Nearrings and Nearfields: Proceedings of the Conference on Nearrings and Nearfields, Hamburg, Germany July 27 - August 3, 2003

NEARRINGS AND NEARFIELDS NEARRINGS AND NEARFIELDS Proceedings of the Conference on Nearrings and Nearfields, Hamburg, ...

Author: Hubert Kiechle | A. Kreuzer | Momme Johs Thomsen

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NEARRINGS AND NEARFIELDS

NEARRINGS AND NEARFIELDS Proceedings of the Conference on Nearrings and Nearfields, Hamburg, Germany July 27–August 3, 2003

Edited by

HUBERT KIECHLE Universität Hamburg, Germany

ALEXANDER KREUZER Universität Hamburg, Germany and

MOMME JOHS THOMSEN Universität der Bundeswehr Hamburg, Germany

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-3390-7 (HB) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-10 1-4020-3391-5 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-13 978-1-4020-3390-2 (HB) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-13 978-1-4020-3391-9 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved © 2005 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

Contents

Foreword Acknowledgments

Part I

vii x

Invited Addresses

On Recent Developments of Planar Nearrings Wen-Fong Ke

3

Some Problems Related to Near-rings of Mappings Carl J. Maxson

25

Some Recent Developments in Group Near-rings John D. P. Meldrum

35

Loop-nearrings Silvia Pianta

57

The Z -Constrained Conjecture Stuart D. Scott

69

Part II Contributed Papers Primeness and Radicals in Near-rings of Continuous Functions Geoﬀrey L. Booth

171

Diﬀerence Methods and Ferrero Pairs Tim Boykett and Peter Mayr

177

Zero-Divisor Graphs of Nearrings and Semigroups G. Alan Cannon, Kent M. Neuerburg, Shane P. Redmond

189

Additive GS-Automata and Syntactic Nearrings Yuen Fong, Feng-Kuo Huang, Chiou-Shieng Wang

201

On the Nilpotence of the s-Radical in Matrix Near-rings John F.T. Hartney, Anthony M. Matlala

217

vi

NEARRINGS AND NEARFIELDS

A Right Radical for Right d.g. Near-rings John F.T. Hartney, Danielle S. Rusznyak

225

From Involution Sets, Graphs and Loops to Loop-nearrings Helmut Karzel, Silvia Pianta, Elena Zizioli

235

Semi-nearrings of Bivariate Polynomials over a Field Kent M. Neuerburg

253

Automorphism Groups Emitting Local Endomorphism Nearrings II Gary L. Peterson

263

Planar Near-rings, Sandwich Near-rings and Near-Rings with Right Identity277 Gerhard Wendt On the f -Prime Radical of Near-rings Satyanarayana Bhavanari, Richard Wiegandt

293

On Finite Goldie Dimension of Mn (N )-Group N n Satyanarayana Bhavanari, Syam Prasad Kuncham

301

Near-rings, Cohomology and Extensions Mirela S ¸ tef˘ f nescu

311

Foreword

This present volume is the Proceedings of the 18th International Conference on Nearrings and Nearﬁelds held in Hamburg at the Universit¨ at der Bundeswehr Hamburg from July 27 to August 03, 2003. This Conference was organized by Momme Johs Thomsen and Gerhard Saad from the Universit¨ at der Bundeswehr Hamburg and by Alexander Kreuzer, Hubert Kiechle and Wen-Ling Huang from the Universit¨ a¨t Hamburg. It was already the second Conference on Nearrings and Nearﬁelds in Hamburg after the Conference on Nearrings and Nearﬁelds at the same venue from July 30 to August 06, 1995. The Conference was attended by 57 mathematicians and many accompanying persons who represented 16 countries from all ﬁve continents. The ﬁrst of these conferences took place 35 years earlier in 1968 at the Mathematische Forschungsinstitut Oberwolfach in the Black Forest in Germany. This was also the site of the second, third, ﬁfth and eleventh conference in 1972, 1976, 1980 and 1989. The other twelve conferences held before the second Hamburg Conference took place in nine diﬀerent countries. For details about this and, moreover, for a general historical overview of the development of the subject we refer to the article ”On the beginnings and developments of near-ring theory” by Gerhard Betsch [3] in the proceedings of the 13th Conference in Fredericton, New Brunswick, Canada. During the last ﬁfty years the theory of nearrings and related algebraic structures like nearﬁelds, nearmodules, nearalgebras and seminearrings has developed into an extensive branch of algebra with its own features. In its position between group theory and ring theory, this relatively young branch of algebra has not only a close relationship to these two more well-known areas of algebra, but it also has, just as these two theories, very intensive connections to many further branches of mathematics. Thanks to the foresight of the early workers in the ﬁeld, a comprehensive classiﬁed bibliography was established and is updated regularly in the Nearring Newsletter. The latest version [9] appeared at the end of

viii 2003. It listed 2485 publications contributed by a total of 708 authors. Within this large number of papers is reﬂected the great diversity of the subject. That the development of nearrings and nearﬁelds has matured to a substantial theory with numerous applications can now be best retraced by studying the ﬁve existing books on the subject. They are written by the authors G. Pilz [11], J.D.P. Meldrum [10], H. W¨ ahling [15], J.R. Clay [4], C. Cotti Ferrero and G. Ferrero [6]. This present volume is the ninth proceedings of a nearring conference following the proceedings [13], [5], [1], [2], [12], [7], [14] and [8]. It contains the written version of ﬁve invited lectures followed by 13 contributed papers. All papers in the volume have been refereed. This Proceedings opens with the invited paper by Wen-Fong Ke which reports on some recent developments of planar nearrings and points out several possible research directions in this area for the future. The second paper is an expanded version of the invited survey talk by Carl J. Maxson on nearrings of mappings which mentions several open questions in this ﬁeld of research. This paper is a continuation of the invited survey paper by the same author on nearrings of homogeneous functions in the Proceedings of the ﬁrst Hamburg Conference of 1995. Next we have the invited paper by John D.P. Meldrum which presents an account of some of the work on group nearrings, emphasizing the parallels with matrix nearrings and the most recent developments. The invited paper of Silvia Pianta on loop-nearrings considers a generalization of the notion of nearring by relaxing the associativity of the addition. Then for these loop-nearrings several generalizations of planarity and corresponding Ferrero pairs are investigated. Just as eight years before, Stuart R. Scott brought to Hamburg from the opposite side of our planet the by far longest paper of this Proceeding “The Z-Constrained Conjecture”. For an overview of this substantial work, we refer to its ﬁrst twelve pages. The topics of the 13 contributed papers are so diverse that, for an overview, we refer to the Table of Contents and the abtracts or introductions at the beginning of each paper.

ix

FOREWORD

References [1] G. Betsch (ed.). Near-rings and near-ﬁelds. Proceedings of the 1985 T¨ u ¨bingen Conference. North-Holland. Amsterdam 1987. [2] G. Betsch, G. Pilz, H. Wefelscheid (eds.). Near-rings and near-ﬁelds. Proceedings of the 1989 Oberwolfach Conference. [3] G. Betsch. On the beginnings and development of near-ring theory. In: NearRings and Near-Fields, Y. Fong et al. (eds.). Kluwer Academic Publishers. Dordrecht 1995, 1-11. [4] J.R. Clay. Nearrings: Geneses and Applications. Oxford University Press. Oxford 1992. [5] C. Cotti Ferrero, G. Ferrero (eds.). Proceedings of the 1981 San Benedetto Conference. 1982. [6] C. Cotti Ferrero, G. Ferrero. Nearrings: Some Developments Linked to Semigroups and Groups. Kluwer Academic Publishers. Dordrecht 2002. [7] Y. Fong, H.E. Bell, Wen-Fong Ke, G. Mason, G. Pilz (eds.). Near-Rings and Near-Fields. Proceedings of the 1993 Fredericton Conference. Kluwer Academic Publishers. Dordrecht 1995. [8] Y. Fong, C. Maxson, J. Meldrum, G. Pilz, A. van der Walt, L. van Wyk (eds.). Near-Rings and Near-Fields. Proceedings of the 1997 Stellenbosch Conference. Kluwer Academic Publishers. Dordrecht 2001. [9] Y. Fong, A. Oswald, G. Pilz, K.C. Smith. Bibliography. Near-Ring Newsletter 24 (2003), 9-171. [10] J.D.P. Meldrum. Near-rings and their links with groups. Research Notes in Mathematics. Vol. 134. Pitman Adv. Publ. Program. Boston, London 1985. [11] G. Pilz. Near-Rings. North-Holland/American Elsevier. Amsterdam 1977. Revised Edition 1983. [12] G. Pilz (ed.). Near-Rings. Proceedings of the 1991 Linz Conference. Contributions to General Algebra 8. H¨ ¨ older-Pichler-Tempsky. Wien 1992. [13] Proceedings of the 1978 Edinburgh Conference. Proc. Edin. Math. Soc. 23 (1980), 1-140. [14] G. Saad, M.J. Thomsen (eds.). Nearrings, Nearﬁelds and K-Loops. Proceedings of the 1995 Hamburg Conference. Kluwer Academic Publishers. Dordrecht 1997. [15] H. W¨ ¨ ahling. Theorie der Fastk¨ ¨ orper. Thales Verlag. Essen 1987.

Momme Johs Thomsen Hamburg, November 2004

Acknowledgments It would be a rare event if a conference and a proceedings of this type were to be organized without the support of many others. Thanks for help are due to many people. We cannot possibly name all of them. The following have provided assistance far above and beyond the call of duty. First we would like to express our gratitude to Herrn Eckhard Redlich, the chancellor of the Helmut-Schmidt-Universit¨ at, Universit¨ ¨at der Bundeswehr Hamburg and to all of his staﬀ involved in making the Conference such a success. In particular, we would like to mention Martina Burmeister who composed the poster for our Conference so nicely, and Konrad Holzen ¨ who arranged the PC-pool for the participants of the Conference. Many thanks are due to the Wilhelm-Blaschke-Gedachtnis-Stiftung ¨ for ﬁnancial support. We are especially indebted to our co-organizers Wen-ling Huang and Gerhard Saad who shared in the organization from the beginnings. Later Marco M¨ oller and Sebastian Rudert joined the eﬀort. Many thanks go to Wolfgang L¨ o¨bnitz who designed and implemented the web-pages. Very helpful was the secretarial work provided by Corina Flegel of the HSU and of Elisabeth Himmler of the UniHH. The enormous engagement of Elisabeth Himmler was really far above and beyond the call of duty. We are also especially indebted to Marta Thomsen for arranging a welcome party at the Thomsen’s home including the catering for almost hundred people, for helping to organize the joint excursion of all those people to Lubeck ¨ and Ratzeburg, and also for planning and implementation of numerous further activities for the accompanying persons. Concerning the Proceedings, we express our gratitude to all the authors. Special thanks go to the referees who took the pain of examining the manuscripts.

x

xi

List of Participants of the Conference Erhard Aichinger, Linz, Austria Nurcan Argac, Izmir, Turkey Howard Bell, St. Catharines, Canada Anna Benini, Brescia, Italy Gerhard Betsch, Weil im Schoenbuch, Germany Andrea Blunck, Hamburg, Germany Geoﬀrey L. Booth, Port Elizabeth, South Africa Tim Boykett, Linz, Austria G. Alan Cannon, Hammond, LA, USA Celestina Cotti Ferrero, Parma, Italy Juergen Ecker, Linz, Austria Giovanni Ferrero, Parma, Italy Yuen Fong, Tainan, Taiwan Lungisile Godloza, Umtata, South Africa Nico Groenewald, Port Elizabeth, South Africa John F.T. Hartney, Johannesburg, South Africa Herbert Hotje, Hannover, Germany Wen-ling Huang, Hamburg, Germany Lucyna Kabza, Hammond, LA, USA Helmut Karzel, Garching bei M¨ u ¨nchen, Germany Hermann Kautschitsch, Klagenfurt, Austria Wen-Fong Ke, Tainan, Taiwan Hubert Kiechle, Hamburg, Germany Alexander Kreuzer, Hamburg, Germany Syam Prasad Kuncham, Manipal, India

xii Mario, Marchi Brescia, Italy Dragan Masulovic, Novi Sad, Serbia Anthony M. Matlala, Johannesburg, South Africa Carl J. Maxson, College Station, TX, USA Peter Mayr, Linz, Austria John D.P. Meldrum, Edinburgh, Scotland Johan H. Meyer, Bloemfontein, South Africa Rainer Mlitz, Wien, Austria Marco Moeller, Hamburg, Germany Kent M. Neuerburg, Hammond, LA, USA Silvia Pellegrini, Brescia, Italy Gary L. Peterson, Harrisonburg, VA, USA Silvia Pianta, Brescia, Italy G¨ u ¨nter Pilz, Linz, Austria Sebastian Rudert, Hamburg, Germany Danielle S. Rusznyak, Johannesburg, South Africa Gerhard Saad, Hamburg, Germany Mohammad Samman, Dhahran, Saudi Arabia Bhavanari Satyanarayana, Andhra Pradesh, India Stuart D. Scott, Auckland, New Zealand Kirby C. Smith, College Station, TX, USA Grozio Stanilov, Berlin, Germany Mirela S ¸ tef˘ fanescu, Constanta, Romania Yohanes Sukestiyarno, Semarang, Indonesia Momme Johs Thomsen, Hamburg, Germany Yulian Tsankov, Soﬁa, Bulgaria Petr Vojtechovsky, Denver, CO, USA

xiii Heinrich Wefelscheid, Duisburg, Germany Gerhard Wendt, Linz, Austria Marcel Wild, Stellenbosch, South Africa Dirk Windelberg, Hannover, Germany Elena Zizioli, Brescia, Italy

I

INVITED ADDRESSES

ON RECENT DEVELOPMENTS OF PLANAR NEARRINGS Wen-Fong Ke∗ Department of Mathematics National Cheng Kung University Tainan 701, Taiwan [email protected]

1.

Introduction

Since the ﬁrst appearance of planar nearrings in 1968, there has been plenty of research results attributed to the understanding and applications of them. However, it appears to us that, at this stage, we have just begun to unearth this beautiful mathematical object. In February 2002, a two year international joint project on research of planar nearrings was established between the research group in Kepler University, Linz and that in National Cheng Kung University, Tainan. This project was supported by the Austrian Science Foundation (FWF) and the National Science Council, R.O.C. (NSC), and has been proven fruitful. The goal of this survey article is to report some recent developments of planar nearrings, and point out some possible research directions for future researches. Some of the directions have been under investigations by the Linz-Tainan cooperation. The materials presented in this article are organized based on the one-hour-talk the author gave at the International Conference on Nearrings and Nearﬁelds, Universit¨ a¨t der Bundeswehr Hamburg and Universit¨ at Hamburg, 27 July–3 August 2003. We would also like to point out that there is a whole chapter in Clay’s book Nearrings: Genesis and Applications (reference item [10]) devoted to this subject that one would like to go through and refer back from time to time.

∗ Supported

by the National Science Council, Taiwan under the project NSC-92-2115-M-006-

001.

3 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 3–23. c 2005 Springer. Printed in the Neatherlands.

4

2.

Wen-Fong Ke

Deﬁnitions and examples

Let (N, +, ·) be a (left) nearring. An equivalence relation ≡m can be deﬁned on N by a ≡m b ⇔ ax = bx

for all x ∈ N .

We say that (N, +, ·) is planar if |N/≡m | ≥ 3, and for each triple a, b, c ∈ N with a ≡m b, the equation ax = bx + c has a unique solution for x in N . It is custom to denote by N ∗ the set of elements not multiplicative equivalent to 0, i.e. N ∗ = {x ∈ N | x ≡m 0}. On the other hand, the set of “zero-multipliers” is denoted by A, i.e. A = {x ∈ N | x ≡m 0}. Certainly all ﬁelds are planar nearrings. It is also true that all ﬁnite nearﬁelds are planar nearrings (cf. [10, Theorem 4.26]). The ﬁrst three nontrivial examples of planar nearrings were given in [1] which we record again in the following. Consider the ﬁeld of complex numbers C. For a, b ∈ C, where a = a1 + ia2 with a1 , a2 ∈ R and i2 = −1, deﬁne a1 · b a ∗1 b = a2 · b

if a1 = 0, if a1 = 0;

a ∗2 b = |a| · b; ⎧ ⎨ a · b if a = 0, a ∗3 b = |a| ⎩0 if a = 0. Then (C, +, ∗1 ), (C, +, ∗2 ), and (C, +, ∗3 ) are planar nearrings which are not rings. These three examples have served as models for many researches on planar nearrings since. It is natural at this point to ask for more examples of planar nearrings. Indeed, one can construct planar nearrings (somewhat) freely when the relationship between planar nearrings and Ferrero pairs is studied.

3.

Planar nearrings and Ferrero pairs

It was shown by G. Ferrero in 1970 [13] that every planar nearring N gives rise to a group of automorphisms Φ of the additive group (N, +) having speciﬁc properties (to be discussed below). On the other hand, if an additive group (G, +) is given together with a group of automorphisms of G satisfying the speciﬁc properties, then G can be turned into a planar nearring through a ﬁxed process.

On Recent Developments of Planar Nearrings

3.1

5

From planar nearrings to Ferrero pairs

Let N be a planar nearring. For a ∈ N ∗ , deﬁne ϕa : N → N ; x → ax for all x ∈ N . Then ϕa ∈ Aut (N, +), and ϕa = 1 if and only if a ≡m 1; ϕa (x) = x if and only if ϕa = 1 or x = 0; −1 + ϕa is surjective if ϕa = 1. Thus, Φ = {ϕa | a ∈ N , a ≡m 0} is a regular group of automorphisms of (N, +) with the property that −1 + ϕa is surjective if ϕa = 1. We call (N, Φ) a Ferrero pair. In general, if Φ is a group acting on another group N as an automorphism group, and for ϕ ∈ Φ \ {1}, −1 + ϕ is bijective, then (N, Φ) is called a Ferrero pair.

3.2

From Ferrero pairs to planar nearrings

Given a Ferrero pair (N, Φ), where N is an additive group. Let C be a complete set of orbit representatives of Φ in N . Let E ⊆ C such that 0 ∈ E and |E| ≥ 2. Then ∪e ∈C\E Φ(e ) , N = ∪e∈E Φ(e) here for an a ∈ N , Φ(a) = {φ(a) | φ ∈ Φ} is the orbit of Φ in N determined by a. Now, deﬁne a binary operation ∗E on N by ϕ(y) e ∈ E, ϕ ∈ Φ, y ∈ N , ϕ(e) ∗E y = 0 otherwise. Then (N, +, ∗E ) is a planar nearring. Notice that (1) the elements in E are exactly the left identities of N , and (2) N is an integral planar nearring if and only if E = C \ {0}. Remark 3.1. (1) Since Φ is a regular group of automorphisms of N , Φ(a) and Φ have the same cardinality for all nonzero a ∈ N . (2) The set E given above is exactly the set of left identities of the planar nearring (N, +, ∗E ). (3) For each e ∈ E, Ne = Φ(e) is a subgroup of the multiplicative semigroup (N, ∗E ) have e as the identity element. Actually, (N Ne , ∗E ) is isomorphic to Φ. (Cf. [10, (4.9)].) Thus, once we have a Ferrero pair, we can easily obtain many planar nearrings.

6

Wen-Fong Ke

3.3

Some examples

First, let us look at the three planar nearrings (C, +, ∗i ), i = 1, 2, 3. Here we ﬁnd that ∗ ), where R ∗ (1) the corresponding Ferrero pair of (C, +, ∗1 ) is (C, R is the group {ϕr | r ∈ R \ {0}}; + ), where R + (2) the corresponding Ferrero pair of (C, +, ∗2 ) is (C, R is the group {ϕr | r > 0}; and where C is (3) the corresponding Ferrero pair of (C, +, ∗3 ) is (C, C), the group {ϕc | |c| = 1}. From the above examples, one immediately obtain the following generalization. Let F be a ﬁeld. Take U ≤ F ∗ = F \ {0} and put = {ϕa | a ∈ U } ≤ Aut (F, +) where each ϕa : F → F is the left U ) is a Ferrero pair. Any planar nearring multiplication by a. Then (F, U constructed from (F, U ) is referred to as ﬁeld generated. Yet, one can generalize this ideal to certain rings. So, if R is a ring with unity, and let U be the group of (multiplicative) invertible elements. If A is a subgroup of U with |A| ≥ 2 and −1 + a ∈ U for all a ∈ A \ {1}, = {ϕa | a ∈ A}, where each ϕa : R → R is the then the group A left multiplication by a, is a regular group of automorphisms of (R, +). is a Ferrero pair. Any planar nearring deﬁned using Moreover, (R, A) will be said to be ring generated. (R, A)

4.

Isomorphism problem

For a given Ferrero pair (N, Φ), there are many ways to choose the set E of orbit representatives of Φ. Each choice of E give rise to a planar nearring. Naturally, one wonders that whether all of these planar nearrings are isomorphic or not? The answer is “no” even if the planar nearrings are integral. Then, the second question would be that “is there a way to distinguish the planar nearrings constructed from (N, Φ)?” The answer to this question is “yes!” Theorem 4.1 ([4]). Let (M, Ψ) and (N, Φ) be Ferrero pairs and let E1 and E2 be sets of orbit representatives of Ψ and Φ in M and N , respectively, with |E1 | ≥ 2. Let (M, +, ·) and (N, +, ) be the planar nearrings deﬁned on M and N using E1 and E2 , respectively. Then an additive isomorphism σ from (M, +) to (N, +) is an isomorphism of the planar nearrings (M, +, ·) and (N, +, ) if and only if σ(E1 ) = E2 and σΨσ −1 = Φ.

On Recent Developments of Planar Nearrings

7

In particular, if (M, Ψ) = (N, Φ), then σ ∈ Aut (N, +) is an isomorphism of (N, +, ∗E1 ) and (N, +, ∗E2 ) if and only if σ(E1 ) = E2 and σ normalizes Φ. of the planar As an illustration, we consider the Ferrero pair (C, C) nearring (C, +, ∗3 ). Let E1 and E2 be two complete sets of orbit repre in C \ {0}. Let σ ∈ Aut (C, +). If σ is an isomorphism sentatives of C of the two planar nearrings, then σ(C) = C. It can be shown that, in this case, σ is either a rotation of the complex plane about the origin or the reﬂection of the complex plane about a line through the origin. Therefore, (C, +, ∗E1 ) and (C, +, ∗E2 ) are isomorphic as integral planar nearrings if and only if E2 = eiθ E1 or E2 = eiθ E 1 for some θ ∈ R, where E 1 denotes the complex conjugate of E1 . Remark 4.2. Theorem 4.1 is valid for a more general class of nearring constructions called Ferrero nearrings. To describe what a Ferrero nearring is, we start with a group G and Φ ≤ Aut G. Let A be a complete set of orbit representatives of Φ in G. Suppose that E ⊆ A. If E = ∅, then we have trivial multiplication on G. If E = ∅, we want E to satisfy ϕ(e) = e for all ϕ ∈ Φ \ {1} and e ∈ E. Put A◦ = A \ E and G◦ = Φ(A◦ ). For x, y ∈ G, deﬁne 0 if x ∈ G◦ , x∗y = ϕ(y) if x = ϕ(e) ∈ Φ(E). Then (G, +, ∗) is called a Ferrero nearring. Note that (G, +, ∗) is a planar nearring if and only if (G, Φ) is a Ferrero pair. Problem 4.3. Study the structure of the planar nearrings constructed from the Ferrero pair (C, C). Problem 4.4. Note that if E is a complete set of orbit representatives of in C\{0}, then the planar nearring (C, +, ∗E ) is a topological nearring C if and only if E is the graph of a continuous curve in C. Is there a way to characterize them?

5.

Characterizations of Planar Nearrings

There have been some results on the characterizations of planar nearrings other than the Ferrero pair construction. Theorem 5.1 ([3]). Let N be a zero-symmetric 3-prime nearring. Let L be an N -subgroup of N . Then there is an e = e2 ∈ N such that L = eN . Let Φ = eN e \ {0}, then (L, Φ) is a Ferrero pair, and L is a planar nearring.

8

Wen-Fong Ke

Theorem 5.2 ([27]). Let N be a nearring. Then the following are equivalent: (1) N is planar. (2) There exists a zero-symmetric nearring M and a left invariant subnearring P of M such that M acts 2-primitively on P via nearring multiplication, (P, Aut M (P )) is a Ferrero pair, and N ∼ = P. In [28] it is shown that a planar nearring is a centralizer nearring in the usual sense, but multiplication is not the usual function composition but rather composition of functions with a suitable sandwich function in between.

6.

Algebraic structure of planar nearrings

The structure of radicals of planar nearrings was completely determined in [14], also lots of facts about ideals in planar nearrings can be found there. For example, Theorem 6.1 ([14, Teorema 1]). Let N be a planar nearring. Then there exists a greatest proper ideal D, which is the sum of all proper left ideals. Using this result, Wendt determines the ideal structure of planar nearrings completely. Theorem 6.2 ([29]). Let N be a planar nearring and D its greatest proper ideal. Then the proper left ideals of N are precisely the additive normal subgroups of N contained in D. Denote by P (N ) and N (N ), respectively, the prime and the nil radicals of a nearring N . Also, let J1 (N ) and J2 (N ), respectively, be the J1 and J2 radicals of N . Theorem 6.3 ([14]). Let N be a planar nearring. Then P (N ) = N (N ) = J1 (N ) = D, D the greatest ideal properly contained in N . In case that J2 (N ) = N , we have that P (N ) = N (N ) = J1 (N ) = J2 (N ). Planar nearrings are very often 2-primitive (without identity). Theorem 6.4 ([29] and also [14]). A planar nearring N is 2-primitive if and only if A does not contain nontrivial subgroups of N . It is shown in [15] that the (nontrivial) homomorphic images of a ﬁnite planar nearring N is again planar. Also, if a planar nearring has a distributive element, it has a very special structure. Theorem 6.5 ([29]). If a planar nearring N has a distributive element d ≡m 0, then A is an ideal of N and N/A is a nearﬁeld.

On Recent Developments of Planar Nearrings

7.

9

Combinatorial designs from planar nearrings

Finite planar nearrings have a natural connection with combinatorial objects called tactical conﬁgurations. The three planar nearrings (C, +, ∗i ), i = 1, 2, 3, provide raw models for such connection. One particular group of tactical conﬁgurations receives most attentions from researchers because they are both structural for studies and practical for real life applications. These tactical conﬁgurations are referred to as BIBDs in short. Deﬁnition 7.1. A ﬁnite set X with v elements together with a family S of k-subsets of X is called a balanced incomplete block design (BIBD) if (i) each element belongs to exactly r subsets, and (ii) each pair of distinct elements belongs to exactly λ subsets. The k-subsets in S are called blocks, and the integers v, b = |S|, r, k, λ are referred to as the parameters of the BIBD.

7.1

B, B− and B∗

Let (N, +, ·) be a ﬁnite planar nearring with corresponding Ferrero pair (N, Φ). Denote Φ0 = Φ ∪ {0} and Φ− = Φ ∪ (−Φ) ∪ {0}. Let B = {N · a + b | a, b ∈ N, a = 0} = {Φ0 (a) + b | a, b ∈ N, a = 0}, B− = {(N · a + b) ∪ (N · (−a)) + b | a, b ∈ N, a = 0} = {Φ− (a) + b | a, b ∈ N, a = 0}, B∗ = {N ∗ · a + b | a, b ∈ N, a = 0} = {Φ(a) + b | a, b ∈ N, a = 0}. We usually denote the set B∗ as BΦ to emphasize the role of Φ. Remark 7.2. These sets get their grounds from the geometrical considerations of the three examples (C, +, ∗i ), i = 1, 2, 3: B and B− are the set of straight lines of the complex plane obtained in (C, +, ∗1 ) and (C, +, ∗2 ), respectively, while B∗ is the set of circles of the complex plane obtained in (C, +, ∗3 ). Now, it is known that (N, B) and (N, B− ) are sometimes BIBDs, and (N, B∗ ) is always a BIBD (cf. [9, Theorems 5.5, 7.14, and 7.99]). Since (N, BΦ ) is always a BIBD, it seems natural to investigate the automorphism group of it. Obviously, every normalizer of Φ in the group Aut (N, +) serves as an automorphism of the design. It is conjectured that the converse is also true. Conjecture 7.3 (Modisett). The automorphism group of (N, BΦ ) is N NAut (N,+)(Φ), where NAut (N,+)(Φ) is the normalizer of Φ in Aut (N, +).

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Since N has an additive group structure and BΦ is obtained from additive translations, it is natural to consider (N, +, BΦ ) as a design group. Namely, N has a group structure, and each of the translations ρa : N → N ; x → x + a, a ∈ N , is an automorphisms of the design. In this case, a mapping N → N is called an automorphism of the design group if it is at the same time an automorphism of the group as well as of the design. With this condition added for abelian N and Φ, or in case that either N or Φ is not abelian, N is large enough, we see that Modisett’s conjecture has aﬃrmative answer. Theorem 7.4 ([17]). Let (N, Φ) be a ﬁnite Ferrero pair such that N and Φ are abelian with |Φ| < |N | − 1. Then Aut (N, +, BΦ ) is the normalizer of Φ in Aut (N, +). Theorem 7.5 ([6]). Let (M, Ψ) and (N, Φ) be ﬁnite Ferrero pair and let σ be an isomorphism from (M, BΨ , +) to (N, BΦ , +). Let |Φ| = k and set s = 2k 2 − 6k + 7. If |N/[N, N ]| > s, then σΨσ −1 = Φ. In particular, if (M, Ψ) = (N, Φ), then σ is a normalizer of Φ. The requirement that N is large enough in case when Φ is not abelian is necessary as the next example shows (cf. [6]). Example 7.6. Let F = GF(73 ) and κ : F → Aut (F ) a coupling on F such that F κ := (F, +, ◦) is a proper nearﬁeld with a ◦ b := a · κa (b). Let Φ ≤ F ∗ of index 2. Since Φ is characteristic, Φκ := (Φ, ◦) is a subgroup of (F κ )∗ . Then Φκ is nonabelian, and so Φ and Φκ are not isomorphic; therefore Φ and Φκ cannot be conjugate to each other. But (F, BΦ ) = (F, BΦκ ).

7.2

Segments

The planar nearring (C, +, ∗2 ) inspires yet another possible construction of interesting geometric objects: the segments. For those who wonder how the segments can be interesting, the paper of Clay [11] oﬀers a surprising construction of “triangles” with measurement of “angles” of the triangles within ﬁelds, and an analog result of the classical Euclidean geometry that the sum of the three angles of a triangle is π. Let (N, Φ) be a Ferrero pair. For distinct a, b ∈ N , deﬁne

a, b = Φ0 (b − a) + a ∩ Φ0 (a − b) + b , and call it a segment with endpoints a and b. Let S = {a, b | a, b ∈ N, a = b}. Note that if one puts S = Φ0 ∩ (1 − Φ0 ), then 1 − S = S and a, b = (b − a)S + a.

