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\ : =

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 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,