Extensions of seminearrings by semigroups of right quotients, Lecture Notes in Mathematics 998 (19831, Proceedings Oberwolfach (1981), 412 - 486. 1151 Weinert, H. J., Partially ordered semirings and semigroups, Proc. First Intern. Symposium on Ordered Algebraic Structures, Marseilles, June 1984, Heldermann-Verlag, Berlin 1986.

Institut fur Mathematik Technische Universitat Clausthal ErzstraBe 1 D-3392 Clausthal-Zellerfeld

Near-ringsand Near-fzlds, G . Betsch (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1987

295

ON SUBDIRECTLY IRREDUCIBLE NEAR-RINGS WHICH ARE FIELDS Richard WIEGANDT Mathematical I n s t i t u t e o f t h e Hungarian Academy o f Sciences Budapest, P f . 127, H-1364, Hungary Herrn H. J. Weinert zum 60. Geburtstag am 26.01.1987

gewidmet

I n t h i s n o t e we s h a l l prove under c e r t a i n c o n d i t i o n s t h a t i f a l e f t i n v a r i a n t subset o f a s u b d i r e c t l y i r r e d u c i b l e n e a r - r i n g s a t i s f i e s a permutation i d e n t i t y , then t h e n e a r - r i n g i s a f i e l d . I n t h e sequel a near-ring w i l l always mean a r i g h t n e a r - r i n g , t h a t i s , i t s a t i s f i e s t h e r i g h t d i s t r i b u t i v e law ( x + y ) z = xz L e t us r e c a l l t h a t an i d e a l group o f

A

I

t

yz.

o f a near-ring

A

i s an a d d i t i v e normal sub-

such t h a t

I A r I and x(y holds f o r a l l

x, Y E A and

symmetric n e a r - r i n g

A

t

i)

-

. By P i l z

iE I A

[31 1.34 ( a ) any i d e a l

A IE I

satisfies also

subdirectly i r r e d u c i b l e , i f

xy E I

. A near-ring

c o n t a i n s a unique minimal i d e a l

i s r e f e r r e d t o t h e heart o f A . We s h a l l c a l l a subset

Zeft invariant subset, i f A L c L we s h a l l denote by Sk t h e s e t s k = Is = tl

holds. For a subset

... tk : tl

f o r any i n t e g e r

k 2 2

L

I . . . ,

tk E

A

I of a 0 i s s a i d t o be

H # 0

which

o f a near-ring

S

o f a near-ring

A

a A

SI

.

I n the f o l l o w i n g k w i l l always stand f o r a f i x e d i n t e g e r 2 2 , and (7 w i l l stand f o r a f i x e d permutation o f k elements such t h a t a ( 1 ) # 1 The n e x t Lemma i s p u r e l y semigroup t h e o r e t i c a l , and i t i s a m o d i f i e d s p e c i a l

.

case o f [ 2 1 Lemma 2. For the sake o f completeness we g i v e i t w i t h p r o o f . LEMMA. If a l e f t invariant subset mutation i d e n t i t y (PI

t,

L

o f a near-ring

... t k = t o ( , ) ... t o ( k l

tiEL,

A

s a t i s f i e s t h e per-

R. Wiegandt

296

then

... tk = y x t l ... tk tlY...'tkEL .

xytl

holds f o r a l l

x, Y E A and

roof. L e t r = yxtl

... tk

i d e n t i t y ( P ) t o the elements

and

xtl,

o(i) = 1 t2,

...) tk

u(1) = j

. Now we apply

and then m u l t i p l y by

y.

the Thus

we g e t

... to ( .-l)Xtu( i)to( i + l )

r = ytu(l)

i

* ''

to( k )

'

Now we use the i d e n t i t y ( P ) i n reverse on the elements Y t 0 ( 1) 'tU(2)

..ta(

Y..

1' - 1 1

SXtu(i

y t u ( i + l

' . * * I

tu(k)

t o obtain

... t k . ,...,tjm1,ytj,

... t J-1 .

y t. t. J J+1 A t t h i s p o i n t we use again ( P ) on the elements tl r=xtl

tj+l

,...,t k

t o obtain r = x y t F i n a l l y , we use ( P ) i n reverse on

o ( 1 ) %(2

t o obtain

tu(,],.,. t o ( k )

r = x y t

1 '

*

tk

Y

and the desired i d e n t i t y has been established. Another proof o f the Lemma has been given by H. J. Weinert ( p r i v a t e commun i c a t i o n ) . The s t r u c t u r e o f near-rings s a t i s f y i n g ( P ) has been i n v e s t i g a t e d by R. Scapellato [41.

be a subdirectly irreducible near-ring w i t h heart H , and A such t h a t H Lk # 0 , If L s a t i s f i e s t h e permutation i d e n t i t y (P), then A i s a commutative ring. If, i n addition, H2 0 , then A i s a f i e Z d . A 0 - s y m e t r i c , subdirectly i r r e d u c i b l e near-ring with commutative idempotent heart i s a f i e l d . THEOREM. Let