On Recent Developments of Planar Nearrings

11

Theorem 7.7 ([25]). If N is a nearﬁeld or a ring, then a, b = c, d if and only if (a, b) = (c, d). The use of the set S in a ﬁnite ﬁeld generated Ferrero pair suggests us to look at a more general construction from ﬁnite ﬁelds. Let F be a ﬁnite ﬁeld and S a subset of F with |S| ≥ 2. Consider S = {Sa + b | a, b ∈ F, a = 0}. Then (F, S) is always a BIBD. When |S| = 3, we are able to compute that full automorphism group of the design without the assumption of the design group structure. Our data also suggests that even for larger S, the full automorphism group of (F, S) should obey this theorem. Theorem 7.8 ([7]). If |S| = 3, then the (F, S) is a 2-(q, 3, λ) design with λ ∈ {1, 2, 3, 6}. Let U = r | {0, 1, r} ∈ S , and let K = U, +, · be the subﬁeld of F generated by U . Then under some mild condition, we have that f ∈ Aut (F, S) if and only if f (x) = T (α(x)) + b (x ∈ F ) for some b ∈ F , α ∈ Aut K (F ), and T ∈ L(F, K) (= linear transformations of the vector space F over K). The last along this line of applications of planar nearrings and Ferrero pairs is to construct partial balanced incomplete block designs, PBIBD in short. We record the deﬁnition of a PBIBD from Clay’s book [10, Deﬁtion 7.107]. Deﬁnition 7.9. Start with a ﬁnite tactical conﬁguration (N, T, ∈) and let P = {A | A ⊆ N, |A| = 2}. Suppose A = {A1 , A2 , . . . , Am } is a partition of P. Then A is an association scheme on N if, given {x, y} ∈ Ah , the number of z ∈ N such that {x, z} ∈ Ai and {y, z} ∈ Aj depends only upon h, i, and j, and not upon x and/or y. That is, there is a number phij such that for {x, y} ∈ Ah , there are exactly phij distinct elements z ∈ N such that {x, z} ∈ Ai and {y, z} ∈ Aj . Association schemes with m = 1 or m = v(v − 1)/2, where v = |N |, are declared ‘uninteresting’. Suppose (N, T, ∈, A) is a ﬁnite tactical conﬁguration with association scheme A. This structure is a partially balanced incomplete block design (PBIBD) if: (a) to each Ai ∈ A, there is a number ni such that for each x ∈ N , there are exactly ni distinct elements y ∈ N such that {x, y} ∈ Ai ; (b) to each Ai ∈ A, there is a number i such that {x, y} ∈ Ai implies x and y belong to exactly i blocks of T. Following the above deﬁnition, Clay invited his readers to take out pens and paper to construct examples of PBIBDs using Hall’s method

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(in eight steps). But here, without the mysterious eight steps, one can have many examples using ring generated planar nearrings. Let (R, +, ·) be a ﬁnite ring with unity and denote by U the group of units of R. Suppose that Φ is a subgroup of U with −1 ∈ Φ. Let {s1 , . . . , sm } be a complete set of orbit representatives of Φ in R\{0}. For each i, let Ai = {{x, y} | x − y ∈ Φ(si )}, and set A = {Ai | 1 ≤ i ≤ m}. Theorem 7.10 ([26]). (1) (R, A) is an associative scheme. (2) For any proper subset S of R with |S| ≥ 2, denote S = {aS + b | a ∈ Φ, b ∈ R}. Then (R, S, A) is a PBIBD.

8.

Circularity and graphs

Clay came up with the concept of circular planar nearrings in [9] when he studied the planar nearring (C, +, ∗3 ). We ﬁrst give the deﬁnition of circular planar nearrings and Ferrero pairs. (Note that this deﬁnition is slightly diﬀerent from the one given in [10, (5.1)].) Deﬁnition 8.1. Let (N, +, ·) be a planar nearring. If for a, c, b, d ∈ N , a ≡m 0 and c ≡m 0, it holds that N ∗ a + b = N ∗ c + d implies that |(N ∗ a + b) ∩ (N ∗ c + d)| ≤ 2, then we say that N is circular. If (N, Φ) is the corresponding Ferrero pair, then N ∗ a = Φ(a) = {ϕ(a) | ϕ ∈ Φ}. So N is circular if |(Φ(a) + b) ∩ (Φ(c) + d)| ≤ 2 for all a, b, c, d ∈ N with a = 0, c = 0 and Φ(a) + b = Φ(c) + d. We also say that the Ferrero pair (N, Φ) is circular in this manner. For example, the planar nearring (C, +, ∗3 ) is a circular planar nearring since for nonzero a, b ∈ C, C∗ a + b is simply the circle which passes through the point a + b and centers at b. Actually, this example was the source for the deﬁnition of circularity of planar nearrings, and also was the inspiration for the connection between circular planar nearrings and graphs. Surprisingly enough, the combinatorial condition of circularity imposed on planar nearrings selects a well-behaved class of nearrings.

8.1

Characterization of ﬁnite circular Ferrero pairs

First we consider Ferrero pairs (N, Φ) with Φ abelian. There are abundant circular Ferrero pairs to be found from ﬁnite ﬁelds. Example 8.2. Let F = GF(p2 ), p a prime, and let Φp+1 be the subgroup of F ∗ order p + 1. Then the Ferrero pair (F, Φp+1 ) is circular. Consequently, if k ≥ 3 and p is a prime with k | (p + 1), then the Ferrero (F, Φk ) is circular.

On Recent Developments of Planar Nearrings

13

The above examples pave the path to the following characterization of circular Ferrero pairs (F, Φ) with F a ﬁnite ﬁeld. It turns the testing for circularity from “combinational” to “numerical.” An algorithm for computing the ﬁnite sets Pk in the theorem was also provided in the cited paper. An improved method for computing Pk can be derived from [2], and we shall state it after the theorem. Theorem 8.3 ([22]). For each k ≥ 3, there is a nonempty ﬁnite subset Pk of prime numbers with the following property: Let q = ps , a power of some prime p, be such that k | (q −1). Then there is a subgroup Φk of the multiplicative group GF(q)∗ of order k, and the Ferrero pair (GF(q), Φk ) is circular if and only if p ∈ Pk . Here, we give our algorithm for computing the sets Pk . Let ζ = e2πi/k ∈ C. For u, v, s, t with 1 ≤ u < v ≤ s ≤ k − 1, 1 ≤ t ≤ k − 1, and v = t and s = t, deﬁne ϕu,v,s,t = (ζ u − 1)(ζ t − 1) − (ζ v − 1)(ζ s − 1) ∈ Z[ζ]. Then ϕu,v,s,t is nonzero and has integer norm Nu,v,s,t = NQ(ζ):Q (ϕu,v,s,t ). It can be seen that if F is a ﬁnite ﬁeld of characteristic p and (F, Φ) is a ﬁeld generated Ferrero pair, then (F, Φ) is circular if and only if the norms Nu,v,s,t are nonzero when considered as elements of F . Thus, the set Pk consists of the prime factors of all such norms Nu,v,s,t . For practical applications (e.g. to construct codes or cryptosystems), a sharp upper bound of Pk (k ≥ 3) in terms of k may be useful. The above method provides us a trivial bound, namely the maximum of all possible Nu,v,s,t . Since ϕu,v,s,t expands to 6 summands of powers of ζ, we see that the norm is less than or equal 6ϕ(k) , where ϕ(k) is the Euler totient function giving the number of automorphisms of the kth cyclotomic ﬁeld. √ Conjecture 8.4. For k ≥ 3, NQ(ζ):Q (ϕu,v,s,t ) ≤ (8 3/3)ϕ(k) . Problem 8.5. Find a better bound for Pk , k ≥ 3. Yet another question one can ask is Problem 8.6. For k ≥ 3, what is the size of Pk ? Next, we consider circular Ferrero pairs (N, Φ) with ﬁnite nonabelian Φ. Since Φ is a regular group of automorphisms of N , the Sylow p-subgroups of Φ are either cyclic or generalized quaternion (cf. [24, (12.6.15) and (12.6.17))]. The circularity of (N, Φ) excludes the second possibility. Namely, if (N, Φ) is circular, then Φ is metacyclic [2, (3.2)]. We note that the converse of the above assertion is not true. One can ﬁnd Ferrero pairs (N, Φ) with metacyclic Φ while (N, Φ) is not circular. Similar to that of Pk , there is a ﬁnite set PΦ of primes which can be used to determine the circularity of (N, Φ) numerically.

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Theorem 8.7 ([2]). Let (N, Φ) be a circular Ferrero pair with ﬁnite Φ. Then there is a nonempty ﬁnite subset PΦ of prime numbers with the following property: Let M be a ﬁnite group such that (M, Φ) is a Ferrero pair. Then (M, Φ) is circular if and only if p ∈ PΦ for all prime divisors p of |M |. Remark 8.8. The assumption that (N, Φ) is circular Ferrero pair is used to guarantee the ﬁniteness of PΦ (see [2] for details). Thus, we said that a given group Φ is a group without ﬁxed points if there is a group N such that (N, Φ) is a Ferrero pair, and is a circular group without ﬁxed points if there is a group M such that (M, Φ) is a circular Ferrero pair. We have just seen that if Φ is a ﬁnite group without ﬁxed points and Φ is circular, then Φ is metacyclic. Conversely, all ﬁnite metacyclic groups are groups without ﬁxed points, but not all of them are circular. Problem 8.9. Let Φ be ﬁnite metacyclic group. (Thus Φ is a group without ﬁxed points.) Under what conditions is Φ circular? Metacyclic groups have very nice presentations as generators and relations. The following is one of the presentations. Theorem 8.10 (Zassenhaus 1936). Let Φ be a metacyclic group. Then Φ∼ = A, B | Am = B n = 1, B −1 AB = Ar , where m > 0, gcd(m, (r − 1)n) = 1, and rn ≡ 1 (mod m). If d is the order of r modulo m, then all irreducible complex representation of Φ are of degree d. Using an extra assumption on Φ so that it can be more easily handled, there is a partial answer to Problem 8.9. Theorem 8.11 ([5]). Let Φ be a metacyclic group with a presentation as in Theorem 8.10 and d = 2. If Φ is embeddable as a subgroup of the multiplicative group of some skew ﬁeld, then Φ is circular. The data we have at hand suggests that this theorem should hold in general. Conjecture 8.12. If Φ is a metacyclic group embeddable as a subgroup of the multiplicative group of some skew ﬁeld, then Φ is circular.

15

On Recent Developments of Planar Nearrings

8.2

Graphs of circular Ferrero pairs

Another idea of Clay on studying the structure of circular planar nearrings is to connect them to graphs. Let (N, Φ) be a Ferrero pair. We will assume that Φ is ﬁnite. For r, c ∈ N \ {0}, deﬁne Ecr = {Φ(r) + b | b ∈ Φ(c)}. Then the set Ecr is a partition of N \ {0}. In fact, it was from an equivalence relation deﬁned on the set {Φ(a) + b | a, b ∈ N , a = 0}, that Clay came up with this partition. For an integer Example 8.13. We consider the Ferrero pair (C, C). of order k is denoted by Φk . k ≥ 3, the subgroup of C r (1) Here are two Ec ’s in (C, Φ5 ) (each × indicates the center of the circle that the ﬁve points (vertices of a pentagon) of a Φ5 r inscribed): • • • • • • • × • ×• • • • × • • • ×• • • • • • • × × • • • • • • • × • • • ×• • × • × • • • • • • (2) Here is an Ecr in (C, Φ6 ): • •

• • ×

• ×

• •

• × •

• × •

• • • × • ×

•

• •

•

To see the structure of an Ecr , a graph G(Ecr ) = (V, E) is assigned to it: the vertex set V = Φ(c) and the edge set E is {c1 c2 | c1 , c2 ∈ Φ(c), c1 = c2 , and (Φ(r) + c1 ) ∩ (Φ(r) + c2 ) = ∅}. An edge c1 c2 is even if |(Φ(r) + c1 ) ∩ (Φ(r) + c2 )| = 2, and is odd if |(Φ(r) + c1 ) ∩ (Φ(r) + c2 )| = 1. The following pictures show the two Ecr

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we have seen above and the graphs they deﬁned. •

• •

• × •

•

•

•

•

• ×• •

• ×•

•

•

•

× •

• ×• • •

•

•

• ×

• ×

• ×

•

•

•

•

• ×

• • ×

• ×

•

• •

•

•

• • •

•

• •

•

•

• •

•

Some properties of the graphs G(Ecr ) can be observed immediately. The most obvious one is that every G(Ecr ) is a regular graph, i.e. all the vertices have the same number of edges connected to them. On the other hand, some properties require some detailed analysis of the graphs. If G(Ecr ) has nonnull edges, then it is a union of even and/or odd basic graphs. Here we illustrate this by two examples. •

•

•

•

•

• = • •

• •

•

• •

•

• •

•

•

•

• + • •

= • •

•

• + • •

•

•

• •

•

•

+ • •

•

•

•

So the leftmost graph on the ﬁrst line is the “disjoint union” of two odd graphs while the leftmost graph on the second line is the “disjoint union” of two even graphs and an odd graph. Some special arrangements of the graphs produce pictures like David’s stars and Prisms (see [10]). After investigation on many examples, we found that there are numbers about the basic graphs that depends on k alone.

17

On Recent Developments of Planar Nearrings

So, let (N, Φ) be a ring generated Ferrero pair with |Φ| = k. Fix an r ∈ N \ {0} and consider all G(Ecr ), where c ∈ N , such that G(Ecr ) has some edges. Then each of such graphs is either a basic graph or a “union” of basic graphs. Let γj denote the total number of the appearances the odd jth basic graphs in these nonnull graphs, and πj the total number of the appearances of the even jth basic graphs. Theorem 8.14 ([19]). If 2 | k, then γj = 1 and πj = k/2 − 1 for any j ∈ {1, 2, . . . , k/2}. The reason for j to stop at k/2 in the statement of the above theorem is that the jth basic graph and the (k − j)th basic graph are identical. Here is a somewhat surprising application of the above counting of basic graphs to ﬁnd solutions of certain equations over ﬁnite ﬁelds. Let F = GF(q) be the Galois ﬁeld of order q. Let k | (q − 1) be such that (F, Φk ) is a circular Ferrero pair. Put m = (q − 1)/k. Denote by n the number of solutions of the equation xm + y m − z m = 1 in F , and by n the number of solutions with xyz = 0. Theorem 8.15 ([18]). (1) If k is even, then ⎧ 3 2 ⎪ ⎨3(k − 1)m + 6m + 3m n = 3(k − 1)m3 + 3m2 + 3m ⎪ ⎩ 3(k − 1)m3 + 3m

if 6 | k; if p = 3; otherwise;

and n = 3(k − 1)m3 . (2) If k is odd, and if (GF(q), Φ2k ) is also circular, then n = (2k − 1)m3 + 2m and n = (2k − 1)m3 . Actually, one can explicitly write down the solutions. This was done by Kiechle in [21]. Next, we observed that when (N, Φ) is a ring generated Ferrero pair with cyclic Φ, some of the basic graphs always appear together in some G(Ecr ) (referred as overlapped graphs). A complete understanding of such behavior is the key to count the total number of graphs G(Ecr ). Using a theorem of vanishing sums [12, Theorem 6], we have a complete and description of this phenomenon in the case of (C, Φk ), Φk ≤ C |Φk | = k ≥ 3. (See [20].) Finally, we have also noticed that the graphs of Ecr ’s occur in the ﬁnite ﬁeld generated case and the complex plane case are the same when the ﬁeld has large enough characteristic. For small characteristic, there are

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more overlapped basic graphs. To explain partly this phenomenon, we note that the overlaps of the basic graphs are in one-one correspondence with the solutions (u, v, s, t) of the equations s ζu − 1 wζ −1 = ζ ζv − 1 ζt − 1

where ζ is a primitive kth root of unity, 1 ≤ u < v ≤ s ≤ k − 1, 1 ≤ t ≤ k − 1, v = t, s = t, and 1 ≤ w ≤ k − 1. Now, consider ζ = e2πi/k ∈ C as before, and put the set OP k the prime factors of the norms of (ζ u − 1)(ζ t − 1) − ζ w (ζ s − 1)(ζ v − 1) for all suitable u, v, s, t, w. Then each OP k is a ﬁnite set. When p is a prime larger than any of that in OP k and k | (p − 1) for some positive integer , the overlaps of the graphs of the Ecr ’s from (GF(ps ), Φk ) and that from (C, Φk ) are the same. Problem 8.16. As we have mentioned, an Ecr is simply an equivalence class of a block Φ(r) + b. Are there any other equivalence on the set {Φ(a)+b | a, b ∈ N , a = 0} which will give use interesting (and hopefully manageable) equivalence classes?

9.

List of ongoing research problems on planar nearrings

We would like to invite more people to join us on exploring the fascinating world of planar nearrings, circular or not. In the following, a list of problems concerning planar nearrings are given. This list came from “Group Discussions” when the author visited Linz in the summers of 2002 and 2003. One realizes easily by scanning through the list that there are much more of planar nearrings to be uncovered! (1) The complex number ﬁeld C. What to study in each individual planar nearring? Characterize all ﬁxed point free automorphism groups Φ on (C, +): Φ ≤ (C∗ , ·), or C as a R2 , or C as a vector space over Q. Note that the descriptions of ﬁnite Φ’s can be found in [30]. Is being algebraically closed important for the study? How continuity may come into play? Are there other constructions similar to Jim Clay’s hyperbolas? (2) The real number ﬁeld R and the rational ﬁeld Q.

On Recent Developments of Planar Nearrings

19

Answer the same questions as for C. What else do we lose? Do we lose circles at all? (3) Designs. Compare the designs which are planar nearring generated and the designs that have been studied in combinatorial design theory. (a) Are there designs in the combinatorial design theory which can also be obtained using planar nearrings? (b) Are there designs from planar nearrings which have not been studied in combinatorial design theory? (4) Make possible visualization of planar nearrings. (5) What more can be done about the triangle constructions of Jim Clay? (6) Make a dictionary among diﬀerence topics: Frobenius groups, Planar nearrings, Design theory, Diﬀerence families (sets). Can one use the techniques or results in one area to apply to the others?2 (7) Structure of planar nearrings. Are there natural generators for planar nearrings or for modules over planar nearrings? When is a planar nearring tame? Can one describe planar nearring modules? (8) Representations of planar nearrings. Is it possible to represent a planar nearring as a nearring of functions on certain nearring modules? (9) Sub-diﬀerence Families. It is often the case in the diﬀerence families constructed from Ferrero pairs using the B ∗ construction that there are several sub-diﬀerence families. That is, subsets of 2 Wendt can construct planar nearrings with certain prescribed properties from a given Ferrero pair using the choices of subgroups to build A and the left units.

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the base blocks that form a diﬀerence family, but with smaller λ. Thus the parameters that we obtain are not the only possible useful results. Check sub-diﬀerence families for small examples. Collect any small results (there are some nonexistence results.) What kind of connections are there between sub-diﬀerence family structure and the Ferrero pair structure? Are we obtaining new BIBDs? (10) Generation. What are the subnearrings generated by one element of a planar nearring? By two elements? How general can the generated nearrings be and when are they planar? When do we obtain the whole planar nearring? (11) Topological planar nearrings. It may be useful to studying topological planar nearrings from the prospect of centralizer sandwich nearrings. (12) Nonisomorphic planar nearrings. Take some small Ferrero pairs (N, Φ) and compute NAut (N ) (Φ). Then compare nonisomorphic planar nearrings to ﬁnd properties that make the individual planar nearrings unique. Note that there is a complete characterization of (N, Φ) with abelian N . (See [8], or [16] for elementary abelian and cyclic N with cyclic Φ.) (13) Matrix nearrings of planar nearrings. Does the matrix nearring Mn (N ), where N = (C, +, ∗E ), have nice properties (eg. behave like a generalization of n dimensional vector spaces)? In general, study matrix nearrings of planar nearrings. Do they possess good properties? (14) Varieties. Which varieties generated by the following classes of nearrings are equal and which are distinct? A ... class of ﬁnite nearﬁelds, B ... class of 0-symmetric nearrings with abelian additive group, Cp ... class of ﬁnite integral planar nearrings, C ... class of ﬁnite planar nearrings, D ... class of ﬁnite zero-symmetric integral nearrings,

On Recent Developments of Planar Nearrings

21

E ... class of ﬁnite zero-symmetric nearrings satisfying xn = x for some natural number n, F ... class of zero-symmetric nearrings. Note that D ⊆ E, and that the subdirectly irreducible elements of E are in D. Thus the varieties generated by D and by E are equal. (15) Here are some more questions related to the previous item. We know that the class of ﬁnite Frobenius kernels generates the variety of all groups, since each ﬁnite p-group can be embedded into a Frobenius kernel. Does the class of Frobenius complements of Frobenius groups also generate the variety of all groups? Determine the variety generated by the multiplicative semigroups of (ﬁnite) planar nearrings. (16) Nearﬁelds. Is there an inﬁnite nearﬁeld whose multiplicative group has ﬁnite exponent? (There is none with exponent 3, 6, 12, or 2n for any natural number n.) (17) Regularity. Study the connection between 2-primitive nearrings and regularity. Regularity of elements (deﬁned in the usual sense for semigroups) seem to play a very vital role in the structure of 2primitive nearrings. For example, if L is a minimal left ideal of a 2primitive nearring N , then L is itself a planar nearring. In particular, the regular elements of L form a (multiplicative) subsemigroup. Do the regular elements of N form a subsemigroup?3 (18) If there is something can be said about 2-primitive nearrings, then it should also be possible to get results on 2-semisimple nearrings. (19) Planar subnearrings. Planar substructures or at least planar like structures arise naturally in 2- and 1-primitive nearrings. What about 0-primitive nearrings, or nearrings with similar conditions imposed on the nearring modules? (20) K-loops. Is there something to say about possible connections between K-loops and planar nearrings? Do techniques for studying planar nearrings carry over to K-loops. (21) Planar quotients. When is a quotient nearring planar? 3 Actually, it is now known that if a planar nearring N satisﬁes descending condition on N subgroups, then the regular elements of a 2-primitive nearring form a semigroup right ideal. See [29]

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Wen-Fong Ke

Acknowledgment I would like to express my gratitude to Gerhard Wendt for the private communication about his results on the algebraic structure of planar nearrings stated in section 6.

References [1] M. Anshel and J. R. Clay. Planar algebraic systems: some geometric interpretations. J. Algebra 10 (1968), 166–173. [2] K. I. Beidar, Y. Fong, and W.-F. Ke. On ﬁnite circular planar nearrings. J. Algebra 185 (1996), 688–709. [3] K. I. Beidar, Y. Fong, and W.-F. Ke. Maximal right nearring of quotients and semigroup generalized polynomial identity. Result. Math. 42 (2002), 12–27. [4] K. I. Beidar, W.-F. Ke. On planar nearrings. Preprint. [5] K. I. Beidar, W.-F. Ke, and H. Kiechle. Circularity of ﬁnite groups without ﬁxed points. Monatshefte f¨ ur Mathematik, to appear. [6] K. I. Beidar, W.-F. Ke, and H. Kiechle. Automorphisms of design groups II. Preprint. [7] K. I. Beidar, W.-F. Ke, C.-H. Liu, and W.-R. Wu. Automorphism groups of certain simple 2-(q, 3, λ) designs constructed from ﬁnite ﬁelds. Finite Fields Appl 9 (2003), 400–412. [8] Ron Brown. Frobenius groups and classical maximal orders Mem. Amer. Math. Soc. 151 (2001), no. 717. [9] J. R. Clay. Circular block designs from planar nearrings. Ann. Discrete Math. 37 (1988), 95–106. [10] J. R. Clay. Nearrings. Geneses and Applications. Oxford University Press. 1992. [11] J. R. Clay. Geometry in ﬁelds. Algebra Colloq. 1 (1994), 289–306. [12] J. H. Conway and A. J. Jones, Trigonometric diophantine equations, Acta Arith. 30 (1976), 229–240. [13] G. Ferrero. Stems planari e bib-disegni. Riv. Mat. Univ. Parma (2) 11 (1970), 79–96. [14] C. Cotti Ferrero, Radicali in quasi-anelli planari, Riv. Mat. Univ. Parma 12 (1986), 237–239. [15] C. Ferrero Cotti, S. Manara Pellegrini, On the homomorphic images of planar nearrings, Sistemi binari e loro applicazioni, Taormina, (me), 1978. [16] W.-F. Ke and H. Kiechle. Characterization of some ﬁnite Ferrero pairs. Nearrings and near-ﬁelds (Fredericton, NB, 1993), 153–160, Math. Appl., 336, Kluwer Acad. Publ., Dordrecht, 1995. [17] W. F. Ke and H. Kiechle. Automorphisms of certain design groups. J. Algebra 167 (1994), 488–500. [18] W. F. Ke and H. Kiechle. On the solutions of the equation xm + y m − z m = 1 in a ﬁnite ﬁeld. Proc. Amer. Math. Soc. 123 (1995), 1331–1339. [19] W.-F. Ke and H. Kiechle. Combinatorial properties of ring generated circular planar nearrings. J. Combin. Theory Ser. A 73 (1996), 286–301.

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[20] W.-F. Ke and H. Kiechle. Overlaps of basic graphs in circular planar nearring. Preprint. [21] H. Kiechle. Points on Fermat Curves over Finite Fields. Contemporary Math. 168 (1994), 181–183. [22] M. C. Modisett. A characterization of the circularity of certain balanced incomplete block designs. Ph. D. dissertation, University of Arizona, 1988. [23] M. Modisett. A characterization of the circularity of balanced incomplete block designs. Utilitas Math. 35 (1989), 83–94. [24] W. R. Scott, Group Theory, Dover, New York, 1987. [25] H.-M. Sun. Segments in a planar nearring. Discrete Math. 240 (2001), 205–217. [26] H.-M. Sun. PBIB designs and association schemes obtained from ﬁnite rings. Discrete Math. 252 (2002), 267–277. [27] G. Wendt. A description of all planar nearrings. Preprint. [28] G. Wendt. Planar nearrings and sandwich nearrings. Preprint. [29] G. Wendt, Planarity in nearrings. Preprint. [30] Joseph A. Wolf. Spaces of constant curvature. (Fifth edition). Publish or Perish, Inc., Houston, TX, 1984.

SOME PROBLEMS RELATED TO NEARRINGS OF MAPPINGS Carl J. Maxson Department of Mathematics Texas A&M University College Station, TX 77843-3368 USA [email protected]

Abstract

In this paper we discuss three areas of research relative to near-rings of mappings and mention several open questions. 2000 Mathematics Subject Classiﬁcation: 16D10; 16Y30; 20K30.

Keywords: Homogeneous functions; Forcing linearity numbers; Rays; Near-rings of mappings.

1.

Introduction

This paper is an expanded version of a talk given at the 18th International Conference on Near-rings and Near-ﬁelds held at the Universit¨ at der Bundeswehr Hamburg, 27 July-3 August, 2003. The main topic of the paper is near-rings of homogeneous functions and this paper may be thought of as a continuation of the paper, “Near-rings of homogeneous functions P 3 ” [15], presented at the 1995 International Conference on Near-rings, Near-ﬁelds and K-loops, also held at the same venue in Hamburg. We discuss three areas of research related to near-rings of mappings. The ﬁrst, forcing linearity numbers, had its origins in the 1980’s, [18] but in the last ﬁve years has been the subject of numerous investigations. The second area, rays, had its origins in the 1995 paper [2] of Albrecht and Hausen and was used heavily in the 2000 paper [5] of Peter Fuchs. The third area of research, subrings of the zero symmetric near-ring of functions on an abelian group, is a very recent research project and the development here will most likely undergo several iterations before reaching a deﬁnitive direction.

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We ﬁx some notation for the remainder of the paper. Let R be a ring, always with identity. An R- module V will always mean a unital R-module and we denote the collection of (left) R-modules by Rmod. A function f : V → W where V, W ∈ R-mod is homogeneous if f (rx) = rf (x), for all r ∈ R, x ∈ V . The additive group of homogeneous functions from V to W is denoted by MR (V, W ) and the near-ring of homogeneous functions on V is denoted by MR (V ). As usual HomR (V, W ) will denote the abelian group of R-homomorphisms from V to W and EndR (V ), the ring of R-endomorphisms on V .

2.

Forcing Linearity Numbers

For any V ∈ R-mod we have MR (V ) ⊇ EndR (V ). For some pairs, MR (V ) = EndR (V ) and when MR (V ) EndR (V ) one would like some type of measure to indicate how close (or how far away) one is to equality. The concept of forcing linearity numbers was introduced to give such a measure. Deﬁnition 2.1. Let C := {W Wα }, α ∈ A be a collection of proper Rsubmodules of an R-module V . We say C forces linearity on V if for f ∈ Wα , V ) for each α ∈ A then f ∈ EndR (V ). MR (V ), whenever f ∈ HomR (W This leads to the next concept. Deﬁnition 2.2 ([17]). To each V ∈ R-mod we assign a number, called the forcing linearity number of V , and denoted by f ln(V ) deﬁned as follows: a) f ln(V ) = 0 if EndR (V ) = MR (V ); b) if f ln(V ) = 0 and there is a ﬁnite collection C of proper submodules which forces linearity then f ln(V ) = inf{|C| | C forces linearity on V }; c) f ln(V ) = ∞ otherwise. Forcing linearity numbers were ﬁrst calculated for vector spaces. Theorem 2.3 ([17]). Let F be a ﬁeld and V ∈ F -mod. i) f ln(V ) = 0 if and only if dimF V = 1. ii) If |F | = ∞ and dimF V > 1 then f ln(V ) = ∞. iii) If |F | < ∞ and dimF V = 2 then f ln(V ) = ∞. iv) If |F | < ∞ and dimF V > 2 then f ln(V ) = |F | + 2.

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Forcing linearity numbers for several pairs (R, V ) have been determined. We mention some of these with appropriate references: 1) All Z-modules, i.e., abelian groups, [4]. 2) Projective modules over commutative Noetherian rings, [16]. 3) Finitely generated modules over commutative Noetherian rings, [20]. 4) Modules over Artinian rings, [19]. 5) Divisible modules over principal ideal domains, [12]. 6) Semisimple modules over integral domains, [22]. 7) Modules over complete matrix rings, [6]. In these situations and in all other known situations the forcing linearity of a module belongs to the set {0, 1, 2, ∞} ∪ {q + 2|q is a power of a prime integer}. This raises the question as to which positive integers can be forcing linearity numbers. In particular we have the speciﬁc question: Problem #1. Is there a ring R and V ∈ R-mod such that f ln(V ) = 3? Suppose R is a commutative ring, not local. Let M1 , M2 be maximal ideals of R and let V be a faithful R-module. If M1 V V and M2 V V then {M M1 V, M2 V } forces linearity. To see this let x, y ∈ V , f ∈ MR (V ) and let w := f (x + y) − f (x) − f (y). Suppose f is linear on M1 V and M2 V . Then for each mi ∈ Mi , i = 1, 2 we have mi w = 0. Since M1 + M2 = R, there exist mi ∈ Mi , i = 1, 2 with m1 + m2 = 1. Thus w = 0 and f is linear on V . So f ln(V ) ≤ 2. In particular if V is a faithfully ﬂat R-module then IV V for each ideal I of R. (See [14].) Theorem 2.4. If R is a nonlocal commutative ring and V is a faithfully ﬂat R-module then f ln(V ) ≤ 2. We continue to let R be commutative and nonlocal. Suppose the faithful R-module V is a generator for R-mod. Then V n ∼ = R ⊕ W for some W ∈ R-mod. From this one can show M1 V V and M2 V V for maximal ideals M1 , M2 of R. Theorem 2.5. If R is a nonlocal commutative ring and V ∈ R-mod is a faithful module and a generator for R-mod, then f ln(V ) ≤ 2.

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Theorem 2.6. Let R be a commutative ring with a minimal ideal L. If the set of zero divisors L(R) is not an ideal of R, then for any faithful module V ∈ R-mod, f ln(V ) ≤ 2. Proof. Since L is a minimal ideal there exists a ∈ L such that L = Ra. Hence the left multiplication map λa : R → Ra, x → xa is an Repimorphism with ker λa = M , a maximal ideal of R and M ⊆ L(R). Since L(R) = M , there exists some b ∈ L(R)\M . Let A = {y ∈ R|yb = 0}. Now M V = V for if this were the case then {0} = aM V = aV , contrary to V being faithful. In the same manner (Rb)V V . Since M + Rb = R we ﬁnd that {M V, (Rb)V } forces linearity. The above results imply that, for many pairs (R, V ) where R is a nonlocal commutative ring, we have f ln(V ) ≤ 2. Problem #2. If R is a nonlocal commutative ring and V a faithful R-module, does one always have f ln(V ) ≤ 2?

3.

Rays

In 1991, Peter Fuchs, G¨ unter Pilz and I considered the following two problems ([6]). (∗) Characterize those rings R such that MR (V ) is a ring for every V ∈ R-mod. Denote this collection of rings by R. (∗∗) Characterize those rings R such that MR (V ) = EndR (V ) for each V ∈ R-mod, i.e., those rings R such that f ln(V ) = 0 for each V ∈ R-mod. Denote this collection of rings by E. We proved the following somewhat surprising result. Theorem 3.1 ([6]). R = E. We then investigated this class of rings. We found that any complete matrix ring, Mn (S), for n ≥ 2 and S any ring, is in E. Further, no commutative ring is in E. In [11], Krempa and Niewieczerzal extended Theorem 3.1 by showing that R ∈ E if and only if MR (V, W ) = HomR (V, W ) for all V, W ∈ Rmod. Further in 2000, Peter Fuchs in [5] showed that a simple Noetherian ring R is in E if and only if R is not a domain. In 1995, [2], Ulrich Albrecht and Jutta Hausen considered the following: If R is a semiprime right Goldie ring and V is a nonsingular R-module, when is f ln(V ) = 0? Recall that a module V is nonsingular if no nonzero element has an essential left annihilator ideal. In the process of answering this question they introduce the concept of “ray”.

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Deﬁnition 3.2. Let R be a ring and C a class of R-modules. We say a module M ∈ R-mod is a ray for C if HomR (M, C) = MR (M, C) for each C ∈ C. If C = R-mod then we say M is a ray. The result of Albrecht and Hausen is Theorem 3.3. Let N be the class of nonsingular modules over a semiprime right Goldie ring, R. Then a module N ∈ N is a ray for N if and only if the simple summands in the Artinian semisimple quotient ring Q(R) of R are complete matrix rings of size at least two. We mention that Fuchs also used the concept of ray in his characterization of simple Noetherian rings in E, ([5]). It is straightforward to verify that being cyclic is a suﬃcient condition for a module to be a ray. It is of interest to know when a ray must be cyclic. In a recent paper [21] the lattice structure of the submodule lattice, L(V ), of an R-module has been used to investigate this problem. Recall that a module V is join irreducible if the sum of all proper submodules of V is a proper submodule of V . Let J (V ) denote the poset of join irreducible submodules of V and let C(V ) denote the poset of cyclic submodules of V . Every join irreducible submodule of V is cyclic for if A ∈ J (V ) and a ∈ A is not in the sum of all proper submodules of A then Ra = A. Problem #3. Characterize those pairs (R, V ) such that J (V ) = C(V ). The rings R for which J (V ) = C(V ), for all V ∈ R-mod are just the local rings. Theorem 3.4 ([21]). Let R be a ring. Then C(V ) = J (V ) for each V ∈ R-mod if and only if R is a local ring. Theorem 3.5 ([21]). Let V ∈ R-mod. a) If V is a ray with C(V ) = J (V ) and C(V ) has a maximal element then V is cyclic. b) If R is a left perfect local ring then V is a ray if and only if V is cyclic. Problem #4. Characterize those rings R such that V ∈ R-mod is a ray if and only if V is cyclic. It is clear that if V is a ray then f ln(V ) = 0. However the converse is not true as is shown in Example II.3 of [20]. On the other hand if a ring R is in E then from the Krempa–Niewieczerzal result, f ln(V ) = 0

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implies V is a ray. Further if R is a ﬁeld, then R ∈ / E but for V ∈ R-mod, f ln(V ) = 0 implies V is a ray. This follows immediately from the fact that if R is a ﬁeld then f ln(V ) = 0 if and only if V is cyclic. Problem #6. Characterize those pairs (R, V ) such that f ln(V ) = 0 implies V is a ray. For V is a semisimple module over R, the above problem has been solved. (See [21], Section IV). We give one part of the solution. Theorem 3.6 ([21]). Let R be a noncommutative, non-Artinian semisimple ring. Then every semisimple module over R is a ray. We conclude this section with an additional relation between the lattice structure of a module and the concept of ray. Theorem 3.7 ([21]). Let R ∈ E. Then every length two R-module is cyclic. As a partial converse, if R is a semiperfect ring, in particular an Artinian ring, such that every length two module is cyclic then R ∈ E. This leads to the ﬁnal problem of this section. Problem #7. Characterize those rings R such that the lattice property of Theorem 3.7 is suﬃcient for R to be in E.

4.

Rings in Near-rings of Mappings

The focus of this research area is to determine subrings of the zero symmetric near-ring, M0 (A), of zero preserving functions on the abelian group A, i.e., M0 (A) := {f : A → A|f (0) = 0}. To the author’s knowledge this is a new research area and there is little doubt but that the development will undergo several iterations before completion. It is well-known that the near-ring M0 (A) is a simple near-ring, (see [3]). In addition, several people have investigated the left ideals, and right ideals of M0 (A). (See [9], [10], and [23].) Recently E. Aichinger and D. Maˇ ˇsulovi´c have found all maximal subnear-rings of M0 (A), where A is a ﬁnite group, [1]. So now we turn to subrings of M0 (A). These do exist, since A is abelian, End A is a subring of M0 (A). Here we will restrict our discussion to subrings of M0 (A) which contain End A. We mention however that, although there has been much research on various properties of End A, there doesn’t seem to be any study of the poset of subrings of End A. We now formulate a general problem. Problem #7. Let A be an abelian group and let M0 (A) be the nearring of zero preserving functions on A. Investigate the partially ordered set of subrings of M0 (A) which contain the ring End(A).

Some Problems Related to Near-rings of Mappings

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We mention that by deﬁnition a subring contains the identity map on A. Further, we give some preliminary results which can be found in [13]. Theorem 4.1. Let A be an abelian group. If R is a subring of M0 (A) then R ⊆ MZ (A) := {f : A → A|f (na) = nf (a), for each n ∈ Z, a ∈ A}. Proof. Suppose f ∈ R. Then f (id + id) = f + f implies f (2x) = 2f (x), x ∈ A. Using induction and ring properties one ﬁnds that f (nx) = nf (x) for all n ∈ Z, x ∈ A. From this theorem, if MZ (A) = End(A) then the only subring of M0 (A) containing End(A) is End(A). From [8] or [9] or [4], we ﬁnd that End(A) = MZ (A), i.e. f ln(Z A) = 0, if and only if A is a subgroup of ⊕Z(p∞ ) or A is a subgroup of the additive group, Q, of rational primes

numbers. Also from [8], MZ (A) is never a ring if A is mixed or if A is torsion ⊕Z(p∞ ). If A is torsion-free it may be that and not a subgroup of primes

MZ (A) is a ring but not equal to End(A). This happens if and only if A is absolutely anisotropic of rank at least two. (For this and the deﬁnition of absolutely anisotropic see [8] or [9].) Suppose MZ (A) is not MZ (A)) is a ring. The collection of distributive elements in MZ (A), Dist(M a subring of MZ (A) and since every f ∈ End A is distributive in M0 (A) we have Dist MZ (A) ⊇ End A. The question remains if these two rings are equal. Theorem 4.2 ([13]). Suppose A is an abelian group with a free summand and MZ (A) is not a ring. Then End A is a maximal subring of M0 (A). Proof. Let A = Z ⊕ Aˆ and suppose A is a subring of M0 (A) which contains End(A). Let f ∈ R, x = (x1 , x2 ), y = (y1 , y2 ) be in A. There ˆ such that α1 (1) = x1 , β1 (1) = exist α1 , β1 ∈ End Z, α2 , β2 ∈ Hom(Z, A) y1 , α2 (1) = x2 and β2 (1) = y2 . Thus α := αα12 00 , β = ββ12 00 are in

End A and α [ 10 ] = [ xx12 ], β [ 10 ] = [ yy12 ]. Since α, β ∈ End A, we have α, β ∈ R so f (α + β) = f α + f β. From this we ﬁnd that f (x + y) = f (x) + f (y), hence f ∈ End A. This gives End A = R. Problem #8. Characterize those abelian groups A such that End(A) is a maximal subring of M0 (A). We say that an abelian group A is E-locally cyclic if for each x, y ∈ A, there exists u ∈ A and α, β ∈ End A such that α(u) = x and β(u) = y. As in the proof of the above theorem, if A is E-locally cyclic and MZ (A)

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is not a ring then End A is a maximal subring of M0 (A). It is not known which abelian groups are E-locally cyclic nor is it known when this property is equivalent to End A being a maximal subring of M0 (A). One can show that any direct sum of cyclic groups (not necessarily ﬁnite) is E-locally cyclic so for any ﬁnitely generated abelian group A such that MZ (A) is not a ring, we have that End A is a maximal subring of End A. We close this section and the paper by indicating other possible subrings of M0 (A) which contain End(A). Suppose B is a fully invariant subgroup of A, i.e, α(B) ⊆ B, for each α ∈ End(A). Let M (B) := {f ∈ M0 (A)|f (B) ⊆ B}, a subnear-ring of M0 (V ). Note that End A ⊆ M (B) since B is fully invariant. Thus for each such B we have Dist(M (B)) a subring of M0 (A) containing End A. Hence if B1 , B2 , . . . , Bn are fully invariant subgroups of A then we have a poset diagram M 0(A)

M(B1 ) ...

Dist(M(B1 ))...

M(Bk )

Dist(M(B Bk ))

M Z (A)

Dist(M MZ (A))

M(Bk +1) ...

Dist(M(B k +1)) ...

M(Bn )

Dist(M(B Bn ))

End(A)

Problem #9. Are there abelian groups with all of these subrings distinct? Are there other subrings?

References [1] Aichinger, E. and Maˇ ˇsulovi´c, D., “Completeness for concrete 0-symmetric nearrings,” preprint. [2] Albrecht, U. and Hausen, J., “Non-singular modules and R-homogeneous functions,” Proc. Amer. Math. Soc., 123 (1995), 2381–2389. [3] Berman, G. and Silverman, R., “Simplicity of near-rings of transformations,” Proc. Amer. Math. Soc., 10 (1959), 456–459. [4] Fuchs, L., “Forcing linearity numbers for abelian groups,” preprint. [5] Fuchs, P., “On modules which force homogeneous maps to be linear,” Proc. Amer. Math. Soc., 128 (2000), 5–15.

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[6] Fuchs, P., Maxson, C.J., and Pilz, G., “On rings for which homogeneous maps are linear,” Proc. Amer. Math. Soc., 112 (1991), 1–7. [7] Hausen, J., “Abelian groups whose semi-endomorphisms form a ring,” Lecture Notes in Pure and Applied Math., Vol. 146, Marcel-Dekker, NY, 1992, 175–180. [8] Hausen, J. and Johnson, J., “Centralizer near-rings that are rings,” J. Austral. Math. Soc., 59 (1995), 173–183. [9] Heatherly, H., “One-sided ideals in near-rings of transformation,” J. Austral. Math. Soc., 13 (1972), 171–179. [10] Johnson, M., “Right ideals and right submodules of transformation near-rings,” J. Algebra, 24 (1973), 386–391. [11] Krempa, J. and Niewieczerzal, D., “On homogeneous mappings of modules,” Contrib. Gen. Alg. Vol. 8, H¨ older–Pichler–Tempsky, Vienna and Teubner, Stuttgart, 1992, 123–135. [12] Kreuzer, A. and Maxson, C.J., “Forcing linearity numbers for injective modules over PID’s,” Archiv der Math., 77 (2001), 476–483. [13] Kreuzer, A. and Maxson, C.J., “When is End A a maximal subring of M0 (A)?,” in preparation. [14] Lam, T.Y., Lectures on modules and rings, Grad. Texts in Math. Vol. 189, Springer-Verlag, NY, 1999. [15] Maxson, C.J., “Near-rings of homogeneous functions, P 3 ,” Near-rings, Nearﬁelds and K-loops, Kluwer Acad. Pub., Amsterdam, 1997, 35–46. [16] Maxson, C.J., “Forcing linearity numbers for projective modules,” J. Algebra, 251 (2002), 1–11. [17] Maxson, C.J. and Meyer, J.H., “Forcing linearity numbers,” J. Algebra, 223 (2000), 190–207. [18] Maxson, C.J. and Smith, K.C., “Centralizer rings that are endomorphism rings,” Proc. Amer. Math. Soc., 80 (1980), 189–195. [19] Maxson, C.J. and Van der Merwe, A.B., “Forcing linearity numbers for modules over rings with nontrivial idempotents,” J. Algebra, 256 (2002), 66–84. [20] Maxson, C.J. and Van der Merwe, A.B., “Forcing linearity numbers for ﬁnitely generated modules,” Rocky Mt. Journal of Math. (to appear). [21] Maxson, C.J. and Wild, M., “When are homogeneous maps linear? A lattice theoretic approach,” preprint. [22] Moch, A., “Forcing linearity numbers of semisimple modules over integral domains,” Results in Math. 42 (2002), 122–127. [23] Pilz, G., Near-rings, Revised Edition, North-Holland, NY, 1983.

SOME RECENT DEVELOPMENTS IN GROUP NEAR-RINGS John D. P. Meldrum University of Edinburgh Scotland, U.K. [email protected]

Abstract

1.

Since their deﬁnition in 1984 and subsequent publication in Meldrum and van der Walt [86], matrix near-rings have proved to be a useful and fruitful generalization of matrix rings. The same basic idea was used to deﬁne group near-rings, culminating in le Riche, Meldrum and van der Walt [89]. Not surprisingly this construction has not attracted as much attention and eﬀort as matrix near-rings: after all matrix rings are much more commonly used than group rings. There are however some interesting results that have been developed and there are intriguing parallels between matrix near-rings and group near-rings. We present here an account of some of the recent work on group near-rings, emphasizing the parallels with matrix near-rings and the most recent developments. Much of this work was done in collaboration with J. H. Meyer.

Fundamentals.

We will now set out the notation we are going to use and give the deﬁnitions and some basic results about matrix near-rings and group near-rings. We will normally write R for a near-ring and our near-rings will be right zero- symmetric near-rings and will contain an identity unless otherwise speciﬁed. We will often be considering the additive group (R, +) as a left module over R, and that will generally be denoted R R. General references for the theory of near-rings are Clay [92], Cotti-Ferrero and Ferrero [02], Meldrum [85] and Pilz[83]. To deﬁne matrix near-rings we need the following concepts. We denote by (R R)n or more often Rn the direct sum of n copies of R R. Then πj : Rn → R is the projection onto the jth component and ιk : R → Rn is the

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injection into the kth component. Finally λr denotes left multiplication by r ∈ R on R R. Deﬁnition 1.1. The elementary n×n matrices are the maps in M0 (Rn ) given by r = [r; i, j]. ιi λr πj = fi,j The near-ring of n × n matrices over R is deﬁned as the subnear-ring of M0 (Rn ) generated by r ; r ∈ R, 1 ≤ i, j ≤ n}. {ffi,j

This is the deﬁnition given in Meldrum and van der Walt [86], and we would also mention as a good survey Meyer [01] which contains a substantial bibliography up to 1997, the date of the conference at which it was presented. We next proceed to the deﬁnition of group near-rings. Let (G, .) be a multiplicative group. Denote by (R R)G or more often G R the complete direct sum of copies of R R indexed by the elements of G. We denote elements of RG by functions from G to R. Deﬁnition 1.2. The element [r, g] of M0 (RG ) is deﬁned by ([r, g]µ)(h) = rµ(hg) for all µ ∈ RG , h ∈ G. Then the group near-ring R[G] of R over G is deﬁned as the subnear-ring of M0 (RG ) generated by {[r, g]; r ∈ R, g ∈ G}. This deﬁnition was introduced in le Riche, Meldrum and van der Walt [89], where a number of fundamental results about these near-rings were established. In particular this deﬁnition reduces to the standard deﬁnition of a group ring when R is a ring. R. L. Fray has worked on these near-rings in his thesis [88] and several papers: [92a], [92b], [92c], [95], [97a], [97b], [00]. Other papers to be mentioned are Meldrum and Meyer [98] and Meyer [04]. In comparing this deﬁnition with the original deﬁnition we must point out that the notation is diﬀerent: we are using [r, g] for the map originally denoted r, g. This deﬁnition is not the standard one in group rings, but is the one used for semigroup rings and that is the reason it is adopted here. The alternative deﬁnition is equivalent to this one. We now mention some basic results from le Riche, Meldrum and van der Walt [89]. Lemma 1.3. Let R be a near-ring, G a multiplicative group. Then both R and G are embedded in the group near-ring R[G] by the maps which send r → [r, e] and g → [1, g] respectively.

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Here and subsequently we will write 1 for the identity in the near-ring R and e for the neutral element in the multiplicative group G. Denote the restricted direct sum of copies of R indexed by the elements of G by (R R)(G) or R(G) . Lemma 1.4. The restricted direct sum R(G) is a faithful R[G] module. This result enables us to simplify some arguments and calculations. When using this result the following conventions are useful. Let µ ∈ R(G) with the support σ(µ) of µ deﬁned as usual by σ(µ) = {g ∈ G; µ(g) = 1}. Then σ(µ) is a ﬁnite set and we can write (n) µ = a(1) g1 . . . agn

where σ(µ) = {g1 , . . . , gn }, gi = gj if i = j, and µ(gi ) = a(i) . This penultimate condition means that the order in which we write down g1 , . . . , gn is immaterial. Then the deﬁnition of the action of [r, g] on µ given in deﬁnition 1.2 becomes [r, g]µ = (ra(1) )g1 g−1 . . . (ra(n) )gn g−1 . This remark enables us to conclude very easily that the following result is true. Lemma 1.5. R(G) is a monogenic R[G] module and is generated by any element of the form rg where r is a unit in R. There is a close link between matrix near-rings and group near-rings apart from the seminal concept. This is exempliﬁed in the next result which can be found in Meyer [04]. Theorem 1.6. Let G be a ﬁnite group with |G| = n, and R a near-ring. Then R[G] is a subnear-ring of Mn (R) sharing the same identity [1, e] 1 + f1 + . . . + f1 . = f1,1 n,n 2,2 The proof is based on the fact that RG is isomorphic to Rn and every element [r, g] can be linked to an element of Mn (R) once the isomorphism between RG and Rn is established. However R[G] is much smaller than Mn (R) as the operations which can be performed on the elements of Rn by elements of Mn (R) are far more varied than those that can be performed on RG by elements of R[G]. We now mention brieﬂy an interesting application of semigroup nearrings.

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John D. P. Meldrum

Polynomial near-rings.

Near-rings of polynomials constructed from rings of polynomials on one variable over a ring with the operations being the usual addition and composition of functions, have been studied quite extensively, the book by Lausch and N¨ obauer [73] being a major early inﬂuence. This is however in a very diﬀerent spirit from the generalization to near-rings of the usual polynomial rings. By using the idea of group near-rings, a deﬁnition of polynomial near-rings in one variable over a near-ring is obtained which reduces to the usual deﬁnition of polynomial rings when the near-ring is a ring. The key to this deﬁnition is the remark that if R is a ring and S is a free monoid on one generator x, then the semigroup ring R[S] is naturally isomorphic to the ring of polynomials in the variable x over the ring R. This can be used in the context of near-rings and the ﬁrst paper on the subject is due to S. W. Bagley [97] following on his thesis [93]. Further papers on the subject were written by M. Farag [01] and [02]. We now come to the deﬁnition. Instead of S, the free monoid on {x}, we use N = {0, 1, 2, . . . , n, . . .}, the set of natural numbers under addition, a semigroup isomorphic to S with zero. Construct RN , the complete direct sum of copies of the near-ring R. Write f ∈ RN as f = (ff0 , f1 , f2 , . . .), an inﬁnite sequence where fi = f (i) ∈ R. Let a ∈ R. Then we deﬁne La ∈ M (RN ) by La (ff0 , f1 , f2 , . . .) = (aff0 , af1 , aff2 , . . .). For i ∈ N we deﬁne xi ∈ M (RN ) by xi (ff0 , f1 , f2 , . . .) = (0, . . . , 0, f0 , f1 , f2 , . . .) where there are i elements 0 at the beginning of the sequence. Deﬁnition 2.1. The polynomial near-ring R[x] in the variable x over R is deﬁned to be the subnear-ring of M (RN ) generated by the set {La , xi ; a ∈ R, i ∈ N }. This ties in with our earlier deﬁnition if we realise that the following result holds. Lemma 2.2. Using the notation above, we have La = [a, 0] and x = [1, 1], where the ﬁrst 1 in [1, 1] is the identity of R and the second 1 is the integer 1 in N . As [a, g] = [a, e][1, g], we see that lemma 2.2 gives us the corollary.

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Corollary 2.3. R[x] is isomorphic to R[S] where S is the free monoid with zero on the element {x}. Polynomial near-rings need very careful handling which is probably why relatively little work has been done on them so far. But for those willing to make the eﬀort to understand fully the basic results on polynomial near-rings, there should be ample reward.

3.

Ideals in matrix near-rings.

In this section we take a quick look at some fundamental results about ideals in matrix near-rings before we embark on the more complicated but related situation in group near-rings. We will give the deﬁnitions and the results which are of interest very compactly as this is not the main thrust of the paper. Throughout this section R is a right zerosymmetric near-ring with identity. The best source for these results is the paper by Meyer [01], and details of proofs can be found in Meldrum and Meyer [96]. Deﬁnition 3.1. Let Mn (R) be a near-ring of n × n matrices over R. Let I be an ideal of R and let I be an ideal of Mn (R). We deﬁne a I + = Idf1,1 ; a ∈ I,

I ∗ = {A ∈ Mn (R); ARn ⊆ I n }, I∗ = {π1 Aι1 (1); A ∈ I}, where IdX means the ideal generated by X. Some basic facts about these deﬁnitions are collected together in the next theorem. Theorem 3.2. Let Mn (R) be a near-ring of n × n matrices over R. Let I be an ideal of R and let I be an ideal of Mn (R). Then we have I ∗ is an ideal of Mn (R); I∗ is an ideal of R; I + ⊆ I ∗; (I + )∗ = I = (I ∗ )∗ ; I∗ )∗ . (I I∗ )+ ⊆ I ⊆ (I + The maps I → I and I → I ∗ are injective and order preserving, and the map I → I∗ is surjective and order preserving. However, unlike the situation in the case of rings, not all ideals in the matrix near-ring are of the form I + or I ∗ , and I + is often strictly contained in I ∗ . This leads to the following deﬁnition.

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Deﬁnition 3.3. An intermediate ideal K of Mn (R) is an ideal K such that I + ⊂ K ⊂ I ∗ for some ideal I of R. From this we can describe the relationship of the set of ideals in a matrix near-ring with the set of ideals in the base near-ring. Theorem 3.4. All ideals of Mn (R) are either of the form I + or I ∗ for some ideal I of R, or are intermediate. If K is an intermediate ideal of Mn (R), then there is a unique ideal I of R such that I + ⊂ K ⊂ I ∗ . So we have, corresponding to each ideal I of R a lattice of ideals in Mn (R), all of which lie between I + and I ∗ . Indeed if K is an intermediate ideal of Mn (R) then (K∗ )+ ⊂ K ⊂ (K∗ )∗ by theorem 3.2. Apart from the fact that the relationship between ideals in R and those in Mn (R) is one to many, there is a strong connection. The ﬁnal result of this section shows that these lattices of ideals in Mn (R) corresponding to a single ideal of R are not restricted in any way. Theorem 3.5. There exist abelian near-rings R with arbitrarily long chains of intermediate ideals corresponding to a single ideal of R. Also there exist d weakly distributive d. g. near-rings R of d weak distributivity class 2 with an arbitrary lattice of intermediate ideals corresponding to a single ideal of R. We refer ahead to deﬁnition 7.3 for an accouont of d weak distributivity. For now it suﬃces to say that these near-rings are very close indeed to being rings. We now turn to our main theme.

4.

Ideals in group near-rings.

In this section we are concerned with the relationship between ideals in a near-ring R and those in the group near-ring R[G]. In some respects it parallels the relationship we have seen exists in the case of matrix nearrings. But it is in fact much more complicated, as might be expected since the relationship in the ring case is more involved for group rings than for matrix rings. See le Riche, Meldrum and van der Walt [89], Meldrum and Meyer [98], Meyer [04] and Meldrum and Meyer [xx] as well as the papers by Fray. We start with those aspects which mirror the matrix near-ring case. Deﬁnition 4.1. Let R[G] be a group near-ring. Let I be an ideal of R and I an ideal of R[G]. We deﬁne I + = Id[a, e]; a ∈ I, I ∗ = {A ∈ R[G]; ARG ⊆ I G },

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I∗ = {r; r = (Aη)(e) for someA ∈ I}, where η is deﬁned by η(e) = 1, η(g) = 0 for all g = e. Note the parallel with deﬁnition 3.1. We also have a parallel version of theorem 3.2. Theorem 4.2. Let R[G] be a group near-ring. Let I be an ideal of R and I an ideal of R[G]. Then we have I ∗ is an ideal of R[G]; I∗ is an ideal of R; I + ⊆ I ∗; (I + )∗ = I = (I ∗ )∗ ; I ⊆ (I I∗ )∗ . The maps I → I + and I → I ∗ are injective and order preserving, and the map I → I∗ is surjective and order preserving. Note the one result from theorem 3.2 that is missing here, namely (I I∗ )+ ⊆ I. As we will see later, this result does not hold for a whole family of ideals in R[G], ideals which even in the case of group rings behave diﬀerently from the ideals of matrix rings. In the next result we list some more properties of ideals in group near-rings which can be easily veriﬁed. Theorem 4.3. Let I and J be ideals of R, and let R[G] be a group near-ring. Then (I ∩ J)∗ = I ∗ ∩ J ∗ , I ∗ + J ∗ ⊆ (I + J)∗ , (I + J)+ = I + + J + , (I ∩ J)+ ⊆ I + ∩ J + . As with matrix near-rings, we have a class of ideals in R[G] lying between I + and I ∗ for ideals I of R. We look at these ﬁrst. Deﬁnition 4.4. An intermediate ideal K of R[G] is an ideal K such that I + ⊂ K ⊂ I ∗ for some ideal I of R. Only part of theorem 3.4 will be valid in the case of group near-rings. Theorem 4.5. Let K be an intermediate ideal of R[G]. Then there is a unique ideal I of R such that I + ⊂ K ⊂ I ∗, and K cannot be an ideal of the form J + or J ∗ for any ideal J of R.

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While I is unique as deﬁned in theorem 4.5, it is an open question as to whether I = K∗ in all cases. These lapses from the satisfying results in matrix near-rings are mainly due to the existence of ideals in R[G] which are not of the form I + or I ∗ for any ideal I of R, nor indeed are they intermediate. Typical such ideals are those linked to subgroups of G, the prime example being the augmentation ideal. These ideals will be treated in some detail in the next section. For the present we will pursue the type of ideals we have just deﬁned. Deﬁnition 4.6. Let w = w(x1 , . . . , xn ) be a word in the free additive group on the set X = {xi ; i ≥ 1}. For a group (G, +) the word subgroup of G determined by w is deﬁned by w(G) = Gpw(g1 , . . . , gn ); gi ∈ G, 1 ≤ i ≤ n. Examples of word subgroups are given by the terms of the derived series of a group and the terms of its lower central series. To be more precise we deﬁne γ2 = δ1 = [x1 , x2 ] = −x1 − x2 + x1 + x2 , δn = [δn−1 , δn−1 ], γn+1 = [γ γn , xn+1 ], and then δn (G) is the nth term of the derived series of G, γn+1 (G) is the nth term of the lower central series of G. We are restricting ourselves in the next result to d. g. near-rings, that is near-rings generated as additive groups by a multiplicative semigroup of elements which all satisfy both distributive laws. One advantage of d. g. near-rings is that in some minor ways they are closer to rings than more general near-rings. A fairly old and well known result is the following. Lemma 4.7. Let w = w(x1 , . . . , xn ) be a word and let R be a d. g. near-ring. Then w(R) is an ideal of R. This enables us to state a result about ideals in group near-rings. Theorem 4.8. Let R be a d. g. near-ring, w = w(x1 , . . . , xn ) be a word and G a multiplicative group. Then w(R)+ ⊆ w(R[G]) ⊆ w(R)∗ .

The nearer we get to rings, the better the situation becomes.