A

L be a l e f t invariant subset of

+

Proof. By the Lemma we have

... t k = y x t , ... t k ,

x y t , that is, (xy x, y E A

for a l l

and

t,,

-

y x ) tl

... tk = 0

...'tk E L . This means t h a t t h e element

i s contained i n t h e a n n i h i l a t o r k k ( 0 : L ) = {a E A : aL I = 0 k o f Lk One can e a s i l y check t h a t ( 0 : L ) i s an i d e a l i n A . k k subdirectly irreducible, either (0 : L ) = 0 or H G (0 : L ) k case we would get H L = 0 c o n t r a d i c t i n g t h e assumption. Thus

.

holds, implying

.

xy = y x

for a l l

x, y E A

. This means t h a t i n

xy

Since

- yx A

is

I n the l a t t e r k (0 : L ) = 0

A

the multi-

291

On subdirectly irreducible near-rings which are fields A

p l i c a t i o n i s commutative, and hence

i s a d i s t r i b u t i v e n e a r - r i n g which i s

a l s o c a l l e d a r i n g w i t h n o t n e c e s s a r i l y commutative a d d i t i o n . As one can prove (see f o r i n s t a n c e [l],151 o r [61), each a d d i t i v e commutator

(x, y E A)

,

i s contained i n the a n n i h i l a t o r

A

i s an i d e a l i n

and

A

( 0 : A)

of

-x

-

y

t

. Since

A

i s subdirectly irreducible, i t follows

x

t

y

,

( 0 : A)

,

( 0 : A) = 0

otherwise namely we would g e t k O # H L S ( O : A ) A = O . Thus we have x t y = y t x x, y E A

for all

A

and t h e r e f o r e

i s a commutative r i n g .

The h e a r t o f a s u b d i r e c t l y i r r e d u c i b l e r i n g i s e i t h e r simple o r a z e r o - r i n g . If

ideal A

, then

H2 # 0

H of

. Since

A

A

i s a s i m p l e commutative r i n g , t h a t i s , a f i e l d . Hence t h e

H

has a u n i t y element and t h e r e f o r e

H

i s a d i r e c t summand o f

i s subdirectly irreducible, i t follows

H = A

.

The l a s t a s s e r t i o n i s s t r a i g h t f o r w a r d . For r i n g s we can prove a s t r o n g e r v e r s i o n o f t h e Theorem. A be a s u b d i r e c t l y i r r e d u c i b l e ring w i t h idempotent heart i s a l e f t i n v a r i a n t subset of A , then H Ln # 0 holds f o r

PROPOSITION. Let H

. If

every

L # 0

.. .

n = 1,2,.

We prove t h e statement by i n d u c t i o n . F i r s t , assume t h a t ( 0 : H)r

right annihilator

( 0 : H)r

Further, yields

of

H in A

i s an i d e a l o f

2 0 # H = H E H(O : H)r = 0

.

n = 1 Next, assume t h a t

valid for

(H Ln-')L = H Ln = 0 (H L"')

. Hence

This y i e l d s f o r every

H L"'

n = 1,2,

,

, so

L

we conclude

, and

H L = 0 hence

. Then t h e (0 : H),#

H C ( 0 : H)r

0.

which

a c o n t r a d i c t i o n . Thus t h e a s s e r t i o n i s

for

n 2 2

the l e f t annihilator

(0 : L )

H L = 0

= 0

A

contains

.

Suppose t h a t

H L # 0

... .

. Then by

(0 : L)

i s a nonzero i d e a l o f

, contradicting

H Ln = 0

.

o f L i n A contains A and so we have H r ( 0 : L ) Thus H Ln # 0 has been proved

The Theorem and t h e P r o p o s i t i o n y i e l d immediately t h e f o l l o w i n g COROLLARY. Let

A

be a s u b d i r e c t l y i r r e d u c i b l e ring w i t h idempotent h e a r t .

I f a l e f t i n v a r i a n t subset (P), then

A

L # 0

of

A

s a t i s f i e s t h e permutation i d e n t i t y

is a f i e l d .

Rermrk. The P r o p o s i t i o n and hence t h e C o r o l l a r y a r e v a l i d a l s o f o r d i s t r i b u t i v e near-rings,

t h a t i s , f o r r i n g s w i t h n o t n e c e s s a r i l y commutative a d d i t i o n .

REFERENCES [ 11 Furtwangler, Ph. and Taussky, Olga, Ober S c h i e f r i n g e , Sitzungsber. Akad.

der Wiss. Wien, Mathem.-natum.

K l a s s e , 145 (1936), 525.

.

296

R. Wiegondt

[ 2 1 Parmenter, M. M., Stewart, P. N. and Wiegandt, R., On the Groenewald man s t r o n g l y prime r a d i c a l , Quuest. Math. 7 (1984), 225-240. [31 P i l z , G. , Near-rings, North-Holland, 1977.

-

Hey-

, Sui q u a s i - a n e l l i v e r i f i c a n t i i d e n t i t a semigruppale C-mobili, BoZZettino U.M.I. (6) 4-8 (1985), 789-799. [5] Taussky, Olga, Rings w i t h non-commutative a d d i t i o n , BUZZ. C Q Z C U t t Q MQth.

[4J Scapellato, R.

SOC., 28 (19361, 245-246. [6] Weinert, H. J., Ringe m i t nichtkomnutativer A d d i t i o n I , Jber. Deutsch. &th.-Verein. 77 H. 1 (1975). 10-27.