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Theorem 4.9. Let R be a d. g. near-ring, G a multiplicative group, and let I be an ideal of R. If I + ⊇ δ1 (R[G]), then I + = I ∗ . From the deﬁnition of δ1 , we know that R[G]/δ1 (R[G]) has an abelian group structure. Together with the fact that R is d. g., and hence that R[G] is d. g., we can deduce that R[G]/δ1 (R[G]) is a ring. The result then follows from group ring theory. However, as soon as we move away from rings, matters are not so simple. Example 4.10. There exists a d. g. near-ring with identity of d-weak distributivity class 2 such that (δ1 (R))+ ⊂ δ1 (R[G]). Once again we refer ahead to deﬁnition 7.3 for the concept of d-weak distributivity. But in zero-symmetric near-rings which are not necessarily d. g., but which have abelian group structure, intermediate ideals can abound. Example 4.11. There exists a zero-symmetric near-ring of polynomials over x, with addition and composition of functions as operations, with zero constant term, denoted Z0 [x]. Take R to be the set of polynomials such that the coeﬃcients of x2 , . . . , x2n−1 are all 0, and let mR be the subnear-ring with all coeﬃcients divisible by m. Let m = 2n , A = mR, A an ideal of R. Then there is a chain of n − 2 ideals strictly between A+ and A∗ . This result, due to Meyer [04], needs careful analysis of the structure of these near-rings. Compare this with theorem 3.5. This result matches the ﬁrst part of that theorem. The second part of theorem 3.5 has not yet been extended to group near-rings and so constitutes another open problem.

5.

Exceptional ideals.

We now consider the ideals arising in group near-rings which are neither of the form I + , nor I ∗ for some ideal I of R, nor intermediate. The examples we have are of ideals associated with subgroups of the multiplicative group G. It is an open question whether these are the only exceptional ideals. The answer is probably not. Deﬁnition 5.1. An ideal I of R[G] which is not of the form I + or I ∗ for some ideal I of R, and is not an intermediate ideal is called an exceptional ideal.

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That these ideals exist and that their existence should not come as a surprise follows from the next example which shows that we can ﬁnd them even in a commutative group ring. This result is due to Meyer [04]. Example 5.2. Let R be a commutative ring, G an abelian group with an element of order 2. Let A = [1, e] + [1, g] ∈ R[G], and let I be the ideal of R[G] generated by A. Then I ⊂ R[G] but (I I∗ )+ = R[G]. It is instructive to see what is happening, so we give details here. Consider (Aη)(e) = (([1, e] + [1, g])η)(e) = η(e) + η(g) = 1. I∗ )+ = R[G]. It remains to show So 1 ∈ I∗ and I∗ must be R, forcing (I G that I ⊂ R[G]. Let δ ∈ R where δ(e) = 1, δ(g) = −1 and δ(h) = 0 for all h ∈ {e, g}. Then Aδ(h) = = = =

(([1, e] + [1, g])δ)(h) δ(h) + δ(hg) 1 − 1 = 0, if h = e, 0 + 0 = 0 if h = e.

So A ∈ AnnR[G] (δ), forcing I ⊆ AnnR[G] (δ). Note that because R[G] is a commutative ring, since R and G are both commutative, we can deduce that AnnR[G] (δ) is not just a left ideal, but is a two sided ideal. Now AnnR[G] (δ) ⊂ R[G] since R[G] contains an identity, [1, e], which does not annihilate δ. Thus I ⊂ R[G]. In particular we can say that (I I∗ )+ ⊆ I showing that the missing part of theorem 3.2 can never be added to theorem 4.2. We now look at a family of ideals of R[G] arising from subgroups of G. These are going to provide us with numerous exceptional ideals, as well as being of interest in their own right. Let H be a subgroup of G. An element µ ∈ RG will be called an H function if µ is constant on right cosets of H, that is Hx = Hy ⇒ µ(x) = µ(y). G . From le Riche, Meldrum and Denote the set of all H functions by RH van der Walt [89] we have the following results. G is an R[G] submodule of the left R[G] module RG . Lemma 5.3. RH

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G by ω(H). Then ω(H) is a two sided ideal of Denote the AnnR[G] RH G . When H is a R[G], the kernel of the representation of R[G] on RH normal subgroup of G we have a more satisfying result.

Theorem 5.4. Let H be a normal subgroup of G. Then R[G]/ω(H) ∼ = R[G/H]. There is one particular example of this family of ideals which is of special interest and merits a deﬁnition of its own, as we will be considering it in some detail later. Deﬁnition 5.5. The ideal ω(G) is called the augmentation ideal of R[G] and will be denoted ∆. There has been a great deal of interest in, and work done on, the augmentation ideal in the case of group rings. It should prove of equal interest and importance in the theory of group near-rings. We present here a few results from le Riche, Meldrum and van der Walt [89] on this topic. Deﬁne the augmentation map f∆ : R[G] → R as follows. For r ∈ R deﬁne εr ∈ RG by εr (g) = r for all g ∈ G. Then it can be shown that for A ∈ R[G] there exists a unique a ∈ R such that Aεr = εar for all r ∈ R. Deﬁne f∆ (A) = a. With a little bit more work we get the following result. Theorem 5.6. The augmentation map f∆ : R[G] → R is a near-ring epimorphism with kernel ∆. There is a convenient set of generators for ∆. Theorem 5.7. The augmentation ideal is generated as an ideal by the set {[1, g] − [1, e]; g ∈ G}. There are two further characterizations of ∆, not available in the original paper, but which are straightforward consequences of the work done there. Theorem 5.8. The augmentation ideal is generated as an ideal by the set of elements {[1, g] − [1, h]; g, h ∈ G} and can also be characterized, when R is d. g., as n n ri = 0, ri ∈ R, gi ∈ G, 1 ≤ i ≤ n}. ∆ = { [ri , gi ]; i=1

i=1

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Before we look at the connection with exceptional ideals, we state one more result from the original paper. Theorem 5.9. The map ω from the lattice of subgroups of G into the lattice of ideals of R[G] is injective, isotone and ω(H ∩K) ⊆ ω(H)∩ω(K) for subgroups H and K of G. We will now consider whether some of these ideals may be exceptional. We start by looking at ω(H) a bit more closely. Lemma 5.10. Let H be a subgroup of G. If k ∈ CorH, the largest normal subgroup of G contained in H, then [1, k] − [1, e] ∈ ω(H). From the deﬁnition of the action in the right regular representation of G on the right cosets of H, we have Hxk = Hx if and only if xkx−1 ∈ H. This holds for all right cosets of H if and only if k ∈ CorH which can be deﬁned as CorH = ∩x∈G x−1 Hx. G , we have So Hxk = Hx for all x ∈ G, k ∈ CorH. Then for µ ∈ RH

(([1, k] − [1, e])µ)(g) = µ(gk) − µ(g) = 0 since Hg = Hgk forcing µ(gk) = µ(g). This shows that G . [1, k] − [1, e] ∈ ω(H) = AnnR[G] RH

From this result we can show easily that ω(H)∗ = R, the key to what we are looking for, so long as CorH is not trivial. Theorem 5.11. Let H be a subgroup of G which contains a non-trivial normal subgroup of G. Then ω(H)∗ = R and (ω(H)∗ )+ = R[G]. Consider η ∈ RG with η(e) = 1, η(g) = 0 for g = e. Then (([1, k] − [1, e])η)(e) = η(k) − η(e) = −1 where we take k = e, k ∈ CorH. Hence 1 ∈ ω(H)∗ and, since it is an ideal of R, we can deduce that ω(H)∗ = R, and, with very little further eﬀort, (ω(H)∗ )+ = R[G]. As an immediate corollary we can state the following result.

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Corollary 5.12. ∆∗ = R and (∆∗ )+ = R[G]. From theorems 5.4 and 5.6, we can see that ω(H) and, in particular, ∆, are proper ideals of R[G]. We will now see that any proper ideal I of R[G] which satisﬁes (I I∗ )+ = R[G] must be exceptional. Lemma 5.13. Let I be a proper ideal of R[G] such that (I I∗ )+ = R[G]. Then I is an exceptional ideal. I∗ )+ = Let I ⊆ J ∗ . Then by theorem 4.2 (J ∗ )∗ = J ⊇ I∗ = R, since (I + + ∗ forces I∗ = R by the injectivity of I → I . Hence J = J = R[G]. So if I is not exceptional then we have I ⊆ J ∗ and J + = J ∗ = R[G] and I = R[G], a contradiction. Hence I must be exceptional. R+

Corollary 5.14. Let H be a subgroup of G containing a non-trivial normal subgroup of G. Then ω(H) is an exceptional ideal of R[G]. In particular ∆ is an exceptional ideal. It is an open question as to whether there are exceptional ideals not of this form. We now come to our ﬁnal topic. We examine when the augmentation ideal is nilpotent, a question that has attracted a lot of interest in the case of group rings. The work involves two new strands. One strand is the theory of generalized distributivity in near-rings, for which see Meldrum [92] and Roberts [83]. The other strand is, perhaps rather surprisingly, the theory of wreath products of groups which exhibit a rather remarkable link with the theory of augmentation ideals in group near-rings. We will cover the necessary background in the next two sections before our ﬁnal section on nilpotent augmentation ideals.

6.

Wreath products of groups.

We will start with describing this method of group construction before showing how this area of group theory ties in with group near-rings. Let A and B be two groups with the group operation written multiplicatively. Write AB for the complete direct product of |B| copies of A indexed by the elements of B. We use A(B) for the restricted direct product of the copies of A indexed by the elements of B. So A(B) ⊆ AB and we have equality when B is ﬁnite. We deﬁne B as a group of automorphisms of AB by f b (b ) = f (b b−1 ) for all f ∈ AB , b, b ∈ B. In eﬀect the action of B is to permute the direct factors of AB . It is immediate that A(B) is invariant under the action of B. This provides the background for our deﬁnition.

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Deﬁnition 6.1. The semidirect product AB .B is called the complete standard wreath product of A by B, denoted A Wr B. The semidirect product A(B) .B is called the restricted standard wreath product of A by B, denoted A wr B. The primary reference for the theory of wreath products of groups is Meldrum [95]. As we will only be using this wreath product, the word standard will normally be omitted. The next step is to establish the link between wreath products of groups and group near-rings. We are going to show that there is a very strong relationship between the group near-ring R[G] and the wreath product (R, +) wr G. Beware: in the deﬁnition of (R, +) wr G we are mixing additive and multiplicative notation. We start with the embedding of G in R[G] (lemma 1.3) which we will denote by θ: θ(g) = [1, g]. Now R[G] is deﬁned as a subset of M0 (R(G) ), so the elements of R[G], and thus of θ(G), have a natural action on R(G) which is given by (θ(g))µ(h) = µ(hg) for µ ∈ R(G) , h ∈ G.

(6.2)

Turning our attention to (R, +) wr G, we have an action by conjugation of G embedded naturally in (R, +)(G) .G on (R, +)(G) . Writing g f for gf g −1 , this action is given by (g f )(h) = f (hg) for f ∈ (R, +)(G) , h ∈ G.

(6.3)

We are identifying G with its embedding in (R, +) wr G. The actions described in (6.2) and (6.3) depend on the fact that R[G] is deﬁned as a set of maps on R(G) and R(G) is embedded, as an additive group, in (R, +) wr G. This fact enables us to state the following result. Lemma 6.4. A subgroup S of (R, +)(G) which is normal in (R, +) wr G corresponds when considered as a subgroup of R(G) to a normal G module of R(G) , and is a Z[G] submodule of R(G) and conversely. Z[G] is here deﬁned as the subnear-ring of R[G] generated by {[1, g]; g ∈ G} and is in fact a ring, the group ring Zn [G] where n is the order of 1 ∈ R, and hence the exponent of R[G]. Corollary 6.5. A R[G] ideal J of R(G) corresponds to a normal subgroup J of (R, +) wr G. This is a key result in our later work. For the next result we need to introduce some notation. We refer back to the deﬁnition of commutators just after deﬁnition 4.6. Let J be a subgroup of G. Then [J, G] = Gp[j, g]; j ∈ J, g ∈ G,

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[J,n+1 G] = Gp[k, g]; k ∈ [J,n G], g ∈ G. Another notation for [J,n G] which is often used is [J,n G] = [J, G, . . . , G] with n factors G on the right hand side. We now use a result from Meldrum [95]. Lemma 6.6. The subgroup [J,n G] in (R, +) wr G is trivial for some positive integer n if and only if J is a p-group of ﬁnite exponent for some prime p and G is a ﬁnite p-group for the same prime p. This result is an easy consequence of lemma 4.1.11 of Meldrum[95]. We can leave wreath products there and go to generalized distributive near-rings.

7.

Generalized distributive near-rings.

We are going to consider zero-symmetric near-rings and extend the idea of distributivity. It is well-known, and easy to prove, that a distributive near-ring, that is a near-ring that satisﬁes both distributive laws, is very like a ring in the sense that R/δ1 (R) is a ring and δ1 (R).R = R.δ1 (R) = {0}. The idea of distributivity can be generalized and there are two useful generalizations, much as in the same way that there are two useful generalizations of commutativity in group theory, namely solubility and nilpotence. These generalizations are based on two series we have already met just after deﬁnition 4.6. There are two corresponding series for generalized distributivity in near-rings. As a reference we quote Meldrum [92] and the references given there. Deﬁnition 7.1. Let R be a near-ring. The distributor ideal of R relative to an ideal J of R is deﬁned by D(R, J) = Ida(b + c) − ac − ab; a ∈ R, b, c ∈ J. Write D(R) for D(R, R). We also deﬁne d(R, R, J) = Ida(b + c) − ac − ab; a, b ∈ R, c ∈ J.

The point of these deﬁnitions is given in the next result. Lemma 7.2. Let R be a near-ring, J and K ideals of R with K ⊆ J. Then R/K distributes over sums of elements of J/K if and only if K ⊇ D(R, J).

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The proof is trivial. These two deﬁnitions of distributor ideals give rise to two series of ideals in a near-ring and to two deﬁnitions of generalized distributivity for near-rings. Deﬁnition 7.3. We deﬁne two distributor series for a near-ring R as follows. The distributor series of a near-ring R is given by D0 (R) = R, Dn+1 (R) = D(R, Dn (R)). The d-distributor series of R is given by d1 (R) = R, dn+1 (R) = d(R, R, dn (R)). We say that R is weakly distributive if Dn (R) = {0} for some n, and is d- weakly distributive if dn+1 (R) = {0} for some n. The least such n is called the weak distributivity (d weak distributivity) class of R. In the case of d. g. near-rings, the structure of these series is much easier to determine and we obtain the following very interesting result. Theorem 7.4. Let R be a d. g. near-ring with identity Then Di (R) = δi (R), di (R) = γi (R).

So in a d.g. near-ring with identity, weak distributivity of R is very closely linked to solubility of (R, +), and d weak distributivity of R is very closely linked to nilpotence of R. In particular the d. g. near-rings of d weak distributivity class 2 mentioned in theorem 3.5 and example 4.10 have additive group nilpotent of class 2, very close to being abelian and hence a ring, since abelian d. g. near-rings are rings. We now have the background needed to present our last set of results.

8.

The nilpotence of augmentation ideals.

We wish to establish necessary and suﬃcient conditions for the augmentation ideal to be nilpotent. We can do this in all but a very special case. The method hinges on examining carefully how ∆ acts on R(G) and using the link deﬁned in section 6. We start by looking at how the link enables us to use results from the theory of wreath products of groups to tell us what happens to ∆.

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Proposition 8.1. Let J be an R[G] ideal of R(G) . Using the link deﬁned in section 6 between R(G) and (R, +)(G) ⊆ (R, +) wr G, we have ∆J ⊇ [J, G] = [G, J]. Here ∆J is deﬁned by ∆J = Idδj; δ ∈ ∆, j ∈ J and is an R[G] ideal of R(G) , and [J, G] = Gp[j, g]; j ∈ J, g ∈ G. Since [j, g] = −[g, j] we have [J, G] = [G, J] and we quote a result of P. Hall which states that if G = GpX, Y , then [X, Y ] = Gp[x, y]; x ∈ X, y ∈ Y is normal in G. This result can be easily obtained from the commutator identities available in Hall [33]. An easy extension of this result shows that if J is normal in (R, +) wr G then [J, G] is also normal in (R, +) wrG. The key part of the proof relies on the fact that [1, g]j ↔ g j = j + [j, −g]. The elements on the left are elements in ∆J, while the elements on the right form a set of generators of [J, G], as j runs through J and g runs through G. Continuing to examine ∆J, we obtain another useful result. Proposition 8.2. Let J be an R[G] ideal of R(G) of the form K (G) , where K is an R ideal of R R. Then ∆J ⊇ δ1 (K)(G) . The proof of this result is quite straightforward and is the group nearring version of a well-known technique from the theory of wreath products of groups. Write ae , be for two elements from K (G) where a, b ∈ K and the only non-identity element is in the factor indexed by e. Then by considering ([1, g] − [1, e])ae + ([1, g] − [1, e])be − ([1, g] − [1, e])(a + b)e we obtain the result. These two results are the key to the group theoretic aspects of the problem. There is a distributivity aspect also, and, as we see now, we have to add a condition on the group G. Proposition 8.3. Assume that exp G = 2. Let J be an R[G] ideal of R(G) of the form J = K (G) , where K is an R ideal of R R. Then ∆J ⊇ D(R, K)(G) .

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We ﬁrst point out that the restriction on G is very weak. All we require is that G contains a non-identity element not of order 2, i. e. there exists g ∈ G such that g −1 = g = e. We will not go into any details of the proof which is fairly long. It involves considering suitable combinations of elements similar to those considered in the proof of the previous result. For those who are interested in the details we refer to Meldrum and Meyer [xx]. Combining two results we have already stated, namely propositions 8.1 and 8.3, we can state the following theorem. Theorem 8.4. Assume that exp G = 2. Then ∆R(G) = [R(G) , G] + D(R)(G) . This provides the ﬁrst step for what will be an induction argument to determine necessary and suﬃcient conditions for ∆n R(G) to be trivial, and hence for ∆n = {0}. To take the next step we need the following. Proposition 8.5. Let J be an R[G] ideal of R(G) . Then ∆n J ⊇ [J,n G]. This can be proved by induction from proposition 8.1. We can now appeal to lemma 6.6 to obtain the ﬁrst stage in our objective, using proposition 8.5 with J replaced by R(G) . Theorem 8.6. Let ∆ be the augmentation ideal of R[G]. If ∆ is nilpotent then R has characteristic a power of a prime p and G is a ﬁnite p group for the same prime p. This result can be obtained without using the theory of wreath products of groups. But then we would have to mirror the proof of lemma 6.6 in the context of group near-rings, and this proof is not that short. The next stage is to look at generalized distributivity properties. We ﬁrst quote a result of Fray [97a] which is needed before we can apply proposition 8.3. Lemma 8.7. Let R[G] be a group near-ring. If K is a left ideal of R, then K (G) is an R[G] ideal of R(G) . We can now obtain a special case of proposition 8.3 which forms a key step in our argument. Lemma 8.8. Assume that exp G = 2. Then ∆Dn (R)(G) ⊇ Dn+1 (R)(G) .

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Again we apply a straightforward induction argument to this result to provide the next necessary condition for ∆ to be nilpotent. Theorem 8.9 Assume that exp G = 2 and that ∆ is nilpotent. Then R is weakly distributive. The induction argument using lemma 8.8 shows that ∆n R(G) ⊇ Dn (R)(G) . The result then follows immediately. So far we have three necessary conditions for the augmentation ideal ∆ of the group near-ring R[G] to be nilpotent under the assumption that exp G = 2: (1) R has characteristic pn for some prime p by theorem 8.6; (2) G is a ﬁnite p group for the same prime p, by theorem 8.6; (3) R is weakly distributive by theorem 8.9. We ﬁnish by showing that these conditions are also suﬃcient. The method we use is to take advantage of the weak distributivity in order to reduce the problem to the case of a group ring, applying induction to complete the process. So the ﬁrst result we quote is from Meldrum [76] (it was probably known before that but I have not been able to pin down a reference). Theorem 8.10. Let R be a ring, G a group. Then the augmentation ideal of R[G] is nilpotent if and only if R has characteristic a power of a prime p and G is a ﬁnite p group for the same prime p. Notice that these are conditions (1) - (3) for group rings and that the restriction on the exponent of G is not needed. That is expected because the restricition was only needed for the distributivity properties of the near-ring. The result which enables us to apply the group ring case, theorem 8.10, in a form which lends itself to an induction argument is the following. Lemma 8.11. Consider the action of R[G] on (Dm (R)/Dm+1 (R))(G) . Then the action is a group ring action by (R/D(R))[G]. We use the fact that R acts distributively on Dm (R)/Dm+1 (R), hence acts in the same way as R/D(R), which is a ring, being a distributive near-ring with identity. It is then easy to conclude that the action of R[G] on (Dm (R)/Dm+1 (R))(G) is that of (R/D(R))[G] and hence is a group ring action. This result enables us to obtain a result relating powers of the augmentation ideal to terms of the weak diatributivity series.

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Lemma 8.12. Let R be a near-ring satisfying condition (1) and G is a group satisfying condition (2). For each positive integer m, there exists a positive integer rm such that ∆rm Dm (R)(G) ⊆ Dm+1 (R)(G) . Using lemma 8.11 the result follows with rm the power of the augmentation ideal of (R/D(R))[G] acting on (Dm (R)/Dm+1 (G))(G) which lies in the kernel of the action. By theorem 8.10 such an rm exists and is less than or equal to the nilpotency class of the augmentation ideal of (R/D(R))[G] which exists since conditions (1) and (2) are satisﬁed. We come to the ﬁnal theorem. Theorem 8.13. Assume that exp G = 2. Then the augmentation ideal of the group near-ring R[G] is nilpotent if and only if the following three conditions are satisﬁed: (1) R has characteristic pn for some prime p; (2) G is a ﬁnite p group for the same prime p; (3) R is weakly distributive. Theorems 8.6 and 8.9 prove the necessity. The suﬃciency follows quickly from lemmas 8.11 and 8.12. The restriction on G, which amounts to saying that G is not an elementary abelian 2-group, is somewhat disappointing. It is only needed to prove that the distributivity condition (condition (3)) is necessary. It is not needed for the proof of suﬃciency and it is very likely that it can be dispensed with. This is the ﬁnal open question that we will mention.

References Bagley, S. W. [93] Polynomial near-rings, distributor and J2 -ideals of a generalized centralizer near-ring, Ph. D. Thesis, Texas A & M University. Bagley, S. W. [97]. Polynomial near-rings: Polynomials with coeﬃcients from a near-ring. In Saad, G. and Thomsen, M. J. (eds) Nearrings, Nearﬁelds and K-loops. Kluwer, Dordrecht, 179-190. Clay, J. R. [92] Nearrings. Geneses and Applications. Oxford University Press. Oxford. Cotti-Ferrero, C. and Ferrero, G. [02] Nearrings. Some developments linked to Semigroups and Groups. Kluwer. Advances in Mathematics. Dordrecht. Farag, M. [01] A new generalization of the center of a near-ring with applications to polynomial near-rings. Comm. Algebra 29 (6), 2377-2387. Farag, M. [02] Ideals in polynomial near-rings. Algebra Colloq. 9 (2), 219-232. Fray, R. L. [88] Ph. D. Thesis, University of Stellenbosch. Fray, R. L. [92a] On group near-ring modules. Quaestiones Math. 15, 213-223.

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Fray, R. L. [92b] On a relationship between group and matrix near- rings. Quaestiones Math. 15, 225-231. Fray, R. L. [92c] On group distributively generated near-rings. J. Australian Math. Soc. (Series A). 52, 40-56. Fray, R. L. [95] On suﬃcient conditions for near-rings to be isomorphic. In Mason G. et al (eds), Near-rings and Near-ﬁelds. Kluwer, Dordrecht. Fray, R. L. [97a] On ideals in group near-rings. Acta Math. Hungar. 74, 155-165. Fray, R. L. [97b] On direct decompositions in group near-rings. In Saad, G. and Thomsen M. J., (eds), Nearrings, Nearﬁelds and K-Loops. Kluwer, Dordrecht. Fray, R. L. [00] A note on pseudo-distributivity in group near-rings. In Fong, Y. et al (eds), Near-rings and Near-ﬁelds. Kluwer, Dordrecht. Hall, P. [33] A contribution to the theory of groups of prime power order. Proc. London Math. Soc. 36, 29-95. Lausch, H. and N¨ ¨ obauer, W. [73] Algebra of polynomials. North Holland/Elsevier, Amsterdam. le Riche, L. R., Meldrum, J. D. P. and van der Walt, A. P. J. [89] Group near-rings. Arch. Math. 52, 132-139. Meldrum, J. D. P. [76] The group distributively generated near-ring. Proc. London Math. Soc. (3) 32, 323-346. Meldrum, J. D. P. [85] Near-rings and their links with groups. Pitman, Research Notes in Mathematics. 134, London. Meldrum, J. D. P. [92] Les g´ ´en´eralizations de la distributivit´e dans les presqueanneaux. Rend. Sem. Mat. e Fis. Milano LIX (1989), 9-24. Meldrum, J. D. P. [95] Wreath products of groups and semigroups. Longman, Harlow, England. Meldrum, J. D. P. and Meyer, J. H. [96] Intermediate ideals in matrix nearrings. Comm. Alg. 24 (5), 1601-1619. Meldrum, J. D. P. and Meyer, J. H. [98] Word ideals in group near- rings. Algebra Colloquium, 5 (4), 409-416. Meldrum, J. D. P. and Meyer, J. H. [xx] The augmentation ideal in group near-rings. Submitted. Meldrum, J. D. P. and van der Walt [86] Matrix near-rings. Arch. Math. 47, 312-319. Meyer, J. H. [01] On the development of matrix near-rings and related near-rings over the past decade. In Fong, Y. et al (eds), Near-rings and Near-ﬁelds, Kluwer, Dordrecht, 23-34. Meyer, J. H. [04] Two sided ideals in group near-rings. J. Austral. Math. Soc. (Series A), to appear. Pilz, G. [83] Near-rings, 2nd Edition. North Holland/Elsevier, Amsterdam. Roberts, I. [83] Generalized distributive near-rings. M. Phil. thesis, University of Edinburgh.

LOOP-NEARRINGS Silvia Pianta Dipartimento di Matematica e Fisica, Universit` a Cattolica, Via Trieste 17, I-25121 Brescia [email protected]

Introduction The notion of nearring, due to H.Zassenhaus (1936, see [23]), can be considered as a generalization of the notion of nearﬁeld. In fact, according to the more recent deﬁnitions (see e.g. [4], [5], [21]), a nearring is a triple (N, +, ·) where N is a non empty set, “+” and “ · ” are binary operations on N such that (N, +) is a group with identity 0 (not necessarily abelian), (N, ·) is a semigroup and one of the two distributive laws of multiplication with respect to addition is satisﬁed. A nearﬁeld is a nearring (N, +, ·) such that (N \ {0} , ·) is a group. Nearﬁelds attained particular relevance in two main areas of investigation. The former area is group theory, dealing with sharply 2-transitive permutation groups: it is well known (see, e.g., [22]) that, if (N, +, ·) is a nearﬁeld and T2 (N ) := {ττa,b : N → N, x → a + bx | a ∈ N, b ∈ N ∗ } is the aﬃne group of N , then T2 (N ) acts sharply 2-transitively on N ; conversely, if G is a sharply 2-transitive permutation group acting on a ﬁnite set N , then N can be provided with the structure of a nearﬁeld (N, +, ·) such that G = T2 (N ). But if the set N is inﬁnite, then the additive structure of (N, +, ·), constructed with the same procedure as in the ﬁnite case, fails to be a group in general (it loses the associative law), and what one gets is a weaker structure, called neardomain (introduced by H.Karzel in 1968, [14]). It is still an open problem whether it is possible to ﬁnd any example of a proper neardomain which is not actually a nearﬁeld. Neardomains result to be special instances of left neoﬁelds, introduced and studied from both algebraic and geometric point of view by D.F.Hsu and A.D. Keedwell in 1984 (see [13]) in order to generalize the older structure of neoﬁeld (N, +, ·), due to L.J.Paige (1949, see [20]), where (N, +) is a loop, (N \ {0} , ·) is a group and both distributive laws hold 57 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 57–67. c 2005 Springer. Printed in the Neatherlands.

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true; a left neoﬁeld is required to fulﬁl only the left distributivity. A left neoﬁeld is a generalization of a nearﬁeld: one relaxes the associativity for the additive structure. Also a nearring is a generalization of a nearﬁeld: one relaxes the group property for the multiplicative structure. Well, why not consider both ways of generalizing a nearﬁeld? We will just do that, introducing the notion of loop-nearring. The latter area of interest for nearﬁelds is geometry, precisely the study of coordinate structures of non desarguesian projective planes: for this purpose nearﬁelds require an additional “key-property”, that is planarity. In this work we want to go through with this concept, that was introduced by J.Clay in 1967 also in the more general nearring structure (see [1], [2]) and intensively studied by G.Ferrero (e.g. [9], [10]), who gave a standard method to construct planar nearrings from Frobenius groups and showed how to obtain diﬀerent classes of designs from ﬁnite planar nearrings. Starting from here, in §2 we extend the deﬁnition of planarity to the loop-nearring case and moreover we show how this notion can be weakened in order to obtain a sort of “graduation” of planarity into diﬀerent steps (0-planarity, semiplanarity, full planarity), each of them being realized in some concrete examples (see §3). Finally, just as planar nearrings can be constructed starting from Ferrero pairs, i.e. pairs of groups (N, Φ), where Φ is a non trivial group of uniform 1 and ﬁxed-point-free automorphisms of the group N , so by a similar construction 0-planar, semiplanar and planar loop-nearrings correspond respectively to weak-, semi- and full l-Ferrero pairs, i.e. pairs (N, Φ), where N is, more generally, a loop, and Φ is a non trivial group of automorphisms of N admitting at least one regular orbit on N in the weak case; the automorphisms of Φ are, in addition, ﬁxed-point-free (or a bit more) in the semi-l-Ferrero pairs and also uniform in the full l-Ferrero pairs.

1.

Preliminary deﬁnitions

Let N be a non empty set, furnished with two binary operations “+” and “ · ”. We call (N, +, ·) a loop-nearring if:

1A

loop (or group) automorphism ϕ : N → N is said to be uniform if for all c ∈ N the mapping c+ ϕ : N → N ; x → c + ϕ (x) admits a ﬁxed point. If (N, +) is a group, this condition is equivalent to saying that the map ϕ − 1 : N → N ; x → ϕ (x) − x is surjective (see [6]).

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L1. (N, +) is a loop, i.e. ∀a, b ∈ N ∃1 (x, y) ∈ N × N such that a + x = b = y + a and there is a neutral element 0 ∈ N such that ∀a ∈ N : a + 0 = 0 + a = a, L2. (N, ·) is a semigroup, i.e. “ · ” is associative, L3. One of the two distributive laws holds, assume ∀a, b, c ∈ N : a (b + c) = ab + ac. In any loop (N, +) consider the maps a+ : N → N ; x → a + x, called left translations, or left additions, for all a ∈ N (in the literature also denoted by λa or L(a)). All these maps are bijective, hence N + := {a+ | a ∈ N } generates a group, the left translation group Tl :=< a+ | a ∈ N >, which is a subgroup of the whole permutation group SymN . Moreover, for all a, b ∈ N , the precession maps

−1 + + a b δa,b := (a + b)+

( = λ−1 a+b λa λb )

are permutations ﬁxing the element 0, and their algebraic meaning is cleared up by the identity a + (b + x) = (a + b) + δa,b (x), which follows directly from the deﬁnition of δa,b . We shall denote by ∆ :=< δa,b | a, b ∈ N > the subgroup of Tl generated by the precession maps, called left inner mapping group (or structure group) of N . Now, in a loop-nearring (N, +, ·) another signiﬁcant mapping set can be regarded: denoting by a· : N → N ; x → a · x the left multiplication by a, for all a ∈ N , the left distributive law (L3) entails that N · := {a· | a ∈ N } is a subsemigroup of End (N, +) := {ϕ : N → N | ϕ (x + y) = ϕ (x) + ϕ (y) , ∀x, y ∈ N } , the endomorphism semigroup of (N, +). In general, we can say that whenever in a loop (N, +) there is deﬁned a map · : N → End (N, +) ; a → a· such that (a· (b))· = a· b· , then one gets a loop-nearring (N, +, ·), where a · b := a· (b) for all a, b ∈ N . Here we want to remark that, if in a loop-nearring (N, +, ·) the loop (N, +) admits a unique inverse to each a ∈ N , i.e. a + b = 0 implies b+a = 0 for all a, b ∈ N , and the precession maps are left multiplications, i.e. {δa,b | a, b ∈ N } ⊆ N · , then (N, +, ·) becomes a complete K-nearring in the sense of Karzel (cf. [15]), which, in turn, becomes a (complete) neardomain exactly when (N \ {0} , ·) is a group; ﬁnally, a neardomain is a nearﬁeld if and only if (N, +) is a group.

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Planarity and its graduation

In a loop-nearring (N, +, ·) the left multiplications are endomorphisms of (N, +) hence a · 0 = a· (0) = 0 holds true for all a ∈ N . On the other hand one cannot say anything, in general, about the products 0 · a, so just as in nearring theory, the loop-nearring is called 0-symmetric if 0· is the 0-endomorphism, i.e. if 0 · a = 0 for all a ∈ N . Now, in order to copy the deﬁnition of planar nearring there are two possibilities: a loop-nearring (N, +, ·) such that |N · | ≥ 3 is called planar on the right, resp. left, side if the following property (P r), resp. (P l), is satisﬁed: (P r) ∀a, b, c ∈ N with a· = b· ∃1 x ∈ N : a · x = b · x + c (P l) ∀a, b, c ∈ N with a· = b· ∃1 x ∈ N : a · x = c + b · x. In the sequel we shall regard to the left side planarity and simply call planar a loop-nearring which satisﬁes (P Pl ).

2.1

0-planarity

It is an easy exercise to check that any loop-nearring (N, +, ·) which is planar on the right or on the left side has all properties stated e.g. by J.Clay in the “main structure theorem” of planar nearrings (cf. [4], (4.9)), in particular it is 0-symmetric. Since the proofs of such properties make use of the planarity equation not in its complete form, but just in the form a · x = c + 0 · x, the idea is to start from such simpler equation in order to introduce a notion which is weaker than planarity, but yields the same structure as described in the cited theorem. We proceed as follows. Given a loop-nearring (N, +, ·), for a, b, c ∈ N with a· = b· let us deﬁne Sa,b,c := {x ∈ N | a · x = c + b · x}. (2.1) The loop-nearring (N, +, ·) is 0-symmetric if and only if for all Sa,0,0 | = 1 (i.e. the equation a · x = 0 · x admits a a ∈ N with a· = 0· : |S unique solution). Proof. From a · 0 = 0 = 0 · 0 and a · (0 · x) = (a · 0) · x = 0 · x = (0 · 0) · x = 0 · (0 · x) it follows that |S Sa,0,0 | = 1 if and only if 0 · x = 0 for all x ∈ N . Denote by 1 the identity of N , by 0 the 0-endomorphism of (N, +) and by A := {a ∈ N | a· = 0} the set of 0-multipliers of (N, +), and set N ∗ := N \ A. (2.2) In a loop-nearring (N, +, ·), if (N ∗ )· ⊆ SymN then either 0· = 0 or 0· = 1.

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Proof. If 0· = 0 then 0 ∈ N ∗ hence 0· ∈ SymN and from 0 =· 0 · 0 = −1 · · · 0 (0) it follows 0 = (0 ) (0), that entails 0 = (0· )−1 (0) . But · · · 0· (0· )−1 (0) = 0· (0· )−1 (0) hence 1 = (0· )−1 (0) = 0· . (2.3) Let (N, +, ·) be a loop-nearring with |N · | ≥ 2. Then (N ∗ )· ⊆ SymN if and only if |S Sa,0,c | = 1 for all a, c ∈ N with a· = 0· . Proof. (⇐) By (2.1) 0· = 0 and this implies that ∀a ∈ N ∗ , ∀c ∈ N the equation a· (x) = c is equivalent to a · x = c + 0 = c + 0 · x, so for |Sa,0,c | = 1, ∃1 x ∈ N : a· (x) = c hence a· ∈ SymN . (⇒) Suppose 0· = 1. Then, for any a ∈ N , a· = a· 0· = (a· (0))· = (a · 0)· = 0· = 1, a contradiction to |N · | ≥ 2. Hence, by (2.2), 0· = 0, so (N, +, ·) is 0-symmetric and Sa,0,c is the set of solutions of the equation a · x = c, i.e. |S Sa,0,c | = 1 for all a, c ∈ N with a· = 0· . Let us say that a loop-nearring (N, +, ·) is 0-planar if |N · | ≥ 2 and Sa,0,c | = 1 holds. for all a, c ∈ N with a· = 0· the condition |S In particular, any 0-planar loop-nearring is 0-symmetric by (2.1). Moreover: (2.4) Let (N, +, ·) be a 0-planar loop-nearring and Φ := (N ∗ )· , then 1. Φ ≤ Aut (N, +); 2. ∀ϕ ∈ Φ \ {1} : F ix ϕ ∩ N ∗ = ∅; 2 3. Φ (A) = A and Φ (N ∗ ) = N ∗ . Proof. 1. By (2.3) we already know (N ∗ )· ⊆ Aut (N, +). Moreover Sa,0,a =: {e} implies a · e = a and a· e· = (a· (e))· = a· , that entails e· = 1 ∈ (N ∗ )· . Finally, let {b} := Sa,0,e , then a · b = e and e· = (a· (b))· = a· b· , that means (a· )−1 = b· ∈ (N ∗ )· . 2. For all a, b ∈ N ∗ , b = ϕ (b) := a· (b) implies a· b· = (a· (b))· = b· , hence, ϕ = a· = 1. 3. For a ∈ N ∗ , b ∈ A, (a· (b))· = a· b· = a· 0 = 0, hence for all ϕ ∈ Φ, ϕ (A) = A, and this implies also ϕ (N ∗ ) = N ∗ . Now we are ready to show the structure of 0-planar loop-nearrings: Structure Theorem. Let (N, +, ·) be a 0-planar loop-nearring, E := E {e ∈ N | e· = 1} and Φ := (N ∗ )· ≤ Aut (N, +) (by (2.4)), then ∅ = and E is a complete set of representatives for the orbits of Φ on N ∗ (i.e. 2 For

a permutation α ∈ SymN we denote by F ix α := {x ∈ N | α (x) = x}.

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N ∗ = ˙ e∈E Φ (e)) and Φ acts regularly on each of these orbits Φ (e) (i.e. Φ is semiregular on N ∗ ). Moreover, for each e ∈ E, (Φ (e) , ·) is a subgroup of the semigroup (N, ·), isomorphic to Φ. Proof. Let x ∈ N ∗ , hence x· ∈ Φ and if e := (x· )−1 (x) then x· (e) = x, so that x· = x· e· , i.e. e· = 1, hence e ∈ E and N ∗ ⊆ Φ (E). Conversely, let, x ∈ Φ (E), hence x = ϕ (e) for some e ∈ E and ϕ ∈ Φ = (N ∗ )· . Then there is an n ∈ N ∗ with n· (e) = ϕ (e) = x, i.e. x· = (n· (e))· = n· e· = · ∗ ∗ ∗ n = 0, hence x ∈ N and we get Φ (E) = N . Thus N = Φ (E) = e∈E Φ (e). Let us prove that the latter union is disjoint: if f ∈ Φ (e) ∩ E then there is a b ∈ N ∗ with b · e = f ∈ E, hence 1 = f · = b· e· = b· implying f = e. By (2.4) Φ acts semiregularly on N ∗ , which means that the action of Φ on Φ (e) is regular for each e ∈ E, thus Φ induces on Φ (e) a group with identity e and, for each x ∈ Φ (e), x−1 is deﬁned −1 structure · (x) = e. by x Now, let (N, +) be a loop and {1} = Φ ≤ Aut (N, +). For any complete set T of orbit representatives of Φ on N (i.e. T is such that N = Φ (T ) and Φ (t) ∩ T = {t} for all t ∈ T ), denote by T ∗ := {t ∈ T | ∀ϕ ∈ Φ : ϕ (t) = t ⇒ ϕ = 1}. Since Φ is non trivial 0 ∈ / T ∗. Then we call (N, Φ) a weak l-Ferrero pair if for a (hence for any) set of orbit representatives T of Φ on N the condition T ∗ = ∅ holds; in other words, if (N, Φ) admits at least one regular orbit. By the previous theorem, if (N, +, ·) is a planar loop-nearring and Φ := (N ∗ )· then the set E can be extended to a complete set T of on orbit representatives

· ∗ ∗ the whole N such that T ⊇ E = ∅, hence (N, +) , (N ) is a weak l-Ferrero pair. Conversely, Theorem 1. Let (N, Φ) be a weak l-Ferrero pair, T a complete set of E ⊆ T ∗ . Deﬁne, orbit representatives of Φ on N with T ∗ = ∅ and ∅ = for each a ∈ N : 0 if a ∈ N \ Φ (E) · a := ϕ if a = ϕ (e) f or e ∈ E, ϕ ∈ Φ, and a · b := a· (b) for all a, b ∈ N . Then (N, +, ·) is a 0-planar loopnearring. Proof. Since Φ ≤ Aut (N, +) ⊆ End (N, +) and 0 ∈ End (N, +), the left distributive law holds in (N, +, ·). The associative law a·(b · c) = (a · b)·c is equivalent to a· b· = (a· (b))· . In order to prove this identity, ﬁrst note that 0 ∈ N \ Φ (E), hence 0· = 0.

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Now if a· = 0 then a· b· = 0b· = 0 and (a· (b))· = (0· (b))· = 0· = 0 for all b ∈ N . If b· = 0 then b ∈ N \ Φ (E) and so a· (b) ∈ N \ Φ (e) which implies · (a (b))· = 0 = a· 0 = a· b· . If a· , b· = 0 then a· , b· ∈ Φ and if e ∈ E is such that b = b· (e) then a· (b) = a· b· (e) ∈ Φ (e), thus (a· (b))· = a· b· . This shows that (N, +, ·) is a loop-nearring with A := N \ Φ (E), N ∗ := Φ (E) and (N ∗ )· = Φ ≤ Aut (N, +). By deﬁnition |N · | ≥ 2, so by (2.3) (N, +, ·) is 0-planar. Let us call “Ferrero derivation” the procedure, described in Theorem 1, to construct a (not uniquely determined) loop-nearring from a weak l-Ferrero pair (N, Φ). One sees immediately that this is the same procedure used in [2], [9], [10] to construct planar nearrings from Ferrero pairs: the main fact here is that the Ferrero derivation keeps on working under the weaker assumptions of Theorem 1, in which (N, +) is just a loop and Φ ≤ Aut (N, +) is only required to admit a regular orbit.

2.2

Planarity and semiplanarity

So far we have used planarity only under the restriction b = 0 or equivalently a = 0 (0-planarity) in the equation a · x = c + b · x. Therefore, we may assume now that, in the full planarity condition, a· is diﬀerent from 0· , i.e. by (2.3) that a· ∈ Aut (N, +), thus condition (P Pl ) becomes equivalent to: (P Pl ) ∀b, c ∈ N with b· = 1, |S1,b,c | = 1, i.e. |F ix (c+ b· ) | = 1. We call a loop N together with an automorphism group Φ ≤ AutN an l-Ferrero pair (N, Φ) if |Φ| ≥ 2 and |F ix (c+ ϕ) | = 1 for all c ∈ N and ϕ ∈ Φ \ {1}. In a loop (N, +) with the inverse property 3 (e.g. in a Moufang loop) it is straightforward to verify that, given a non trivial automorphism ϕ ∈ AutN \{1}, the mapping ϕ−1 : N → N ; x → ϕ (x)−x is injective if and only if |F ix (c+ ϕ) | ≤ 1, and surjective if and only if |F ix (c+ ϕ) | ≥ 1 hence bijective exactly when |F ix (c+ ϕ) | = 1, for all c ∈ N . 3 In a loop (N, +), for each a ∈ N , let −a be the unique solution of the equation a + x = 0. The left inverse property (L.i.p.) is expressed by the identity −a + (a + x) = x (for a, x ∈ N ). When this property holds, the right inverse −a is also the left inverse of a, i.e. −a + a = 0, so the loop has unique inverses, and one can express also the right inverse property (R.i.p.) by the identity (x + a)−a = x (for a, x ∈ N ). The loop is said to have the inverse property (I.p.) if both L.i.p. and R.i.p. hold. For further algebraic properties of loops and the deﬁnition of Moufang loops, see e.g. [3].

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Now if (N, Φ) is an l-Ferrero pair then all ϕ ∈ Φ \ {1} act ﬁxed-pointfree (i.e. F ix ϕ = {0}) on N , hence in this case, chosen any complete set T of orbit representatives of Φ on N , T ∗ = T \ {0} ⊆ N \ {0} and T ∗ = ∅ since |N | ≥ 2. Therefore any l-Ferrero pair (N, Φ) is a weak l-Ferrero pair as well, so by Theorem 1 the loop-nearring (N, +, ·) obtained by the Ferrero derivation is 0-planar and, indeed, planar because (P Pl ) is fulﬁlled. Thus we have proved the following theorem, that provides a direct generalization to the loop case of the classical procedure to construct planar nearrings starting from Ferrero pairs of groups: Theorem 2. If (N, Φ) is an l-Ferrero pair, then the “Ferrero derivation” of Theorem 1 gives rise to a planar loop-nearring. Let us consider the class of complete K-nearrings introduced in [15] by H.Karzel, i.e. loop-nearrings (N, +, ·) characterized by the properties: K1. (N, +) is a loop with unique inverses; K2. For all a, b ∈ N the precession map δa,b is a left multiplication, i.e. δa,b ∈ N · ∩ SymN . So the loop (N, +) is a so called weak K-loop (cf. e.g. [15], [18]). Here planarity becomes so strong that it forces the additive loop to be a group. In fact we can show the following Theorem 3. Any planar complete K-nearring is a nearring. Proof. First observe that ∀a ∈ N \{0} , a = a+(−a + a) = (a + (−a))+ δa,−a (a) = δa,−a (a), hence δa,−a = 1 since any non trivial left multiplication is ﬁxed-point-free by hypothesis. Pl ) Now assume δa,b = 1 for some a, b ∈ N. Since δa,b ∈ (N ∗ )· , by (P there exists an x such that x = (a + b) + δa,b (x) = a + (b + x); but we have also x = (a − a) + x = (a + (−a)) + δa,−a (x) = a + (−a + x), hence b + x = −a + x, that implies b = −a, so δa,b = δa,−a = 1, in contradiction with the original assumption. Owing to the previous observations, it is quite of interest to consider also a condition which, at least in the inﬁnite case, is a bit weaker than planarity. A loop-nearring (N, +, ·) is called semiplanar if it is 0-planar and |S1,b,c | ≤ 1, i.e. |F ix (c+ b· ) | ≤ 1, for all b, c ∈ N with b· = 1. One can construct also semiplanar loop-nearrings via the “Ferrero derivation” as before, starting now from semi-l-Ferrero pairs (N, Φ), where N is a loop and Φ ≤ AutN an automorphism group such that for all c ∈ N and ϕ ∈ Φ, ϕ = 1, |F ix (c+ ϕ) | ≤ 1.

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We remark that, if (N, Φ) is a semi-l-Ferrero pair then there is a Ferrero derivation (cf. Th. 1) such that for the obtained semiplanar loop-nearring (N, +, ·) we have A = {0}, hence N ∗ = N \ {0}. Note that |F ix (c+ ϕ) | ≤ 1 for all c ∈ N and ϕ = 1 always implies (set c = 0) that ϕ is ﬁxed-point-free, while the converse, that can be easily proved if N is a group, does not hold in general for a loop, even when the inverse property holds. In the following section we will show an example of a proper semiplanar loop-nearring, which is a complete K-nearring.

2.3

Examples

Many examples of 0-planar loop-nearrings can be constructed starting from weak l-Ferrero pairs. In particular we get properly 0-planar complete K-nearrings starting from pairs (K, Φ), where K is a K-loop (for the deﬁnition, see e.g. [18]) Φ a non trivial subgroup of the left inner mapping group ∆ and the right nucleus Nr := {a ∈ K | ∀x, y ∈ K : x + (y + a) = (x + y) + a} is non trivial and coincides with F ix δx,y for all x, y ∈ K (examples can be found e.g. in [11], [16], [19]). If we restrict to 0-planar nearrings, then a rich class of examples is furnished by those pairs (N, Φ) where N is a group and Φ ≤ AutN is such that for all ϕ, ϑ ∈ Φ, F ixϕ = F ixϑ ≤ N (expressed by condition (K)’ in [12]) as for instance when N is the additive group of a vector space and Φ is a subgroup of elations with a ﬁxed hyperplane as axis, or a subgroup of rotations with ﬁxed axis of a euclidean space N with dimN ≥ 3. Semiplanar loop-nearrings include inﬁnite neoﬁelds, left neoﬁelds, neardomains and nearﬁelds. An interesting semiplanar complete K-nearring can be constructed from the semi-l-Ferrero pair (K, ∆) where K is the K-loop of hyperbolic translations of the real hyperbolic plane (cf. e.g. [17]) and ∆ is the whole structure group of K, consisting of all hyperbolic rotations with centre the point 0 of K: it is straightforward to verify that for each a ∈ K and δ ∈ ∆ the isometry a+ δ has exactly one ﬁxed point except the case of “limit rotations” for which F ix (a+ δ) = ∅, hence the general condition |F ix (a+ δ) | ≤ 1 holds for all a ∈ K, δ ∈ ∆. Finally, any planar nearring or nearﬁeld, any ﬁnite (left) neoﬁeld can be regarded to as an instance of a planar loop-nearring.

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Acknowledgment The author in much indebted to prof. Helmut Karzel for the stimulating discussions and fruitful suggestions that contributed to give rise to the ideas developed in the present work.

References [1] M.Anshel, J.R.Clay: Planar algebraic systems: some geometric interpretations. J.Algebra 10 (1968), 166-173. [2] G.Betsch and J.R.Clay: Block designs from Frobenius groups and planar nearrings. Proceedings of the Conference on Group Theory, Park City, Utah, 1973. [3] R.H.Bruck: A survey on binary systems. Springer Verlag, Berlin 1966. [4] J.R.Clay: Nearrings. Geneses and applications. Oxford Univ. Press, New York 1992 [5] C.Cotti Ferrero, G.Ferrero: Nearrings. Some developments linked to semigroups and groups. Kluwer Academic Publ., Dordrecht 2002. [6] M.Curzio: Sugli automorﬁsmi uniformi nei gruppi a condizione minimale. Ricerche Mat. 9 (1960), 248-254. [7] G.Dembowski: Finite geometries. Springer Verlag, Berlin (1968). [8] A.A.Drisko: Loops with transitive automorphisms. J. Algebra 184 (1996), 213229. [9] G.Ferrero: Stems planari e BIB-disegni. Riv. Mat. Univ. Parma 11 (1970), 7996. [10] G.Ferrero: Qualche disegno geometrico. Le matematiche 26 (1971), 1-12. [11] L.Giuzzi, H.Karzel: Co-Minkowsky spaces, their reﬂection structure and K-loops. Discr. Math. 255 (2002), 161-179. [12] H.Hotje: Einige Anmerkungen zu einer Konstruktion von Marchi/Zizioli. Procedings of the 4th International Congress of Geometry (Thessaloniki ’96), 192197. Giachoudis. Giapulis, Thessaloniki 1997. [13] D.F.Hsu, A.D.Keedwell: Generalized complete mappings, neoﬁelds, sequenceable groups and block designs. I. Paciﬁc J. Math 3 (1984), 317-332. ..

[14] H.Karzel: Zusammenhange zwischen Fastbereichen, scharf zweifach transitiven Permutationsgruppen und 2-Strukturen mit Rechteckaxiom. Abh. Math. Sem. Univ. Hamburg 32 (1968), 191-206. [15] H.Karzel: From nearrings and nearﬁelds to K-loops. In G.Saad and M.J.Thomsen (eds.): Nearrings, Nearﬁelds and K-loops, 1-20. Kluwer Academic Publ., Dordrecht 1997. [16] H.Karzel, S.Pianta, E.Zizioli: K-loops derived from Frobenius groups. Discrete Math. 255 (2002), 225-234. [17] H.Karzel, H.Wefelscheid: A geometric construction of the K-loop of a hyperbolic space. Geometriae Dedicata 58 (1995), 227-236. [18] H.Kiechle: Theory of K-loops. Lecture notes in Mathematics 1778. Springer Verlag, Berlin 2002.

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[19] H.Kiechle, G.P.Nagy: On the extension of involutorial Bol loops. Abh. Math. Sem. Univ. Hamburg 72 (2002), 235-250. [20] L.J.Paige: Neoﬁelds. Duke Math. J. 16 (1949), 39-60. [21] G.Pilz: Near-rings. North Holland/American Elsevier, Amsterdam 1983. ..

[22] H.Wahling: Theorie der Fastkorper. Thales Verlag, Essen 1987. ..

..

[23] H.Zassenhaus: Uber endliche Fastkorper. Abh. Math. Sem. Univ. Hamburg 11 (1936), 187-220.

THE Z -CONSTRAINED CONJECTURE Stuart D. Scott Department of Mathematics, University of Auckland, Auckland, New Zealand [email protected]

Abstract

A Z-constrained N -group is one without central minimal factors. Zconstrained tame nearrings with DCCR are deﬁned diﬀerently but in accordance with this deﬁnition. A 3-tame nearring N has a unique maximal locally N -nilpotent right N -subgroup L(N ). This right N subgroup is an ideal. It is shown that if, for a compatible nearring N with DCCR, N/L(N ) is Z-constrained, then all Fitting factors of faithful compatible N -groups are N -isomorphic. A number of other formulations of this theorem are possible. It is a big result that requires investigation into many other areas. One important concept, on which the proof rests, is that of realisations. 2000 Mathematics Subject Classiﬁcation: 16Y30

Keywords: compatibility, Z-constraint, Fitting factors, realisations

Throughout this paper all nearrings will be zero-symmetric and left distributive. As is normal in nearring theory, groups will be written additively. This is not taken to imply commutativity. Also, unless indicated otherwise, our nearrings will have an identity and N -groups will be unitary. There are a couple of places in this paper where this assumption is discarded. One is in § 6 and the other in § 12. Notiﬁcation will be given when this is the case. In the ﬁrst few sections of what follows we shall be using terms and notation that the reader might be unfamiliar with. Their deﬁnition can be found in the body of the paper. If N is a ring, then faithful N -modules are plentiful. As a general rule a ring N has a large number of distinct (up to N -isomorphism) faithful N -modules. The question arises as to whether or not there is any reasonable assumption on a faithful N -group (tame, 2-tame or compatible) that ensures some degree of uniqueness of the N -group. This type of behaviour of N -groups, does indeed occur. It illustrates just how diﬀerent from ring type situations things can become. 69 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 69–168. c 2005 Springer. Printed in the Neatherlands.

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The ﬁrst example of uniqueness of N -groups is contained in [13]. It is shown there, that when a faithful 2-tame N -group is r-constrained (has no minimal factors which are ring modules), it is unique. A ﬁniteness assumption is used to establish this. The result illustrates well, how substantial theorems become possible, when radical departure from ring type situations, is employed. The N -groups involved are not only unique but they are ﬁnite. In some quite real sense [13] introduced a new perspective. If this sort of thing happened as generally as was proved there, then did it hold under far wider circumstances? Such questioning was also stimulated by a result of G. Peterson. In [3] he proves that if Ni , i = 1, 2, are isomorphic nearrings generated by groups of automorphisms of ﬁnite groups Vi , i = 1, 2, then there is often a type of uniqueness. When the groups of automorphisms contain V1 ) and the inner automorphisms and the Vi are perfect, then V1 /Z + (V + V2 ) are isomorphic. This tended to illustrate that something V2 /Z (V more general than the result of [13] held true. Perhaps it was possible to replace r-constraint by something much weaker and still obtain some type of uniqueness. The result of Peterson illustrates that, r-constraint may be much stronger, than what is required. However, it deﬁnitely appears Z-constraint (see § 25) is necessary. The need for Z-constraint is reinforced by [19]. There it is shown that for N semiprimary (with DCCR) all faithful compatible N -groups are N -isomorphic if, and only if, one is Z-constrained. Certainly this emphasises the essential nature of Zconstraint. It would be a large step forward if the semiprimary assumption could be discarded. At ﬁrst this looked like quite a real possibility. Evidence was supplied by [5]. There it was shown, that the Z-constraint of one faithful compatible N -group, implied the Z-constraint of all such N -groups. This provided me with motivation to make attempts on a proof of the conjecture. The failure of these attempts threw more light on things. As understanding developed it appeared that the conjecture asked for too much. It had to be discarded. What looked more likely to hold was suggested by properties of the Fitting submodule (F (V ), V a compatible N -group). In this way the real picture emerged. If the N group V (faithful, compatible and N with DCCR) was Z-constrained, then maybe all Fitting factors of faithful compatible N -groups were N isomorphic. This paper proves this is the case. It is a remarkable result that tends to press such uniqueness theorems to the boundary of what is possible. Indeed in a quite general special case it is an ‘if, and only if’ result. The size of this paper should not be a deterrent to those seeking to explore it further. Most of the sections are reasonably self contained.

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The whole work can be seen as a witness to the fact that Tame Theory is reaching maturity. It is hoped that, what is set forth here, will attract other nearringers into research in this area. If it attracts even a few young researchers toward the mysteries of tame theory, I will be satisﬁed. Many of the questions surrounding tameness are deep. They have bearing on nearly all nearrings that arise naturally (see § 5). It can be argued that such structures should be investigated before more exotic nearrings. It seems to me that the steps from groups to rings to nearrings are natural ones and, as far as the last step goes, that from rings to tame nearrings, almost demands attention.

1.

Introduction

This section is of an introductory nature. It looks at the paper as a whole. Without going into elaborate detail, it seeks to give some feeling for what is to come. Sections 2, 3 and 4 discuss each section in more detail, but here we provide an overview of the paper. There is a feature of the way this paper is written that needs mentioning. Although its motivation is the proof of a modiﬁed form of the Z-constrained conjecture, there is much subsidiary material requiring attention. This is dealt with by ensuring each section handles a small corner of material required. Some sections are linked by common features, but on the whole they do not run together much. Each section tends to stand on its own. Certainly those occuring later draw on earlier ones, but still the main common thread is that they are all related to the goal set out in the abstract. Some are quite elementary in nature, while others are quite deep. Much of interest arises along the way. Sections have been included to cover such side issues. Many details come together within the domain required to prove 39.2. Roughly speaking each section handles some detail of this kind. The four sections following 39 are more concerned with tying up loose ends. They constitute an examination of what has been proved. The statement of the main result of this paper (see 39.2) is about the implications of Z-constraint. There are at least three ways of formulating this theorem. These are given in § 39, where other weaker formulations are also made. The theorem states that, with Z-constraint present, certain large factor N -groups are all N -isomorphic. The factors we are looking at are those formed modulo maximal locally N nilpotent submodules. Thus, there are two things brought together into strong relationship by the theorem. These are Z-constraint and local N -nilpotency. The picture does not ﬁnish there. In other important results, Z-constraint and local N -solubility are brought together into close

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relationship. The next two paragraphs will discuss how this paper handles these concepts. The ﬁrst covers how understanding of Z-constraint is unravelled. After that we look at the display presented in relation to local N -nilpotency and solubility. As far as Z-constraint goes, theory relating to it is developed in 25, 26 and 27. What is ﬁrst covered is some of the more elementary implications of this concept. A feature of Z-constraint is that it implies certain N -groups are cyclic. This holds rather more generally than might be thought. After establishing that, three equivalent conditions are unearthed. One is Z-constraint in a nearring. Such a condition is rather diﬀerent from what one would expect. The equivalence of the three conditions is established under the 2-tame assumption. An important result on Z-constraint is then proved. This is about what to expect when compatibility is present. Here we have eight equivalent conditions. They are very enlightening. Local N -nilpotency and N -solubility are covered in sections 33 to 36. What exists in the literature on these concepts is insuﬃcient to meet our needs. Local N -nilpotency must be looked at in 3-tame nearrings and not just in 3-tame N -groups. The same applies to N -solubility. Quite general deﬁnitions are required. Theory based on them is developed. It is also required that, understanding of the semi-Fitting submodule be supplied. The fact that, in situations of interest to us, this is just the maximal locally N -soluble submodule, is proved. All this material enters into formulating and proving the substantial results of this paper. There is nothing deep about it, but it supplies essential ingredients. The proof of 39.2 rests on two major results. Because F (V ) (the Fitting submodule) is an intersection of centralisers, V /F (V ) is a subdirect sum. It is not just any old subdirect sum. The N -groups involved are U1 /U U2 ), where U1 /U U2 are minimal factors of V . those of the form V /C CV (U Establishing the uniqueness of such factors is one aspect of the proof. More precisely, it is shown that for perfect faithful 3-tame N -groups Vi , CV1 (W W1 /W W2 ) and V2 /C CV2 (H1 /H H2 ) are N -isomorphic, wheni = 1, 2, V1 /C W2 and H1 /H H2 are N -isomorphic (DCCR necessary). This ever W1 /W is a big step forward. All that remains is to show that the subdirect sum involved is unique. This undertaking requires compatibility and Zconstraint. When these two assumptions are made the subdirect factors of V /F (V ) ﬁt together in such a way as to ensure uniqueness. In what remains of this section, it will be explained how these two major aspects of the proof of 39.2 are accomplished. Proving uniqueness of the subdirect factors of V /F (V ) is approached through realisations. There is another manner in which this could be done. It could be approached through faithful hulls. The ﬁrst way has

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been chosen because it ﬁts in well with the overall development. A realisation of a perfect faithful 3-tame N -group has a number of helpful features. A result on decomposition is possible (see § 28). It is inspired by what is known to happen for some faithful hulls. This and other results allow a subdirect factor to be expressed as M/C CM (R) (M a realisation and R a minimal submodule of M ). Diﬀerent N -groups appear to give rise to diﬀerent M and R. Thus, there is some diﬃculty with matching up the realisations involved. This is taken care of by the concept of conjugacy. As far as proving 39.2 goes, this is the main use made of conjugacy. There is also some diﬃculty in matching up the minimal submodules R. This is catered for by the fact that the left units of realisations are well behaved. The proof of the uniqueness of the subdirect factors of V /F (V ) is outlined in the previous paragraph. It is by far the biggest undertaking in the proof of 39.2. Many sections are directly or indirectly devoted to establishing this. There are quite a number of sections that consider realisations. There are also quite a number that consider forms of complete reducibility. Two sections consider conjugacy. Material from all these sections and a number of others is brought together to establish the uniqueness of the subdirect factors of V /F (V ) (see § 32). As far as piecing together the subdirect factors, things go relatively smoothly. It can be shown that, rather primitive subdirect sums of two N -groups, constitute a diagonal N -subgroup which is unique. This happens because the N -groups involved are perfect and have no nontrivial N -endomorphims. In a less primitive situation uniqueness may not hold. However, some form of uniqueness does. The subdirect sums are determined by their intersection with the two components. What is required is for there to exist some reasonable circumstances under which such intersections are unique. At this stage compatibility comes into play. It is one of the two assumptions needed to ensure the intersections are uniquely speciﬁed. The other stems from what is, in certain circumstances, known about compatible N -groups. With compatibility Z-constraint has important features. Results go through nicely for two Z-constrained compatible N -groups. Moreover, the scenario of two components can be extended. A subdirect sum, which is in accordance with above conditions, is shown to be uniquely determined. This is the second major requirement in the proof of 39.2. The fact that it is such a natural feature of the assumptions is very pleasing.

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First Sections

This section and the next two cover in more detail the material of this paper. Here we deal with § 5 to § 17. Five is, amongst other things, an introduction to tameness. All the principal deﬁnitions related to and derived from this concept are presented. Compatibility is used to illustrate how frequently it arises. After this we move to a discussion of chain conditions. This is included for two reasons. The ﬁrst is to clarify notation. The second is that the humble chain condition be given more prominence. In my view these are important assumptions and deserve that. After dealing with chain conditions, factors, minimal factors, two sided factors and minimal two sided factors, are deﬁned. In section six centrality is introduced. Elementary theory is ﬁrst supplied. After that six moves on to supplying examples. The ﬁrst occurs U1 + U2 )/U U2 when Ui , i = 1, 2, are submodules of an N -group V and (U is central in V /U U2 . The next is submodules of a direct sum, intersecting components trivially. Then central sums are speciﬁed and the example that N -endomorphisms provide is presented. The ﬁnal illustration of centrality comes from compatible N -groups with a certain type of redundancy. General understanding of N -solubility and N -nilpotency is supplied in seven. The theory is elementary. It is mentioned under what circumstances direct products of such N -groups are well behaved. The N -subgroups V (n) and V n of an N -group are deﬁned. The relationship between these N -groups is speciﬁed. Eight ﬁrst covers the deﬁnition of minimal N -groups and the radical. It is stated how N/J(N ) (N with DCCR) decomposes. After that the section moves on to deﬁning complete reducibility. Conditions equivalent to complete reducibility are given. At this stage, tame series are deﬁned. The fact that tame N -groups (N with DCCR) have a tame series, is mentioned. A theorem is then covered giving equivalent conditions that such an N -group be ﬁnitely generated. Also, the fact that such an N group has maximal and minimal submodules, is covered. Finally two results holding in tame situations are stated. Nine is the ﬁrst section of a less elementary nature. In it the Frattini N -subgroup is deﬁned. It is shown that one of the properties of a radical holds. Under slightly restricted circumstances, the other does also. Property q is introduced. It is shown that, for an N -group V with property q, V /Φ(V ) (having DCCS) is a ﬁnite direct sum of minimal N -groups.

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Ten is about aspects of complete reducibility. The ﬁrst question considered is when is an N -subgroup of a direct sum of minimal N -groups, a direct sum of some of them. After that, equivalent conditions for a completely reducible N -group to be tame, are given. The example of V /Φ(V ) (V a tame N -group and N with DCCR), is covered. Perfect N -groups are deﬁned and, in certain tame situations, the cyclic nature of such an N -group is established. Eleven introduces realisations. The fact that they are self monogenic is established. It is shown that, in situations of interest to us, they have property q. The section ﬁnishes by proving that, for a cyclic tame N group V (N with DCCR), the minimal factors V /U and M/H (M a realisation) are N -isomorphic. Twelve is somewhat deeper than previous sections. It is diﬀerent in that it does not use tameness. The main result is about any nearring with DCCN . It is the purpose of twelve, to show that a nearring N with a minimal ideal A intersecting Z(N ) non-trivially, is necessarily central. This is accomplished by showing that, if M is in mr(A), then M ∩ A is central in M . The theorem is then proved, by lifting the centrality of M ∩ A in M to that of A in N . Section thirteen ﬁnds necessary and suﬃcient conditions that two tame N -groups U and W (N with DCCR) be such that (0 : U ) + (0 : W ) = N . This is a result having bearing on a number of other situations. The condition is that no minimal factor of U be N -isomorphic to one of W . Preliminaries require we look at what happens when N is a sum A1 + A3 and A2 + A3 of ideals Ai , i = 1, 2, 3. The proof of the main theorem is substantial. After that, a theorem related to the main result is established. Fourteen is about the covering and avoidance of minimal factors of a 2-tame N -group (N with DCCR). It is the minimal factors which are not ring modules that are of interest. A cover of such factors will cover all N -isomorphic to them. The intersection of two covers (of the type outlined above) is another such cover. This means that minimal covers exist. This section also exhibits a considerable amount of notation that will be of use later. In ﬁfteen it is shown that, if V is a 2-tame N -group (N with DCCR), S a subset of tnr(N ) and W a minimal cover of V (S), then a minimal factor W/U of W , is an element of V (S). This allows us to conclude that W is perfect, cyclic and W/Φ(W ) is a ﬁnite direct sum of minimal N groups, which are not ring modules and where no two are N -isomorphic. Using this, it is proved that, W/Φ(W ) is N -isomorphic to M/Φ(M ) (M a realisation of W ).

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Sixteen is directed toward proving a quite important theorem. The set up is that of two faithful 2-tame N -groups V and W (N with DCCR). Lemmas are proved about the minimal covers V1 and W1 of V (S) and W (S), respectively (S a subset of tnr(N )). The theorem establishes wM , over all w in W , coincides that, if M is a realisation of V1 , then with W1 . Section seventeen is a carry on from sixteen. With the notation of the W2 of W1 , is N -isomorphic to previous paragraph, a minimal factor W1 /W V2 of V1 and visa versa. Not only are V1 /Φ(V V1 ) and a minimal factor V1 /V W1 ) N -isomorphic, but they are N -isomorphic to both M1 /Φ(M M1 ) W1 /Φ(W M2 ) (M M2 a realisation of W1 ). The (M M1 a realisation of V1 ) and M2 /Φ(M very important fact that (0 : V1 ) = (0 : W1 ) is proved.

3.

Middle Sections

In this section we cover, in some detail, the material of § 18 to § 30. In eighteen factors of an N -group that centralise a subset are deﬁned. Hence central factors are deﬁned. It is shown that, if for a tame N -group V , all minimal factors of N which are N -isomorphic to a given minimal factor of V are central, then the given factor of V is central. Next it is shown that, in 2-tame situations, N -isomorphic factors of an N -group centralise the same subset. A partial converse of a previous result is obtained. If a faithful 2-tame N -group (N with DCCR) has a central minimal factor, then all minimal factors of N , N -isomorphic to it, are central. Nineteen is about minimal faithfuls. These exist in faithful tame N groups where N has DCCR. For V a 2-tame N -group with a faithful N -subgroup H, V /H is often N -nilpotent. A result on the intersection of all minimal faithfuls is stated. When N has a faithful perfect 2-tame N -group, a characterisation of the minimal faithful of a faithful 2-tame N -group W , can be given. This minimal faithful coincides with W ω or W (ω) . Twenty deﬁnes conjugacy. These are systems of right N -subgroups, which are all, in some sense, N -isomorphic. The ﬁrst lemma of that section gives a condition, like N -isomorphism but stronger, that links together certain right N -subgroups. Some discussion follows that proves that, for a tame nearring N with DCCR and collection S of minimal ideals of N , mr(S) is a conjugacy class. The concept of conjugacy is modiﬁed in the next section to that of weak conjugacy. Although conjugacy is not used elsewhere a signiﬁcant use of weak conjugacy is made.

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Twenty one deﬁnes weak conjugacy. Again the elements of a weak conjugacy class are right N -subgroups linked together in a certain way. One quite major use will be made of weak conjugacy later. It is used to show that realisations of perfect 2-tame N -groups can be identiﬁed. As far as § 21 goes, two examples of weak conjugacy are supplied. The ﬁrst states that min(∆) is a weak conjugacy class. The next speciﬁes re(S) as one also. Roughly speaking re(S) is a certain set of realisations associated with minimal covers of a subset S of tnr(N ). If V is a tame N -group, then what can be expected about centrality in the socle soc V of V ? Twenty two addresses this question. It is ﬁrst shown that, for such an N -group V , soc(V, ∆) intersects Z(V ) nontrivially if, and only if, soc(V, ∆) ≤ Z(V ). At the other extreme where ∆ is in tnr(N ), soc(V, ∆) is either {0} or a minimal N -group. The ﬁnal result of that section proves, that when N has DCCR and soc(V, ∆) is an inﬁnite direct sum of minimal N -groups, soc(V, ∆) ≤ Z(V ). Some properties of soc(V, ∆), outlined in the previous paragraph, are not shared by soc(N, ∆). Section twenty three looks at this. What is required is a notion of complete reducibility that encompasses submodules of an N -group V . We need to be able to specify what it means for such a submodule to be completely reducible in V . It is shown that how, in many situations, soc N decomposes into the direct sum soc(N, ∆) ⊕ soc(N, ∆ ). An elementary proposition is then stated. At that stage, it is indicated how soc(N, ∆) is an ideal of N . In twenty four, the case of a tame N -group U ⊕ W (U minimal) is examined. All minimal factors of U ⊕ W , N -isomorphic to U are central if, and only if, U is a ring module. In this case (faithful and DCCR assumed) all minimal factors of N , N -isomorphic to U , are central. The case where U is not a ring module is also examined. It is shown U ⊕ W is cyclic if, and only if, W is cyclic. In twenty ﬁve Z-constraint is deﬁned in the case of N -groups. Such N -groups are perfect. In tame situations with DCCR, they are cyclic. Next, Z-constraint in a tame nearring with DCCR, is deﬁned. This is rather diﬀerent from what one might initially expect. However the deﬁnition does, in a sense, relate well to the N -group one. It is shown that if such a nearring is tame on V , then V is cyclic. This is quite a substantial result. In section twenty six we prove a single theorem. It is a rather beautiful characterisation, in the 2-tame case (N with DCCR), of Z-constraint. It relates N -group Z-constraint to nearring Z-constraint and the Zconstraint of one N -group to others. The result applies to N (with DCCR) being 2-tame on V . There are three equivalent conditions. The ﬁrst is that V is Z-constrained. Secondly, we have the condition, that

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N is Z-constrained. The third is that all faithful 2-tame N -groups are Z-constrained. Twenty seven starts by proving that, for a compatible N -group, one can directly add copies of a central factor and still obtain a compatible N -group. After that an enlightening theorem is proved. With DCCR, eight equivalent conditions are produced. One is that N is compatible on a Z-constrained N -group. Three of them (including the one mentioned) are similar to those occuring in the previous paragraph. Twenty eight is an important contribution to our study of realisations. Expressed simply, the main theorem states that, if a realisation M of a perfect 2-tame N -group (faithful and DCCR assumed) is expressible as a sum H + R, where H an N -subgroup of M and R a submodule of M such that H ∩ R = {0}, then HR = {0}. Twenty nine is about a certain lifting of left units. Suppose M is a realisation of a perfect 2-tame N -group (faithful and DCCR assumed) and M = H + R, where H and R are as in the previous paragraph. The main result of twenty nine states that, if γ is a left unit of H, then there exists α in R such that γ + α is a left unit of M . Section thirty is, in some respects, quite technical. With M a realisation of a perfect 2-tame N -group (faithful and DCCR assumed), A a minimal ideal of N and R a minimal submodule of M contained in M ∩ A, it is shown that M ∩ A = βR, where the sum is over all left units β of M . A number of other issues are dealt with in the process of showing this.

4.

Final Sections

In this section we discuss, in some detail, the contents of § 31 to § 43. In thirty one we cover the important notion of centralisers. Not only are they present in 3-tame N -groups, but also in 3-tame nearrings. The section begins by establishing their existence in 3-tame N -groups. This is a quite well known result, but the approach is, in some respects, diﬀerent. The fact that, N -isomorphic factors of such an N -group, have centralisers that coincide, is established. Some additional notation is also dealt with. The section moves on to looking at centralisers in 3-tame nearrings. There is a connection between them and those in N -groups. They are right ideals of the nearring. After all this, some discussion is given covering drawbacks and strengths of what has been shown. In many situations centralisers exist in a more real way. Also normalisers often exist. Thirty two is a key section of our paper. In the ﬁrst lemma it is established that, when N (with DCCR) is 3-tame on the perfect N -

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group V and U is a minimal submodule of V , then V /C CV (U ) is often N -isomorphic to a certain well deﬁned N -group. This is M/C CM (R), where M is a realisation of V and R a certain minimal submodule of M . This allows us, in the next lemma, to establish that, when Vi , i = 1, 2, are faithful perfect 3-tame N -groups and Ui , i = 1, 2, minimal CV1 (U U1 ) is often N -isomorphic N -isomorphic N -subgroups of Vi , then V1 /C CV2 (U U2 ). The theorem after proves that, in fact, if U1 /U U2 is a to V2 /C W2 of V2 , minimal factor of V1 , N -isomorphic to the minimal factor W1 /W CV1 (U U1 /U U2 ) is always N -isomorphic to V2 /C CV2 (W W1 /W W2 ). then V1 /C Section thirty three is about local N -nilpotency. The deﬁnition given is general. It is needful to apply it to 3-tame nearrings. For a 3-tame nearring N , L(N ) is taken as the right N -subgroup of N generated by all locally N -nilpotent right N -subgroups. It is shown that L(N ) is the sum of all N -nilpotent right ideals of N . This means, amongst other things, L(N ) is locally N -nilpotent. It can also be shown that it is an ideal of N . Understanding does not ﬁnish there. If N is 3-tame on V , then L(N ) ≤ (L(V ) : V ). This is a result that does not go as far as is desirable. What is really required is that L(N ) = (L(V ) : V ). This holds under circumstances suﬃcient to carry out later developments. In the case where L(V ) is N -nilpotent it certainly holds that L(N ) = (L(V ) : V ). Thirty four is, in some respects, similar to thirty three. In it we give a general deﬁnition of local N -solubility. It is needful to apply this deﬁnition to tame nearrings. For a tame nearring N , S(N ) is taken as the right N -subgroup generated by all locally N -soluble right N -subgroups of N . It is shown that S(N ) is the sum of all N -soluble right ideals of N . It can also be shown to be an ideal of N . Once again (see the previous paragraph), if N is tame on V , then S(N ) ≤ (S(V ) : V ). Again what is really required is equality. In situations where this is needed the result holds. When S(V ) is N -soluble, S(N ) = (S(V ) : V ). Thirty ﬁve looks at the semi-Fitting submodule. This submodule is an intersection of centralisers of certain minimal factors. It can be deﬁned for tame V . The minimal factors are those which are not ring modules. It is denoted by sF (V ). The results of thirty ﬁve are quite elementary. The ﬁrst states that (sF (V )+H)/H ≤ sF (V /H) (H a submodule of V ). After that, it is shown that sF (sF (V )) = sF (V ). Many of the properties of sF (V ) are reminiscent of what happens for F (V ) (V 3-tame). Section thirty six looks at conditions for sF (V ) to be N -soluble and S(V ) = sF (V ). This holds for tame V and N with DCCR. It also holds for compatible V , where N has DCCI. This last result uses a substantial result from [12].

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Thirty seven is about subdirect sums. The type of subdirect sum of two N -groups Vi , i = 1, 2, intersecting Vi trivially can often be very restricted. If such a subdirect sum is 2-tame and perfect, then it is, in fact, unique. This can be proved by showing that two such subdirect sums establish an N -automorphism of V1 or V2 and the perfect nature of Vi gives the result. This result has an implication. Any perfect 2tame subdirect sum of the Vi , i = 1, 2, is completely determined by its intersection with the Vi . Thirty eight is about compatible Z-constrained subdirect sums of N groups Vi , i = 1, . . . , k. Such subdirect sums are unique. This is shown by making use of the results of § 37. The way Z-constraint is used is fairly standard. In the compatible case it rules out redundancy. Thirty nine covers our main theorem. It consists of showing that, two faithful compatible N -groups Vi , i = 1, 2, with N/L(N ) Z-constrained, Vi ). In order to show this, it may be assumed have N -isomorphic Vi /F (V the Vi are perfect. The section concludes with ﬁve corollaries. Three of these are restatements of the main theorem. The other two are weaker formulations. Forty is, in many respects, similar to § 39. It covers an important result of this paper. It consists in showing that, two faithful compatible N -groups Vi , i = 1, 2, with N/S(N ) Z-constrained, have N -isomorphic Vi ). In order to show this, it may be assumed the Vi are perVi /sF (V fect. The section concludes with ﬁve corollaries. Three of these are restatements of the theorem just mentioned. The other two are weaker formulations. Forty one is about how our main results relate to the semiprimary situation. If N is compatible on a semiprimary N -group, then it is semiprimary. With Z-constraint and DCCR a converse holds. The main result of § 41 is that, if N is a compatible semiprimary nearring with DCCR, then N is Z-constrained if, and only if, all faithful compatible N -groups are N -isomorphic. Section forty two deals with a loose end. Throughout the paper realisations have been used extensively. In the type of situations that are of interest to us, they are faithful hulls. What is shown in § 42 is that the realisation of a faithful perfect 2-tame N -group (N with DCCR) is a faithful hull. The ﬁnal section (forty three) is a discussion relating to the whole paper. How investigations in these areas started is covered in the ﬁrst paragraph. The second paragraph covers how developments then blossomed into this paper. In the third and fourth paragraphs we ask questions concerning P0 (V ) (V an Ω-group). The ﬁfth tells us about how strong our main result is, while the sixth looks at the question of a con-

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verse. Diﬃculties in weakening the DCCR assumption are discussed in the eighth, while the nineth tells us why compatibility, rather than a polynomial approach, is best. The ﬁnal paragraph of this paper covers acknowledgments.

5.

Tameness, etc

Back in 1968, when I was playing around with algebraic structures that interested me, it began to become clear that certain N -groups had a property on which many theoretic developments could be based. There was one simple feature that these N -groups had in common, that tended to dominate most other considerations relating to them. It seemed like sense to set aside the examples that portrayed this feature, for an investigation into the general theory they indicated. In this way Tame Theory was born. In my thesis [11] theory relating to tame N -groups and nearrings was developed. This was not a superﬁcial exposition. Much of the theory was quite deep. It is very pleasing that tame investigations have continued to ﬂourish. What is the feature that tame N -groups have? As is the case with many of the more fundamental mathematical deﬁnitions, it is easy to state. An N -group V is called tame if all N -subgroups are submodules. This can be restated by saying all cyclic N -subgroups are submodules. Consequently, an alternative deﬁnition is possible. The N -group V is tame if, and only if, for any given v and w in V and α in N , there exists β in N , such that (v + w)α − vα = wβ. This characterisation is important. As we shall see below it allows the deﬁnition of n-tame and compatible N -groups. Deﬁnitions do not stop with tame N -groups. Tameness allows the establishment of signiﬁcant results about N . Generally this is the case when V is faithful. Faithfulness alone, means many signiﬁcant facts can be derived for N . Consequently we call a nearring N tame (or tame on V ) if N has a faithful tame N -group (V say). This is not taken to imply N is tame as an N -group. In many situations, much more theory becomes possible if the notion of tameness is strengthened. The elementwise characterisation of tameness (see above) allows this. If n ≥ 1 is an integer, then an N -group V is called n-tame, if for any given v in V , n-tuple (w1 , . . . , wn ) of elements of V and α in N , we can ﬁnd β in N such that (v + wi )α − vα = wi β, for i = 1, . . . , n. 1-tame N -groups are precisely the tame N -groups. For integers n ≥ m ≥ 1, V (an N -group) is n-tame means it is m-tame. As with tame nearrings, we deﬁne a nearring N as n-tame (n-tame on V )

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if, N has a faithful n-tame N -group (V say). This is not meant to imply the N -group N is necessarily n-tame. Many of the examples of n-tame N -groups satisfy a stronger requirement. Indeed, as seen below when examples are provided, nearrings that arise naturally satisfy such an assumption. The condition is what is called compatibility (see [11]). An N -group V will be called compatible if, for any given v in V and α in N , there exists β in N such that (v + w)α − vα = wβ, for all w in V . As with tame or n-tame nearrings, the nearring N is called compatible (compatible on V ) if it has a faithful compatible N -group (V say). Once again care must be taken as to how this deﬁnition is used. Calling a nearring N compatible, does not mean the N -group N is compatible. Examples of compatible nearrings and N -groups are easy enough to come by. Nearrings that arise naturally in other mathematical contexts provide illustrations. The ﬁrst example is any ring (with identity). This is compatible on itself. Next we may take the nearring C0 (V ) of all zeroﬁxing continuous self maps of a topological group V . This is compatible on V . Another instance is that of a nearring generated by a set S of endomorphisms of a group V . When Inn(V ) ⊆ S, N is compatible on V . Examples do not ﬁnish here. Two more important ones will be provided. If we take an Ω-group V and P0 (V ) as the nearring of all zero-ﬁxing polynomial self maps of V , then P0 (V ) is compatible on V . A ﬁnal interesting example will complete this paragraph. Take V as a group, S as a collection of normal subgroups of V and N as all zeroﬁxing self maps α of V , such that (v + H)α ⊆ vα + H, for all v in V and H in S. With this notation, N is a nearring which is compatible on V . The center part of this section is about chain conditions. The study of nearrings owes much to the use of these conditions. It is easy to underestimate the immense value they play. They often give considerable insight as to what is and is not possible, in more general situations. Because nearrings have quite a number of diﬀerent types of subobjects, a signiﬁcant collection of diﬀerent conditions exist. Those that most readily present themselves are DCCN , DCCR, DCCI, ACCN , ACCR and ACCI. The ﬁrst three are, respectively, the descending chain condition on right N -subgroups, right ideals and ideals. Also ACCN , ACCR and ACCI, respectively, denote the corresponding ascending chain conditions. There are also the DCCS and ACCS assumptions. These relate to N -groups. The DCCS is the descending chain condition on submodules, while ACCS is a similar ascending version. The corresponding assumptions on N -subgroups will be denoted by DCCN and ACCN . They are not used in this paper and therefore there is no confusion with above notation.

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Many semi-classical results have made use of DCCN and DCCR (on a nearring). More recently deep use of DCCI and ACCI has been made. This is very satisfying because these two conditions are quite weak. What is proved to hold with DCCI or ACCI present, will hold for a wide class of nearrings. The fact that DCCI and ACCI are quite weak conditions is best illustrated by looking at the nearring M0 (V ) of all zero-ﬁxing self maps of the group V . In this case we see that because M0 (V ) is simple, both chain conditions hold. However, it is not diﬃcult to show DCCR and ACCR both imply V is ﬁnite. So the diﬀerence here is very marked. The weaker conditions are not really conditions at all and the stronger ones are as strong as is possible. Indeed, in the second case because M0 (V ) is ﬁnite, all imaginable ﬁniteness conditions hold. Can those conditions, which for M0 (V ) say nothing, be expected to yield theory? Perhaps it is too much to expect them to have very real application. However, such limited hopes appear to mean very little. The extent to which beautiful theory is possible seems unbounded (see [12] and [17]). The chain condition that this paper uses more than any other is DCCR. This is particularly suitable for tame situations. Much theory has been developed with the use of this assumption (on a tame nearring). The papers [16] and [12], provide examples of this. Moreover, according to 5.7 of [16], DCCR is rather special. It implies each of the conditions DCCN , DCCI, ACCN , ACCR and ACCI. The ﬁnal part of this section deals with what I have called factors. These occur so frequently that their value cannot be underestimated. They give much insight into the manner in which N -groups and nearrings hang together. Very little more will be said about them. Much of this paper is a witness to their value. It simply remains to supply deﬁnitions. U2 , where U1 ≥ U2 are If V is an N -group, then an N -group U1 /U U2 is also a minimal submodules of V , will be called a factor of V . If U1 /U U2 will be called a minimal factor of V . submodule of V /U U2 , then U1 /U It follows that an N -group R1 /R2 , where R1 ≥ R2 are right ideals of N , is a factor of N (as an N -group) and is minimal if R1 > R2 and there are no right ideals properly between R2 and R1 . In the case where both Ri , i = 1, 2, are ideals of N , we call R1 /R2 a two sided factor of N . Furthermore, if R1 > R2 are ideals of N with no ideals properly between them, we call R1 /R2 a minimal two sided factor of N .

6.

Examples of Centrality

The notion of centrality in N -groups is simple enough, but in many regards, it is also fundamental. There are a number of examples where

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central submodules arise quite naturally. These instances of centrality crop up often. It is important that there is clear understanding as to how they arise. Before venturing into considering these particular cases, elementary deﬁnitions and results are covered. An N -subgroup U of an N -group V is called a central N -subgroup of V if (v + u)α = vα + uα, for all v in V , u in U and α in N . By taking α = −1 in this deﬁnition, we see that U ≤ Z + (V ) (the additive center of V ). As a consequence the deﬁning relationship can be replaced by one of three others. These are those obtained by the interchange of v and u on the left and/or the right. This deﬁnition has been made for N with identity and V unitary. Without this assumption a central N subgroup requires slightly more to specify it. Not only do we require that (v +u)α = vα+uα, for all v in V , u in U and α in N , but we also require that U ≤ Z + (V ). In both situations (N with identity and V unitary or otherwise), it follows readily that U is a submodule of V . Thus the term central submodule is equally appropriate for such N -subgroups. For the ﬁrst half of this section it will not necessarily be assumed that N has an identity and N -groups are unitary. The reader will be notiﬁed when these assumptions again come into play. Two elementary facts about centrality are covered in the next proposition. Proposition 6.1. If U is a central submodule of the N -group V , then an N -subgroup of U is central in V and U µ is central in V µ, where µ is any N -homomorphism on V . An important feature of central submodules is that they are additive. Proposition 6.2. If V is an N -group, then any sum of central submodules of V is central in V . Proposition 6.2 means that the sum of all central submodules of an N -group V , is the unique maximal central submodule of V . It is denoted by Z(V ) and has the property that any N -subgroup U of V is central if, and only if, U ≤ Z(V ). In 6.1 we see that N -homomorphic images of central submodules are central. There is one situation where the inverse N -homomorphic image of a central submodule is central. This is an important instance of the transfer of centrality. The result is as follows:Proposition 6.3. If V is an N -group and Ui , i = 1, 2, submodules of U2 is central in V /U U2 if, and only if, U1 /(U U1 ∩ U2 ) is V , then (U U1 + U2 )/U central in V /(U U1 ∩ U2 ). Proof: Suppose U1 /(U U1 ∩ U2 ) is central in V /(U U1 ∩ U2 ). Under the natural N -homomorphism of V /(U U1 ∩ U2 ) onto V /U U2 , U1 /(U U1 ∩ U2 ) is

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mapped to (U U1 + U2 )/U U2 . By 6.1, it follows that (U U1 + U2 )/U U2 is central in V /U U2 . U2 is central in V /U U2 . If this is the case, Now suppose (U U1 + U2 )/U then −v + u1 + v ≡ u1 mod U2 , for all v in V and u1 in U1 . This U1 ∩ U2 ) is contained in means −v + u1 + v ≡ u1 mod U1 ∩ U2 and U1 /(U U1 ∩ U2 )). Also, if α is in N , then (v + u1 )α ≡ vα + u1 α mod U2 Z + (V /(U and (v +u1 )α−u1 α−vα is in U1 ∩U U2 implying, U1 /(U U1 ∩U U2 ) is contained in Z(V /(U U1 ∩ U2 )). The proof is complete. Proposition 6.3 may be regarded as an example of centrality. With V , U1 and U2 as in that proposition, there is no obvious reason that the U2 should imply that of U1 /(U U1 ∩ U2 ). The fact centrality of (U U1 + U2 )/U that it does is very useful. Another example of centrality comes about from direct sums. A submodule that intersects direct summands trivially is always central. This is fully covered in:Proposition 6.4. Suppose the N -group V is a direct sum of the submodules Vi , i = 1, 2. If U is a submodule of V such that Vi ∩ U = {0}, for i = 1, 2, then U ≤ Z(V ). Proof: First we note that because Vi ∩ U = {0}, U is in the additive centraliser of Vi , i = 1, 2. Thus U ≤ Z + (V ). Now for v = v1 + v2 in V (vi , i = 1, 2, in Vi ), u in U and α in N , it follows that (v1 + v2 + u)α − uα − v2 α − v1 α = (v + u)α − uα − vα (= x say). However, V2 ∩ U = {0} and (v2 + u)α = v2 α + uα, so that x is in V1 . A reasonably similar argument shows x is in V2 . Thus x = 0, (v + u)α = vα + uα and U ≤ Z(V ). The proposition is proved. Another example of centrality occurs with central sums. Suppose the N -group V is a sum V1 + V2 of the submodules Vi , i = 1, 2, in such a V2 ) of V2 in V way that V1 is contained in the additive centraliser CV+ (V V1 )) and, for all vi in Vi , i = 1, 2, (this is the same as requiring V2 ≤ CV+ (V and α in N , we have (v1 + v2 )α = v1 α + v2 α. If this is the case, then V is said to be a central sum of Vi , i = 1, 2. Central sums occur reasonably often in nearring theory. In particular they make themselves manifest in tame theory. They give rise to central submodules. Proposition 6.5. If the N -group V is a central sum of the submodules Vi , i = 1, 2, then V1 ∩ V2 ≤ Z(V ). Proof: Because V1 ∩ V2 is in the additive centraliser of V1 and V2 , it follows that V1 ∩ V2 ≤ Z + (V ). If v = v1 + v2 is in V (vi , i = 1, 2, in Vi ),

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u in V1 ∩ V2 and α in N , then (v + u)α = (v1 + v2 + u)α = v1 α + (v2 + u)α = v1 α + v2 α + uα = vα + uα. It follows that V1 ∩ V2 ≤ Z(V ) and the proposition is proved. The standard example of central sums comes about from direct sums. If we take an N -group V1 ⊕ V2 (a direct sum of the submodules Vi , i = 1, 2), then an N -homomorphism µ on V1 ⊕ V2 , gives rise to a central sum. In this situation, (V V1 ⊕ V2 )µ is a central sum of the submodules V1 µ and V2 µ. Furthermore, all central sums arise in this manner. If N has an identity and V is a unitary N -group, then V is a central sum of the submodules Vi , i = 1, 2, if and only if (v1 + v2 )α = v1 α + v2 α, for all vi in Vi and α in N . This follows because, on taking α = −1, we V2 ). This is another situation where, assuming unitary V , see V1 ≤ CV+ (V streamlines deﬁnitions. From now on we shall specialise considerations to unitary V . The next and only time, that non-unitary V make an appearance, is in section 12. Central sums arise in tame theory. The main way this happens is through N -endomorphisms. If V is a 2-tame N -group and µ an N endomorphism on V , then 1 − µ (obvious notation) is, according to 7.3 of [16], another N -endomorphism on V . Furthermore, according to 1.4 of [9], we have:Proposition 6.6. If V is a 2-tame N -group and µ an N -endomorphism on V , then V is a central sum of the submodules V µ and V (1 − µ). Corollary 6.7. With V and µ as in 6.6, V µ ∩ V (1 − µ) ≤ Z(V ). Proof: By 6.5. The above coverage of central sums is suﬃcient for our needs. The ﬁnal example of centrality occurs with compatible N -groups. This is an instance of centrality that has proved important. From 6.1 of [17], it follows that:Proposition 6.8. If V is a compatible N -group and U a submodule of V such that (U : V ) = (0 : V ), then U ≤ Z(V ).

7.

N -solubility and N -nilpotency

In § 3 of [12], N -solubility is deﬁned for a tame N -group. The deﬁnition is not quite as general as we would like it to be. In § 34 we shall be requiring N -solubility in a tame nearring N . Since N , as an N -group, is not necessarily tame, something more is required. In this section a general but elementary account of N -solubility is supplied.

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An N -group V is called N -soluble if there exist an integer r ≥ 0 and N -subgroups {0} = V0 ≤ V1 ≤ V2 ≤ · · · ≤ Vr = V, where Vi , i = 0, . . . , r − 1, is a submodule of Vi+1 and Vi+1 /V Vi is a ring module. The phrase ‘ring module’ will be taken as meaning elements of N distribute over such an N -group. In the above deﬁnition, the smallest integer r ≥ 0 is called the length or N -solubility length of V . Two elementary propositions are given. Proposition 7.1. If V is an N -soluble N -group of N -solubility length r, then an N -subgroup of V is N -soluble of length ≤ r. Proposition 7.2. If V is an N -soluble N -group of N -solubility length r and µ is an N -homomorphism on V , then V µ is N -soluble of length ≤ r. From propositions 7.1 and 7.2, we see that the class of all N -soluble N -groups, is closed under taking subobjects and N -homomorphic images. The question arises as to whether or not it is a variety. This is not the case. On taking the direct product of N -soluble N -groups of increasing N -solubility length, we obtain an N -group which is not N soluble. However, if the N -groups involved have restricted N -solubility length, then all is well. Proposition 7.3. If Vi , i ∈ I,is a family of N -soluble N -groups of length ≤ r (integers ≥ 0), then Vi is N -soluble of N -solubility length ≤ r. If V is an N -group, then V (1) is deﬁned as the intersection of all submodules U of V , where V /U is a ring module. Clearly, V (1) is a submodule of V and V /V (1) is a ring module. V (1) is the smallest submodule of V with this property. It could be deﬁned in a diﬀerent manner. V (1) is, in fact, the submodule of V generated by all (v1 + v2 )α − v2 α − v1 α, where vi , i = 1, 2, is in V and α in N . With V (0) taken as V , V (1) as above and V (n) (n ≥ 1 an integer) deﬁned, we set V (n+1) as (V (n) )(1) . It follows that, V (0) = V ≥ V (1) ≥ V (2) ≥ . . . and V (r+1) is a submodule of V (r) , for all r ≥ 0. Proposition 7.4. The N -group V is N -soluble if, and only if, V (n) = {0}, for some integer n ≥ 0. Furthermore, the smallest such integer n is the N -solubility length of V . As far as this section goes, N -solubility has been dealt with. There remains the need to consider N -nilpotency. In this direction, some basics

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can be found in [12]. It is however desirable to have a complete enough account. The rest of this section provides this. An N -group V is called N -nilpotent, if there exists an integer r ≥ 0 and submodules {0} = V0 ≤ V1 ≤ V2 ≤ · · · ≤ Vr = V Vi , i = 0, . . . , r − 1, is contained in Z(V /V Vi ). In this of V , where Vi+1 /V deﬁnition, the smallest integer r ≥ 0 is called the class or N -nilpotency class of V . As with N -solubility, the following propositions hold. Proposition 7.5. If V is an N -nilpotent N -group of N -nilpotency class r, then an N -subgroup of V is N -nilpotent of class ≤ r. Proposition 7.6. If V is an N -nilpotent N -group of N -nilpotency class r and µ is an N -homomorphism on V , then V µ is N -nilpotent of class ≤ r. From propositions 7.5 and 7.6, we see that the class of all N -nilpotent N -groups, is closed under taking subobjects and N -homomorphic images. As with N -solubility, the question arises as to whether or not, this class is a variety. Again this possibility is not fulﬁlled. By taking the direct product of N -nilpotent N -groups of increasing N -nilpotency class, we obtain an N -group which is not N -nilpotent. The similarity with N -solubility continues. Indeed, if the N -groups involved in the direct product have restricted N -nilpotency class, then expectations are met. Proposition 7.7. If Vi , i ∈ I, is a family of N -nilpotent N -groups of class ≤ r (integers ≥ 0), then Vi is N -nilpotent of N -nilpotency class ≤ r. If V is an N -group and W a submodule of V , then W (V ) is taken as the intersection of all submodules U ≤ W of V , for which W/U ≤ Z(V /U ). Clearly W (V ) is a submodule of V contained in W . Furthermore, if v is in V , w in W and α in N , then (v + w)α − wα − vα is in U , for all submodules U ≤ W of V with W/U ≤ Z(V /U ). This means (v + w)α − wα − vα is in W (V ) and W/W (V ) ≤ Z(V /W (V )). Thus W (V ) is the smallest submodule of V with this property. It can be deﬁned in a diﬀerent manner. W (V ) is, in fact, the submodule of V generated by all (v + w)α − wα − vα, where v is in V , w in W and α in N. For a given N -group V , V 0 is taken as V and V 1 is taken as V (V ) (see above). With V n deﬁned (n ≥ 0 an integer), V n+1 is taken as (V n )(V ). We have V 0 = V ≥ V 1 ≥ V 2 ≥ ...,

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and it is not diﬃcult to establish that:Proposition 7.8. The N -group V is N -nilpotent, if and only if, V n = {0}, for some integer n ≥ 0. Furthermore, the smallest such integer n is the N -nilpotency class of V . Before leaving this section we establish a relationship between V (n) and V n . Clearly V (0) = V 0 . If for an integer r ≥ 0, V (r) ≤ V r then, because V r /V r+1 ≤ Z(V /V r+1 ), it follows that (V (r) + V r+1 )/V r+1 is a ring module and so is V (r) /(V (r) ∩ V r+1 ). This means V (r+1) ≤ V (r) ∩ V r+1 and V (r+1) ≤ V r+1 . It has been shown that:Proposition 7.9. If V is an N -group, then for any positive integer n, V (n) ≤ V n .

8.

Tame Series

In tame theory, type 2 N -groups play a signiﬁcant role. These are simply N -groups without non-trivial proper N -subgroups. They will be called minimal N -groups. It is almost true to say that, as far as this paper goes, these are the only type 0 N -groups, making an appearance. There is an exception to this rule. In section 21, considerations have been broadened to non-tame situations. Other type 0 N -groups will brieﬂy crop up there. The radical J(N ) of a nearring N , will be taken as the intersection of the annihilators of all minimal N -groups. It is the only radical that will concern us. Although in § 9 we brieﬂy mention radicals of N -groups, this is only in passing. Many properties of J(N ) are well known. For example, when N has DCCR, a decomposition of N/J(N ) becomes available. Theorem 8.1. If N is a nearring with DCCR, then N/J(N ) is a ﬁnite direct sum of minimal right ideals which are minimal N -groups. Furthermore, any minimal N -group is N -isomorphic to such a direct summand. Corollary 8.2. If N is a nearring with DCCR, then the number of N -isomorphism types of minimal N -groups is ﬁnite. The next stage up, from consideration of minimal N -groups, is to develop understanding of direct sums of such N -groups. An N -group V will be called completely reducible if it can be expressed as a direct sum of minimal N -groups. This concept occurs frequently throughout this paper. There are a couple of conditions that are equivalent to complete reducibility. It is well known that:Theorem 8.3. If V is an N -group, then the following are equivalent

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(i) V is completely reducible, (ii) V is a sum of submodules which are minimal N -groups and (iii) for any N -subgroup H of V , there exists a submodule W of V , with H + W = V and H ∩ W = {0}. The condition that V is completely reducible is restrictive. In section 23, we shall broaden this concept. For the moment, theorem 8.3 and the result that follows, meet our needs. Information about N -subgroups of completely reducible N -groups is required. Here we have:Proposition 8.4. An N -subgroup of a completely reducible N -group, is completely reducible. Proof: This follows readily from (iii) of 8.3. Completely reducible N -groups form the building blocks of tame N groups. If V is a tame N -group, r ≥ 0 an integer and {0} = V0 ≤ V1 ≤ · · · ≤ Vr = V a sequence of submodules of V such that Vi+1 /V Vi , i = 0, . . . , r − 1, is completely reducible, then such a sequence of submodules of V will be called a tame series (ie. tame series of V ). Clearly if V = {0}, the inclusions Vi ≤ Vi+1 , i = 0, . . . , r − 1, can be taken as strict. It is well known that:Theorem 8.5. If V is a tame N -group and N has DCCR, then V has a tame series. A less well known consequence of 8.5 is the following:Theorem 8.6. If V is a tame N -group and N has DCCR, then the following are equivalent (i) V is ﬁnitely generated, (ii) V has ACCS, (iii) V has DCCS and (iv) V has a composition series. Proof: The reader can show that (iv) implies (iii) and (ii). Now a completely reducible N -group with ACCS or DCCS, is a ﬁnite direct sum of minimal N -groups. Since by 8.5, V has a tame series {0} = V0 ≤ V1 ≤ · · · ≤ Vr = V, Vi , (r ≥ 0 an integer), we see that ACCS or DCCS on V , implies Vi+1 /V i = 1, . . . , r − 1, is a ﬁnite direct sum of minimal N -groups and V has

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a composition series. Thus (ii) to (iv) are equivalent. It is easy enough to see that (iv) implies (i). Now, by 3.1 and 3.4 of [16], it follows that a ﬁnitely generated tame N -group has ACCS (see 5.7 of [16]). Thus (i) to (iv) are equivalent. The result is proved. Non-zero completely reducible N -groups have maximal and minimal submodules. Using tame series it can be shown that:Proposition 8.7. If V = {0} is a tame N -group where N has DCCR, then V has maximal and minimal N -subgroups. If V is a tame N -group (N with DCCR) and H < V an N -subgroup of V , then V /H is a tame N -group. As a corollary of 8.7, we have:Corollary 8.8. If V is a tame N -group, where N has DCCR and H < V an N -subgroup of V , then there exists a minimal factor W/H of V and a minimal factor V /K (with K ≥ H) of V . We ﬁnish this section by stating two further well known consequences of tameness. Proposition 8.9. If V is a tame N -group and M a right N -subgroup of N , then for any v in V , M + (0 : v) is a right ideal of N . Proposition 8.10. If V is a faithful tame N -group ((N with DCCR) and M a minimal N -group, then there exists a minimal factor of V , N -isomorphic to M .

9.

The Frattini N -subgroup

If V is a group then, as is well known, the Frattini subgroup is taken as V when V has no maximal subgroups and as the intersection of all maximal subgroups of V , when such subgroups exist. An analogous concept can be deﬁned for an N -group. If V is an N -group, then the Frattini N -subgroup of V is taken as V when V has no maximal N subgroups and as the intersection of all maximal N -subgroups of V , when such N -subgroups exist. This N -subgroup is denoted by Φ(V ). The main purpose of this section is to consider some elementary properties of Φ(V ). This study of Φ(V ) is not intended to be exhaustive. It is only results that have bearing on later sections that are developed. One feature of Φ(V ) (V an N -group) that makes for problems is as follows. For a group the Frattini subgroup is normal (in fact characteristic). In the case of N -groups there is no analogue of this. In general it cannot be assumed that the Frattini N -subgroup is a submodule. However, in many cases of interest to us, this will be so. One of the ﬁrst things to be noted is that, in some respects, Φ(V ) (V an N -group) behaves like a radical.

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Proposition 9.1. If V is an N -group and µ is an N -homomorphism on V , then Φ(V )µ ≤ Φ(V µ). Proof: If V µ has no maximal N -subgroups, then Φ(V µ) = V µ ≥ Φ(V )µ. Suppose V µ has maximal N -subgroups. Such an N -subgroup is of the form Hµ, where H is a maximal N -subgroup of V . Thus Φ(V )µ ≤ Hµ, where Hµ can be any maximal N -subgroup of V µ. It follows that Φ(V )µ ≤ Φ(V µ). The proof is complete. On p.84 of [2], radical maps on nearrings are deﬁned. This deﬁnition need not be conﬁned to nearrings. It is also available for N -groups. From 9.1, it can be seen that the second property such a map is expected to have, is fulﬁlled in the case of Φ (Φ is the map which assigns to an N group V , the N -subgroup Φ(V )). In order for Φ to be a radical map, it would have to satisfy the ﬁrst condition. This is not always a meaningful requirement. Φ(V ) may not be a submodule of V . However, whenever it is, the ﬁrst condition is satisﬁed. Proposition 9.2. If V is an N -group and Φ(V ) is a submodule of V , then Φ(V /Φ(V )) = {0}. Proof: If Φ(V /Φ(V )) > {0}, then Φ(V /Φ(V )) = H/Φ(V ) where H is an N -subgroup of V such that H > Φ(V ). However, Φ(V ) is the intersection of all maximal N -subgroups of V and there must exist a maximal N -subgroup H1 of V such that H1 ∩ H < H. Now H1 /Φ(V ) is a maximal N -subgroup of V /Φ(V ). Thus H/Φ(V ) ≤ H1 /Φ(V ), yielding the contradiction that H ≤ H1 . The proposition holds. Proposition 9.1 is concerned with a general feature of Φ(V ) (V an N group). Proposition 9.2 presents a specialisation. In it Φ(V ) is assumed to be a submodule. In situations of interest to us, this is frequently the case. Indeed, a much stronger condition is often fulﬁlled. This is property q (now deﬁned). An N -group V is said to have property q if, either V has no maximal N -subgroups, or all such N -subgroups are submodules. Clearly, when this is the case it follows that:Proposition 9.3. If V is an N -group with property q, then Φ(V ) is a submodule of V . The ﬁnal goal of this section, will be to say something deﬁnite about certain N -groups of the form V /Φ(V ), where V is an N -group with property q. First we establish another elementary feature of this property. Proposition 9.4. If V is an N -group with property q and µ is an N -homomorphism on V , then V µ has property q.

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Proof: If V µ has no maximal N -subgroups, there is nothing to prove. A maximal N -subgroup of V µ is of the form Hµ, where H is a maximal N -subgroup of V . Thus H is a submodule of V and Hµ is a submodule of V µ. The proposition is proved. It can now be proved that:Theorem 9.5. If V is an N -group with property q and V /Φ(V ) has DCCS, then V /Φ(V ) is a ﬁnite direct sum of minimal N -groups. Proof: If V /Φ(V ) = {0}, there is nothing to prove. When V /Φ(V ) > {0}, we have, by 9.2, V /Φ(V ) has maximal N -subgroups all of which are submodules (see 9.4). Now V /Φ(V ) has a minimal submodule H1 and clearly we can ﬁnd a maximal N -subgroup K1 of V /Φ(V ), with H1 ≤ K1 (see 9.2). Since K1 is a submodule, H1 ⊕ K1 = V /Φ(V ) and H1 is a minimal N -group (otherwise K1 is not maximal). If K1 = {0}, then it contains a minimal submodule H2 of V /Φ(V ) and we may ﬁnd an N -subgroup K2 of V /Φ(V ) which is maximal and such that H2 ≤ K2 . Clearly K1 ∩ K2 is a submodule of V /Φ(V ) and it is easily seen that V /Φ(V ) = H1 ⊕ H2 ⊕ K1 ∩ K2 , K2 is maximal in K1 ). Continuing where H2 is a minimal N -group (K1 ∩K in this way, constructing H1 , H2 , . . . , and K1 , K2 , . . . , we see from the fact that K1 > K1 ∩ K2 > . . . , that there exists an integer n ≥ 1 with K1 ∩ K2 ∩ · · · ∩ Kn = {0}, and V /Φ(V ) is a ﬁnite direct sum of minimal N -groups. The theorem is proved.

10.

Aspects of Complete Reducibility

This section has two purposes. One is to say more about complete reducibility. The other is to introduce perfect N -groups (see below) and develop some related theory. Complete reducibility (see section 8) is an elementary notion. However, its study involves more than ﬁrst impressions indicate. We start by asking a question. Suppose the N -group V is a direct sum of minimal N -groups. When is an N -subgroup of V necessarily a sum of some of these N -groups? Theorem 10.1. Suppose V is a direct sum of minimal N -groups. Any N -subgroup H of V is a direct sum of some of these minimal N -groups if, and only if, no two are N -isomorphic. Proof: Suppose V is the direct sum Hi , i ∈ I, of minimal N -groups Hi , where no two are N -isomorphic. By 8.4, H is a direct sum of minimal

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N -groups. If any such direct summand U of H is one of the Hi , i ∈ I, then the ‘if’ implication will follow. Now U must have a non-zero Hj say, where j is in I). If the projective projection onto one of the Hi (H image of U into Hi , i ∈ I\{j}, (K say), is non-zero, then K is N isomorphic to U and Hj . However, in this case, K projects onto some Hk with k in I\{j} and K is N -isomorphic to Hk . This is a contradiction (H Hk cannot be N -isomorphic to Hj ) and the ﬁrst part of the proof is complete. Now consider the case where V is the direct sum Hi , i ∈ I, of minimal N -groups Hi and every N -subgroup of V is a direct sum of some of these minimal N -groups. If Hj is N -isomorphic to Hk (j and k in I), then we may take a diagonal N -subgroup ∆ of Hj ⊕ Hk . This is a minimal N -subgroup of V (it is N -isomorphic to Hj ). Clearly, it follows that ∆ is one of the Hi . However, ∆ does not coincide with any Hl (l ∈ I\{j, k}), because ∆ ≤ Hj ⊕ Hk . Also, ∆ does not coincide with either Hj or Hk . This contradiction completes the proof. A minimal N -group is certainly tame. The question arises, as to when a completely reducible N -group is tame. Here we have:Theorem 10.2. Let V be a completely reducible N -group. The following are equivalent. (i) V is expressible as a direct sum of minimal N -groups, where no two which are not ring modules are N -isomorphic, (ii) every N -subgroup of V is a submodule, and (iii) V has property q. Proof: First we show that (i) implies (ii). Thus V is expressible as a direct sum of minimal N -groups, where no two which are not ring modules are N -isomorphic. Collecting those which are not ring modules together we see V is a direct sum of the submodules K1 and K2 , where K1 is a direct sum of minimal N -groups which are not ring modules (no two N -isomorphic) and K2 is a direct sum of minimal N -groups which are ring modules. Clearly K2 is a ring module. An N -subgroup H of V is, by 8.4, a direct sum of minimal N -groups. If X is such a minimal N group, then the projective image of X into K1 is either {0} or a minimal N -group of the direct sum of K1 (see 10.1). In the second case, it is not a ring module and cannot project non-trivially into K2 . Thus X ≤ K1 or X ≤ K2 and H = H ∩ K1 ⊕ H ∩ K2 . Now H ∩ K2 is a submodule of V and, by 10.1, H ∩ K1 is also a submodule of V . It follows that H is a submodule of V and (ii) holds. It is obvious that (ii) implies (iii).

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The proof will be complete if it is shown that (iii) implies (i). If every maximal N -subgroup of V is a submodule, then the same is true for V /U , where U is a submodule of V (see 9.4). Suppose V is a direct sum of minimal N -groups two of which (H Hi , i = 1, 2, say) are not ring modules and N -isomorphic. Clearly, because H1 ⊕ H2 is an N -homomorphic H2 is a submodule. However, this image of V , every N -subgroup of H1 ⊕H is a contradiction because a diagonal N -subgroup of H1 ⊕ H2 intersects each Hi trivially. It is N -isomorphic to H1 and cannot be a submodule, since otherwise it would be central in H1 ⊕ H2 (see 6.4) implying H1 is a ring module. Thus (i) holds and the theorem is completely proved. If V is a tame N -group (N with DCCR), then 10.2 applies to the factor V /Φ(V ) of V . Proposition 10.3. If V is a tame N -group and N has DCCR, then V /Φ(V ) satisﬁes (i) to (iii) of 10.2. Proof: If V = {0}, then by 8.7, it has maximal N -subgroups. If H is such an N -subgroup, then V /H is a minimal N -group and V J(N ) ⊆ H. Thus V J(N ) ⊆ Φ(V ) and V /Φ(V ) is an N/J(N )-group. Now, N/J(N ) is a sum of right ideals which are minimal N -groups. It follows that, for any v in V /Φ(V ), v(N/J(N )) is a sum of minimal N -groups and so therefore is v(N/J(N )) over all v in V /Φ(V ). This N -group is just V /Φ(V ) and the complete reducibility of V /Φ(V ) follows (see 8.3). Because V /Φ(V ) is tame, (ii) of 10.2 holds and, by that theorem, so does (i) and (iii). The proposition is proved. We now come to the second feature of this section. This concerns certain perfect N -groups. An N -group V is called perfect if V (1) = V , or equivalently V has no proper submodule H with V /H a ring module. Proposition 10.4. If V is a perfect tame N -group and N has DCCR, then V /Φ(V ) is a ﬁnite direct sum of minimal N -groups which are not ring modules and where no two are N -isomorphic. Proof: By 10.3 and 10.2, V /Φ(V ) can be expressed as a direct sum of minimal N -groups where, amongst those which are not ring modules, no two are N -isomorphic. If some minimal N -group, in the expression of V /Φ(V ) as a direct sum, is a ring module (H1 say), then V /Φ(V ) = H2 a submodule of V /Φ(V )) and, taking H as the N -subgroup H1 ⊕ H2 (H H ≥ Φ(V ) of V such that H/Φ(V ) = H2 , we see V /H is N -isomorphic to H1 . This means V (1) ≤ H and V is not perfect. Thus V is a direct sum of minimal N -groups which are not ring modules and where no two are N -isomorphic. If this direct sum is ﬁnite the result will follow. However, by DCCR, N/J(N ) is a ﬁnite direct sum of minimal N -groups and, as

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is easily checked, any minimal N -group is N -isomorphic to one of them. Clearly the proposition has been proved. An important fact about perfect N -groups as in 10.4 is:Proposition 10.5. If V is a perfect tame N -group where N has DCCR, then V is cyclic. Proof: If V = {0}, the result is true. For non-zero V it follows, from 8.7, that V has maximal N -subgroups and Φ(V ) < V . By 10.4, V /Φ(V ) is a ﬁnite direct sum H1 ⊕ · · · ⊕ Hk of minimal N -groups, where k ≥ 1 is an integer and the Hi , i = 1, . . . , k, are not ring modules and no two are N isomorphic. Take hi , i = 1, . . . , k, as non-zero elements of Hi . By 10.1, if the N -subgroup (h1 + · · · + hk )N of V /Φ(V ) were proper, then at least one of the hi would be zero. Consequently (h1 + · · · + hk )N = V /Φ(V ) and with b in V such that b+Φ(V ) = h1 +· · ·+hk , we see bN +Φ(V ) = V . If bN < V , then since 8.8 ensures the existence of a maximal N -subgroup H ≥ bN of V , we arrive at the contradiction that H = V (H ≥ Φ(V )). Thus bN = V and the proof is complete.

11.

Realisations

Let N be a nearring. N can often be quite successfully studied in terms of properties of N -groups. One instance of this is tame N -groups. Here, it is often possible to use the tame N -group to obtain information about N . An elementary example follows. If V is a tame N -group and M a right N -subgroup of N then, for any v in V , M + (0 : v) is a right ideal of N (see 8.9). Tameness is not the only example of this type of thing. There are quite a number of other situations where N -groups give insight into N . Sometimes these N -groups will also be tame. Even when they are not, they may still provide valuable transfer of information. The cyclic N -groups are an example of this. Associated with them are certain right N -subgroups of N . Such right N -subgroups occur reasonably generally. They are particularly valuable in tame situations. It will turn out that these right N -subgroups, play an extensive and important role in this paper. We shall be deﬁning them shortly. First we notice one manner in which they arise. Proposition 11.1. If V is a cyclic N -group and N has DCCN , then there exists a right N -subgroup M of N , which is minimal such that vM = V , for some v in V . We are now ready to make the deﬁnition on which so much depends. If V is a cyclic N -group, then a right N -subgroup M of N , minimal for the property that there exists v in V with vM = V , will be called a realisation of V . As indicated in 11.1, when N has DCCN , V (cyclic)

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has a realisation. By [16], a tame nearring N with DCCR has DCCN . Thus, in this situation also (the one that will mostly concern us), such right N -subgroups are available. Before specialising to tame situations with DCCR we prove:Proposition 11.2. If V is a cyclic N -group and N has DCCN , then a realisation of V is self monogenic. Proof: Let M be a realisation of V . There exists v in V such that vM = V . Thus, there exists α in M such that vα = v and consequently vαM = vM = V . The minimality of M forces αM to equal M . The proposition is proved. In tame situations, realisations enjoy another important property. Theorem 11.3. If V is a cyclic tame N -group and N has DCCR (thus DCCN ), then a realisation of V has property q. Proof: If M is a realisation of V , then there exists v in V such that vM = V . Take a maximal right N -subgroup M1 of M . By the minimality of M , v[M M1 + (0 : v)] = vM M1 < vM and M1 + (0 : v) cannot contain M . Since (M M1 + (0 : v)) ∩ M (= L say) is M or M1 , L must coincide with M1 . By 8.9, M1 + (0 : v) is a right ideal of N and therefore L is a submodule of M . The theorem is proved. The realisation of an N -group (tame and cyclic and N with DCCR), is more closely related to the N -group than might at ﬁrst be expected. Theorem 11.4. Suppose that V is a cyclic tame N -group, where N has DCCR and M is a realisation of V . Now, if M/H is a minimal factor of M , then it is N -isomorphic to a minimal factor V /W of V . Furthermore, if V /U is a minimal factor of V , then it is N -isomorphic to a minimal factor M/K of M . Proof: Since M is a realisation of V , there exists v in V such that vM = V . Taking H as a maximal submodule of M , we see that the natural N -homomorphism µ of M/H onto vM/vH is an N -isomorphism. This is because M/H has no proper submodules and (M/H)µ = {0} or µ is an N -isomorphism. The ﬁrst case is excluded, by the minimality of M because, we would then have the contradiction that vM = vH. Take U as a submodule of V , such that V /U is minimal. The natural N -homomorphism µ mapping M onto vM (= V ) and then onto V /U has kernel K, with M/K, N -isomorphic to V /U . Thus M/K is a minimal factor of M , N -isomorphic to V /U . The theorem is proved.

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12.

Minimal Ideals

Throughout this section it is not assumed our nearrings necessarily have an identity. The material of this section has a feature that others do not share. All others tend to focus on tame considerations. Although the theorem proved here, has application to tame nearrings, it is desirable to establish the result more generally. It is nearrings with DCCN that will be the focus of attention. What we are seeking to prove is as follows. For a nearring N with DCCN , the center Z(N ) of the N -group N , may intersect a minimal ideal A non-trivially. It looks quite possible, when this is the case, that A ≤ Z(N ). There is no obvious reason for such inclusion but it is plausible. What is proved below will establish this. In order to carry out the proof we must look at elements of the set mr(A), of all right N -subgroups of N minimal for not annihilating A from the left. The collection mr(A) is empty precisely when N A = {0}. By 6.1 of [15], elements of mr(A) (if they exist) are self monogenic and N -isomorphic. By [8], such a right N -subgroup has a left identity. The ﬁrst thing to observe is that Proposition 12.1. Let N be a nearring with DCCN and a minimal ideal A. If M is in mr(A) and e is a left identity of M , then eA = M ∩ A = M A. Proof: This follows because M ∩ A = e.M ∩ A ≤ eA ⊆ M A ⊆ M ∩ A. Next note that:Lemma 12.2. Let N be a nearring with DCCN , A a minimal ideal of N , M in mr(A) and α in M . If R ≤ A is a right ideal of N , then αR is a submodule of M . Furthermore, if R ≤ Z(N ), then αR is a central submodule of the N -group M . Proof: Since α is in M , it follows that αN = M or αN < M . In the ﬁrst case αR is clearly a submodule of M . Also if R ≤ Z(N ), then by 6.1, αR ≤ Z(M ). If it is shown that when αN < M , αR = {0}, then the result will follow. If αN < M , then αN A = {0} and N A ≤ (0 : α). Thus if R(N A) is the right ideal of N generated by N A, then R(N A) ≤ (0 : α). However, N.N A ⊆ R(N A) and N.R(N A) ⊆ R(N A), proving R(N A) is an ideal. Clearly R(N A) = A (N A = {0}) and A ≤ (0 : α). Thus, in the case where αN < M , it follows that αR = {0}. The lemma is proved. An important step in proving the theorem indicated above now follows.

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Lemma 12.3. Let N be a nearring with DCCN and a minimal ideal A such that A ∩ Z(N ) = {0}. If M is in mr(A), then M ∩ A is a central submodule of M . Proof: Take R = A ∩ Z(N ). Clearly R is a right ideal of N . For all α in M , αR is by 12.2, a submodule of M . Let H = αR, taken over all α in M . Now H = {0}, otherwise M R = {0} and M.Id(R) (Id(R) the ideal generated by R) is zero. This would imply M A = {0}, contrary to M being in mr(A). Also, for each γ in M and α in M , γαR is contained in H and M H ⊆ H. Thus M.Id(H) ⊆ H and, since H is non-zero and contained in A, M A ⊆ H. It follows, by 12.1, that M ∩ A ≤ H. Clearly, for all α in M , αR ≤ M ∩ A and H ≤ M ∩ A. It has been shown that M ∩ A = H. By 12.2, each αR (α in M ) is in Z(M ) and H ≤ Z(M ). Thus M ∩ A is a central submodule of M . The centrality of an N -subgroup of a faithful N -group can be transferred to the nearring. Proposition 12.4. If V is a faithful N -group and U a central submodule of V , then (U : V ) ≤ Z(N ). Proof: Let α and γ be in N and β in (U : V ). We have that v(−α + β + α) = −vα + vβ + vα = vβ, for all v in V because vβ is in Z(V ). Thus −α + β + α = β and β is in Z + (N ). Now v[(α + β)γ − βγ − αγ] = (vα + vβ)γ − vβγ − vαγ = vαγ + vβγ − vβγ − vαγ = 0, for all v in V because vβ is in Z(V ). Thus (α + β)γ − βγ − αγ is zero and (U : V ) ≤ Z(N ). The proof is complete. Theorem 12.5. If N is a nearring with DCCN and A a minimal ideal of N such that A ∩ Z(N ) = {0}, then A ≤ Z(N ). Proof: If mr(A) = ∅, then N A = {0} and, in particular, N.A∩Z(N ) = {0}. Hence, A ∩ Z(N ) is a non-zero ideal of N contained in A and A = A∩Z(N ). Thus, in this case, the result follows. Suppose mr(A) = ∅ and let M be in mr(A). Now (0 : M ) ∩ A = {0} and N/(0 : M ) is faithful on the N/(0 : M )-group M . It follows readily, by 12.3, that M ∩ A is a central submodule of this N/(0 : M )-group. However, since M.[A + (0 : M )] = M.A, we have, by 12.1, M.([A + (0 : M )]/(0 : M )) ⊆ M ∩ A

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and [A + (0 : M )]/(0 : M ) ≤ (M ∩ A : M ) (in N/(0 : M )). By 12.4, [A + (0 : M )]/(0 : M ) is a central submodule of N/(0 : M ). It follows, from 6.3, that A/{0} is a central submodule of N/{0}. Clearly, the theorem is fully proved.

13.

Annihilators

This section is similar to section two of [14]. It is included here for two reasons. The ﬁrst is that it has some bearing on results of this paper. The second is that, it appears that some researchers in tame theory, are not aware of the signiﬁcance of the results contained here. We start by establishing a simple proposition. Proposition 13.1. If N is a nearring and Ai , i = 1, 2, 3, are ideals of N , such that N = A1 + A3 = A2 + A3 , then N = A1 ∩ A2 + A3 . Proof: Because N has an identity, it follows that A2 = A2 .N = A2 .A1 + A2 .A3 , and A2 is contained in A1 ∩ A2 + A3 . However, this last ideal contains A3 and therefore A1 ∩ A2 + A3 contains A2 + A3 . It has been shown that N = A1 ∩ A2 + A3 and the proposition holds. For a non-empty subset S of a nearring N , S r (r ≥ 1 an integer) is the set of all products α1 α2 . . . αr , with αi , i = 1, . . . , r, in S. The ideal of N generated by S will be denoted by Id(S). Proposition 13.2. If N is a nearring with A and B ideals of N such that A + B = N , then N = Id(Ar ) + B for any integer r ≥ 1. Proof: Because N has an identity, it follows that A = A.N ⊆ A2 + A.B ⊆ A2 + B. Also N = A + B ⊆ A2 + B + B = A2 + B. Proceeding further, we see that A = A.N = A.(A2 + B) ⊆ A3 + B and N = A + B ⊆ A3 + B + B = A3 + B. It follows that N = Ar + B, for any integer r ≥ 1. Thus N = Id(Ar ) + B and the proposition holds.

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Theorem 13.3. Let N be a nearring with DCCR and U and W tame N -groups. The ideals (0 : U ) and (0 : W ) are such that (0 : U ) + (0 : W ) = N if, and only if, no minimal factor of U is N -isomorphic to a minimal factor of W . Proof: It can be assumed that U = {0} and W = {0}. First suppose that no minimal factor of U is N -isomorphic to a minimal factor of W . By 8.2, the number of distinct N -isomorphism types of minimal factors of U is a ﬁnite integer n ≥ 1. Let m be the corresponding number of N -isomorphism types of minimal factors of W . Take L1 , . . . , Ln as n minimal factors of U of distinct N -isomorphism type and K1 , . . . , Km , as m minimal factors of W of distinct N -isomorphism type. Set A = ∩ni=1 (0 : Li ) and B = ∩m j=1 (0 : Kj ). From 8.5, U has a tame series U0 = {0} < U1 < U2 < · · · < Ur = U of length r, where r ≥ 1 is an integer. Now A annihilates all minimal factors of U and it follows readily that (U Ui+1 /U Ui ).A = {0} (i = 0, . . . , r − 1) and U.Ar ⊆ Ur−1 .Ar−1 ⊆ · · · ⊆ U1 .A = {0}. Thus Ar ⊆ (0 : U ) and Id(Ar ) ≤ (0 : U ), while an entirely similar argument shows that there exists an integer s ≥ 1 with Id(B s ) ≤ (0 : W ). Once it is shown that Id(Ar ) + Id(B s ) = N , it will follow that (0 : U ) + (0 : W ) = N . In order to prove this we ﬁrst show that A + B = N . Now, for i in {1, . . . , n} and j in {1, . . . , m}, (0 : Li ) = (0 : Kj ). Indeed, if for some i and j, (0 : Li ) = (0 : Kj ), then Li is a minimal N/(0 : Li )-group and Kj a minimal N/(0 : Li )-group. Because N/(0 : Li ) is primitive and has DCCR, we see by 4.46 of [4], that Li is N/(0 : Li )-isomorphic to Kj . This means Li is N -isomorphic to Kj , contrary to our assumptions. It has been shown that (0 : Li ) = (0 : Kj ). Now we note that, by 4.46 of [4], N/(0 : Li ) and N/(0 : Kj ) are simple and (0 : Li ) and (0 : Kj ) are maximal ideals of N . Thus (0 : Li ) + (0 : Kj ) = N and continued application of 13.1 gives A + (0 : Kj ) = N , for all j in {1, . . . , m}. Once again, continued application of 13.1, yields the fact that A + B = N . From 13.2, Id(Ar ) + B = N and applying that result again gives Id(Ar ) + Id(B s ) = N . From what was indicated above, (0 : U ) + (0 : W ) = N . To ﬁnish the proof it must be shown that if, (0 : U ) + (0 : W ) = N , then U does not have a minimal factor N -isomorphic to one of W . We lead the assumption that, U has a minimal factor L, N -isomorphic to a

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minimal factor K of W , to a contradiction. For α in (0 : U ), Lα = {0} and (0 : U ) ≤ (0 : L). Similarly (0 : W ) ≤ (0 : K). Because (0 : L) = (0 : K), N ≤ (0 : L), which is clearly false. The theorem is proved. The above result is contained in the 1995 publication [14]. It is an obvious consequence that when N (with DCCR) is tame on V1 ⊕ V2 , (0 : V1 ) ⊕ (0 : V2 ) = N if, and only if, no minimal factor of V1 is N isomorphic to one of V2 . However, E. Aichinger in [1] (2001) has gone to some length to establish this in the ﬁnite compatible case. What can be said when N is tame on a subdirect sum? Here we have the following:Theorem 13.4. Suppose the nearring N has DCCR and is tame on V . If Ui , i = 1, 2, are submodules of V such that U1 ∩ U2 = {0} and no minimal factor of U1 is N -isomorphic to one of U2 , then (U U1 ⊕ U2 : V ) = (U U1 : V ) ⊕ (U U2 : V ).

Proof: From 13.3, it follows that (0 : U1 ) + (0 : U2 ) = N . This means 1 can be expressed in the form e1 + e2 , where e1 is in (0 : U1 ) and e2 in (0 : U2 ). Now α in (U U1 ⊕ U2 : V ) is of the form αe1 + αe2 and, for v in V , vα = u1 + u2 , where ui is in Ui , i = 1, 2. Also, vαe1 = u2 e1 and, since u2 = u2 e1 + u2 e2 = u2 e1 , we have vαe1 = u2 . It has been shown U2 : V ). Similarly αe2 is in (U U1 : V ). It follows that α is that αe1 is in (U U2 : V ) and we conclude that in (U U1 : V ) + (U U1 : V ) + (U U2 : V ). (U U1 ⊕ U2 : V ) ≤ (U U1 ⊕ U2 : V ) and Clearly the (U Ui : V ), i = 1, 2, are contained in (U the reverse inclusion holds. It is easy to establish that the sum (U U1 : U1 : V ) ∩ (U U2 : V ), then V ) + (U U2 : V ) is direct. Indeed, if β is in (U V β ⊆ U1 ∩ U2 = {0}, and from the faithfulness of V we have β = 0. It follows that U1 : V ) ⊕ (U U2 : V ) (U U1 ⊕ U2 : V ) = (U and the theorem is completely proved.

14.

Covers

Experience has shown that an N -group V is often best studied in terms of factors of V . The factors of V that occur most frequently are minimal factors. This approach to N -groups is also relevant to the tame situation. Again minimal factors play a basic role. Unlike the case of arbitrary V , they are necessarily minimal N -groups. Having some

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notation for such factors is desirable. Thus, if V is a tame N -group, we let m(V ) denote the set of all minimal factors of V . However, m(V ) can have proper subsets which are almost as basic as m(V ). One such important subset is nr(V ). This is just the set of all minimal factors of V , which are not ring modules. In group theory, the notions of covering and avoidance make themselves felt. They apply to normal subgroups H1 ≥ H2 of a group G. The H2 , if W + H2 ≥ H1 . At the other extreme subgroup W of G covers H1 /H H2 . This happens when W ∩ H1 ≤ H2 . This concept W can avoid H1 /H has an easy adaptation to N -groups. It is the particular case of tame N -groups that will concern us. The situation is restricted still further U2 in m(V ), then to minimal factors. If V is a tame N -group and U1 /U U2 is covered by the N -subgroup W of V when W + U2 ≥ U1 . It is U1 /U U2 , precisely when W ∩ U1 covers an elementary fact that W covers U1 /U U2 . U1 /U Proposition 14.1. An N -subgroup W of a tame N -group V , covers the U2 of m(V ), if and only if, W ∩ U1 > W ∩ U2 . Furthermore, element U1 /U U2 is N -isomorphic to (W ∩ U1 )/(W ∩ U2 ). in this situation, U1 /U U2 , then W + U2 ≥ U1 and W ∩ U1 + U2 ≥ U1 . Proof: If W covers U1 /U Consequently W ∩ U1 > W ∩ U2 (otherwise U1 ≤ U2 ). If W ∩ U1 > W ∩ U2 , then W ∩ U1 is not contained in U2 and is contained in U1 . U2 . Thus W ∩ U1 + U2 ≥ U1 and W covers U1 /U U2 , then (W ∩U U1 )/(W ∩U U2 ) is certainly N -isomorphic If W covers U1 /U U2 (= U1 /U U2 ), and the proposition is proved. to (W ∩ U1 + U2 )/U The notion of avoidance will be handled diﬀerently from that set out above. This is possible because we are dealing only with minimal factors. It is desirable to see that, avoidance holds precisely when covering does not. An N -subgroup W of a tame N -group V , will be said to avoid the U2 of m(V ) if W + U2 ≥ U1 . Since W covers U1 /U U2 precisely element U1 /U U2 , we have W avoids U1 /U U2 precisely when W ∩U U1 = when W ∩U U1 > W ∩U W ∩ U2 . This last statement makes the use of the word ‘avoid’ clearer. For a tame N -group V , it can happen that the N -subgroup W of U2 of m(V ) but avoids an element H1 /H H2 of V , covers an element U1 /U U2 . There is nothing very special about such m(V ), N -isomorphic to U1 /U behaviour. It can occur in the 2-tame or compatible case. Even when U2 to be in nr(V ), it still appears possible, in the tame we require U1 /U case, for this type of thing to happen. It would seem that, in general, U2 , necessarily being a there is not much likelihood of a cover of U1 /U H2 . That is why it is surprising that, in the 2-tame case, cover of H1 /H covers of elements of nr(V ), also cover N -isomorphic elements of nr(V ).

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Developments of this paper tend to rest on this important observation. In order to establish it, we ﬁrst prove:U2 and Theorem 14.2. Suppose V is a 2-tame N -group with U1 /U H2 , N -isomorphic factors of V . H1 /H H2 ≤ Z(U U1 /H H2 ). If H1 ≤ U2 , then H1 /H H2 has U1 /U U2 as a natural N -homomorphic image Proof: Since U1 /H H2 ) and U1 /U U2 is N -isomorphic to H1 /H H2 , it follows that, (kernel U2 /H H2 is mapped by an N -endomorphism µ onto H1 /H H2 . By 6.7, U1 /H H2 )µ ∩ (U U1 /H H2 )(1 − µ) ≤ Z(U U1 /H H2 ). (U U1 /H H2 ≥ H1 /H H2 and (H1 /H H2 )(1 − µ) = H1 /H H2 . However, kerµ ≥ U2 /H H2 )µ and (U U1 /H H2 )(1 − µ) contain H1 /H H2 , it follows that Since both (U U1 /H H2 ≤ Z(U U1 /H H2 ) and the proof is complete. H1 /H Now for the result mentioned above. U2 in nr(V ). Theorem 14.3. Suppose V is a 2-tame N -group and U1 /U U2 , it covers all elements of nr(V ), If the N -subgroup W of V covers U1 /U U2 . N -isomorphic to U1 /U H2 , in nr(V ), is N -isomorphic to U1 /U U2 , but W Proof: Suppose H1 /H H2 . In this case W ∩U U1 > W ∩U U2 and W ∩H1 = W ∩H H2 . fails to cover H1 /H U2 , while We have (W ∩ U1 )/(W ∩ U2 ) is N -isomorphic to U1 /U H2 + W ) = (H1 + H2 + W )/(H H2 + W ), (H1 + W )/(H is N -isomorphic to H1 /(H1 ∩[H H2 +W ]), which is N -isomorphic to H1 /H H2 . Since W ∩ U1 ≤ H2 + W and the factors (W ∩ U1 )/(W ∩ U2 ) and (H1 + W )/(H H2 + W ) are N -isomorphic, we obtain the contradiction (see 14.2) that (W ∩ U1 )/(W ∩ U2 ) ≤ Z[(H1 + W )/(W ∩ U2 )]. This is a contradiction because U1 /U U2 is in nr(V ). The theorem is proved. Theorem 14.3 is a basis for further understanding. Certainly an N subgroup W of the 2-tame N -group V , covering an element of nr(V ), covers all elements of nr(V ) which are N -isomorphic to it. However, more is true. U2 an Proposition 14.4. Suppose V is a 2-tame N -group and U1 /U element of nr(V ). If Wi , i = 1, 2, are right N -subgroups of V covering U2 , then so is W1 ∩ W2 . U1 /U W2 ∩ U2 ) is N -isomorphic to U1 /U U2 . By 14.3, Proof: Now (W W2 ∩ U1 )/(W this can only mean W1 covers this minimal factor of V and (W W1 ∩ W2 ∩ U1 )/(W W1 ∩ W2 ∩ U2 ) is N -isomorphic to U1 /U U2 .

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This means W1 ∩ W2 covers U1 /U U2 and the proof is complete. Previous results of this section make it desirable to have more notation. Theorem 14.3 indicates that the covering of minimal factors is, in a number of important situations, much the same as covering of N isomorphism types of minimal factors. We need to be able to look more at such classes of minimal N -groups. In order to do this we deﬁne t(N ) (N a nearring) as the set of all N -isomorphism types of minimal N groups. Certainly t(N ) is a set. Although, its elements are classes, t(N ) has some cardinality. Indeed, in the case where N has DCCR, t(N ) is a ﬁnite set. This is because, N/J(N ) is a ﬁnite direct sum of minimal right ideals which are minimal N -groups, and every minimal N -group is N -isomorphic to such a right ideal. t(N ) is of value to us, but so also is an important subset of t(N ). This is denoted by tnr(N ). It is simply the set of all N -isomorphism types of minimal N -groups, which are not ring modules. We shall have need to deal with subsets of tnr(N ). If S is a subset of tnr(N ) and V a tame N -group, then V (S) will denote the U2 in nr(V ), that are in an element of S. Loosely speaking, set of all U1 /U V (S) is the set of all minimal factors of V of type S. Certainly V (S) could be deﬁned, for subsets of t(N ). However, it is subsets of tnr(N ) which constitute our chief interest. We now set about deﬁning certain covers. If V is a tame N -group and S a subset of tnr(N ), then a cover of V (S) is an N -subgroup W of V , that covers all elements of V (S). By 14.3, when V is 2-tame, W is a cover of V (S) if, and only if, it covers an element of V (S), of each distinct N -isomorphism type. Also, it is clear from 14.4, that if Wi , i = 1, 2, are covers of V (S), then so is W1 ∩ W2 . In tame situations where W is a cover of V (S), such that any N -subgroup H < W of V , fails to cover V (S), we shall call W a minimal cover of V (S). Theorem 14.5. Let V be a 2-tame N -group and S a subset of tnr(N ). If V (S) has a cover with DCCS (minimal condition on submodules), then it has a minimal cover contained in any cover of V (S). Furthermore, this cover has DCCS. Proof: Let W be a cover of V (S) with DCCS. Let Wi , i ∈ I, be all other covers of V (S). Thus {W } ∪ {W Wi : i ∈ I} (= K say) is the set of all covers of V (S). However, L = (∩i Wi ) ∩ W is a ﬁnite intersection of elements of K (by DCCS in W ). By 14.4, L covers V (S), L is a minimal cover and L is contained in any cover of V (S). Since L ≤ W it has DCCS. The theorem is proved. For V a 2-tame N -group where N has DCCR, minimal covers as in 14.5, always exist. If S is a subset of tnr(N ), then S is ﬁnite and with Wi , i = 1, . . . , k, (k ≥ 1 an integer) as representatives of each element Ui /W

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of V (S), we may take ui in Ui not in Wi . Clearly u1 N + · · · + uk N covers Wi and is ﬁnitely generated. It therefore (see 8.6) has DCCS. each Ui /W Thus it certainly follows that:Theorem 14.6. Let V be a 2-tame N -group and S a subset of tnr(N ). If N has DCCR, then V (S) has a minimal cover contained in any cover of V (S). Furthermore, this cover has a composition series (see 8.6).

15.

Minimal Covers and Realisations

Suppose V is a 2-tame N -group (N with DCCR) and S a subset of tnr(N ). In the last section we looked at the minimal cover of V (S). Developments of this paper depend on such minimal covers. They enjoy certain properties. One is that they are perfect. Another is that they are cyclic. Both these properties are established by looking at certain principal minimal factors. The fact that such covers are cyclic, opens up possibilities. They have realisations. How close to the N -subgroup is a realisation associated with it? The last theorem of this section states that their Frattini factors are N -isomorphic. This is a start in establishing other relationships between such covers and their realisations. It is theorem 15.5, that is our present goal. Theorem 15.1. Suppose V is a 2-tame N -group ((N with DCCR), S a subset of tnr(N ) and W the minimal cover of V (S). If U is a maximal N -subgroup of W , then W/U is an element of V (S). Proof: Because W is a minimal cover of V (S), U avoids some minimal W2 of V (S). Thus U ∩ W1 = U ∩ W2 ≤ W ∩ W2 and because factor W1 /W W2 (see 14.3) W covers W1 /W U ∩ W 1 = U ∩ W2 ≤ W ∩ W2 < W ∩ W1 . The only possibility ((W ∩ W1 )/(U ∩ W1 ) is a minimal factor of V ) is that, W ∩ W2 = U ∩ W1 and the two factors (W ∩ W1 )/(U ∩ W1 ) and (W ∩ W1 )/(W ∩ W2 ) coincide. The ﬁrst is N -isomorphic to W/U and the second to W1 /W W2 . Since W1 /W W2 is an element of V (S), the theorem is proved. Corollary 15.2. If V , N , W and S are as in 15.1, then W is perfect. Proof: If W (1) < W then, by 8.8, we can ﬁnd a maximal N -subgroup U ≥ W (1) of W . This is a contradiction, because by 15.1, W/U is not a ring module. It follows that W is perfect. Corollary 15.3. If V , N , W , and S are as in 15.2, then W is cyclic. Proof: This follows from 10.5.

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Corollary 15.4. If V , N , W and S are as in 15.3, then W/Φ(W ) is a ﬁnite direct sum of minimal N -groups which are not ring modules and where no two are N -isomorphic. Proof: This follows from 10.4. A minimal N -group direct summand of W/Φ(W ) (as in 15.4) is necessarily N -isomorphic to an element of an element of S (see 15.1). However, it is possible that there exist elements of an element of S not N -isomorphic to such a direct summand. One reason that this is true is that not all elements of S may contain an element of V (S). However, there is a more profound reason. Not all elements of V (S) are necessarily N -isomorphic to such a direct summand. These factors are those which occur ‘below’ Φ(W ). Now with V , N , W and S as in 15.1, we see from 15.3, that W has a realisation M . Since, by 11.3, M has property q, so also does M/Φ(M ). By 9.5 and 10.2, M/Φ(M ) is a ﬁnite direct sum of minimal N groups, where amongst those which are not ring modules no two are N isomorphic. If, in such a direct sum, ring modules occur, then M/Φ(M ) contains a maximal submodule H, with (M/Φ(M ))/H a ring module. This is not possible because of 15.1 and 11.4. Thus (see 10.4) both W/Φ(W ) and M/Φ(M ) are ﬁnite direct sums of minimal N -groups, which are not ring modules and where no two are N -isomorphic. If W/Φ(W ) = W1 ⊕ · · · ⊕ Wk and M/Φ(M ) = H1 ⊕ · · · ⊕ Hs are two such direct decompositions then, by 10.1, W/Φ(W ) has k maximal N -subgroups (submodules) Ui , i = 1, . . . , k, (direct complements Ui are N -isomorphic. Similarly, Wi say), where no two of (W/Φ(W ))/U M/Φ(M ) has s maximal N -subgroups (submodules) Li , i = 1, . . . , s, (direct complements Hi say), where no two of (M/Φ(M ))/Li are N isomorphic. Because, by 11.4, each (W/Φ(W ))/U Ui is N -isomorphic (to Wi ) to an (M/Φ(M ))/Lj (to Hj ) and vice versa, it follows readily that k = s and W/Φ(W ) is N -isomorphic to M/Φ(M ). It has been shown that:Theorem 15.5. With V , N , W , S and M as above, W/Φ(W ) is N isomorphic to M/Φ(M ).

16.

A Theorem

The main goal of this paper is to establish a relationship (N -isomorphism) between certain N -groups. In this connection realisations are a valuable asset. This is because, when the N -groups involved are cyclic, questions of a relationship between them, are transferred to the nearring.

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Thus, no longer are we working with independent objects. Instead, we are considering reasonably well behaved subobjects of the same structure (the nearring). How this works in practice is diﬀercult to explain. Indeed, that is what much of this paper is about. This section deals with a small but important corner. We are looking at minimal covers in faithful 2-tame N -groups. These are covers of corresponding minimal factors. An attempt is being made (via realisations), to relate one minimal cover to the other. Let V and W be two faithful 2-tame N -groups (N with DCCR). It is too much to expect there will be a signiﬁcant connection between these N -groups. Evidence for this can be seen in the case of rings. Faithful ring modules allow the construction of many others. However, if considerations are restricted to certain N -subgroups of V and W , things look brighter. What we must look at is ‘non-ring’ subobjects of V and W . The right subobjects appear to be the minimal covers (V V1 and W1 respectively) of V (S) and W (S), where S is a subset of tnr(N ). These covers have features that allow analysis. Indeed, approaching them as set out in the ﬁrst paragraph, is rewarding. This is because both V1 and W1 are cyclic and have realisations. It would be a very big step forward, if a realisation of V1 was also one of W1 . The fact that this is indeed true will not concern us at the moment. A preliminary step in this direction is what this section undertakes. It is shown here, that a realisation M1 of V1 relates to W1 in an important manner. Indeed, the sum wM M1 , over all w in W , coincides with W1 . Establishing this theorem is now undertaken. Two lemmas are proved. These embody most of the content of the theorem. This section concludes with its statement and proof. Lemma 16.1. Suppose V and W are faithful 2-tame N -groups ((N with DCCR) and S is a subset of tnr(N ). If M is a realisation of the minimal W2 an element of W (S), then there exists w in cover of V (S) and W1 /W W2 . W such that wM covers W1 /W Proof: Let U be a minimal cover of V (S). By 8.10, V (S) has an element U1 /U U2 , N -isomorphic to W1 /W W2 . Let M be a realisation of U U2 (see 15.3). There exists u in U such that uM = U . Now U ∩U U1 > U ∩U and, (U ∩ U1 )/(U ∩ U2 ) is N -isomorphic to U1 /U U2 . If λ is the natural N homomorphism of M onto uM (= U ), then the respective inverse images U1 and U ∩U U2 are submodules of M such that H1 and H2 under λ, of U ∩U H2 is N -isomorphic to (U ∩ U1 )/(U ∩ U2 ). Take a minimal factor H1 /H U1 )/(U ∩U U2 ), where L1 is chosen to be L1 /L2 of M , N -isomorphic to (U ∩U minimal. Now L1 does not contain a submodule K < L1 of M such that K + L2 = L1 , otherwise L1 /L2 is N -isomorphic to K/(K ∩ L2 ). Since W is faithful, there exists w in W , such that L1 ∩ (0 : w) < L1 . Thus

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L1 ∩ (0 : w) ≤ L2 and wL1 > wL2 (otherwise L1 = L2 + L1 ∩ (0 : w)). U2 and wM covers wL1 /wL2 . Hence, wL1 /wL2 is N -isomorphic to U1 /U W2 . By 14.3, as required it covers W1 /W Now for the second lemma. Lemma 16.2. Suppose V and W are faithful 2-tame N -groups ((N with DCCR) and S a subset of tnr(N ). If M is a realisation of the minimal cover V1 of V (S), then for all w in W , wM ≤ W1 , where W1 is the minimal cover of W (S). Proof: Let a maximal N -subgroup U of wM , have inverse image L in M , under the natural N -homomorphism of M onto wM . Thus M/L is N -isomorphic to wM/U . However, by 11.4, M/L is N -isomorphic to V2 of V1 . Certainly V1 /V V2 is an element of V (S) a minimal factor V1 /V (see 15.1). It has been shown that, for any maximal N -subgroup U of wM , wM/U is an element of W (S). Now wM is minimal for covering all wM/U (U a maximal N -subgroup of wM ). Indeed, if it were not, then some submodule X < wM , covers all wM/U and with U1 ≥ X maximal in wM , X covers wM/U U1 . Clearly this is not the case. Thus, as stated above, wM is minimal for covering the elements wM/U of W (S). Since W1 is minimal for covering all elements of W (S), it follows that wM ≤ W1 . The lemma is proved. Everything is now in place to prove a fundamental theorem. Theorem 16.3. Suppose V and W are faithful 2-tame N -groups ((N with DCCR), S a subset of tnr(N ) and M a realisation of the minimal cover of V (S). If W1 is the minimal cover of W (S), then the sum wM , over all w in W , coincides with W1 . Proof: For W2 /W W3 a minimal factor of W in W (S), it follows from 16.1, W3 . Thus wM that there exists w1 in W such that w1 M covers W2 /W M (w in W ) is contained in covers W (S). However, by 16.2, each w 2 2 wM ≤ W1 . Thus W1 = wM and the theorem is proved. W1 and

17.

Applications

Suppose V and W are faithful 2-tame N -groups (N with DCCR) and S a subset of tnr(N ). The ﬁnal theorem of the last section, establishes a relationship between respective minimal covers V1 and W1 of V (S) and W (S). The result uses a realisation of V1 and, although it may not look to be a substantial contribution to our understanding, it opens up further possibilities. We are seeking to bring V1 and W1 into closer relationship. This is where applications of 16.3 can be made. The ﬁrst V2 of V1 , are is that, the N -isomorphism types of minimal factors V1 /V

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the same as similar principal N -isomorphism types of minimal factors of W1 . It is then shown that, for respective realisations M1 and M2 of M1 ) is N -isomorphic to M2 /Φ(M M2 ). This will allow, V1 and W1 , M1 /Φ(M at a later stage, the conclusion that M1 is N -isomorphic to M2 . Indeed, a stronger relationship than N -isomorphism exists between them. It is hard to compare V1 and W1 , if as faithful N/(0 : V1 ) and N/(0 : W1 )-groups (respectively), there is little similitude between the nearrings N/(0 : V1 ) and N/(0 : W1 ). This problem is fully overcome in the last result of this section (also an application of 16.3). The rather surprising fact that the annihilators (0 : V1 ) and (0 : W1 ) coincide, is established. This is a result used in a very meaningful way later. Lemma 17.1. Suppose V and W are faithful 2-tame N -groups ((N with DCCR) and S is a subset of tnr(N ). If V1 is the minimal cover of V (S) W2 of and W1 the minimal cover of W (S), then a minimal factor W1 /W V2 of V1 and vice versa. W1 is N -isomorphic to a minimal factor V1 /V W2 of W1 is Proof: Clearly, if it is shown that a minimal factor W1 /W V2 of V1 , then it will follow that N -isomorphic to a minimal factor V1 /V V2 of V1 is N -isomorphic to one W1 /W W2 of W1 . a minimal factorV1 /V wM over all w in W , where M is a realisation of By 16.3, W1 = V1 . Thus there exists w1 in W , such that w1 M is not contained in W2 . Now w1 M ∩ W2 is a maximal N -subgroup of w1 M . The inverse image of w1 M ∩ W2 in M , under the natural N -homomorphism of M onto w1 M , yields a maximal submodule L of M with M/L, N -isomorphic to V2 of V1 with w1 M/(w1 M ∩ W2 ). By 11.4, there is a minimal factor V1 /V V1 /V V2 , N -isomorphic to w1 M/(w1 M ∩ W2 ). Since w1 M/(w1 M ∩ W2 ) is W2 (= W1 /W W2 ) the lemma is proved. N -isomorphic to (w1 M + W2 )/W The discussion preceeding theorem 15.5 was used to show W/Φ(W ) was N -isomorphic to M/Φ(M ). Similar arguments can be used to show V1 ) is N -isomorphic to W1 /Φ(W W1 ). We that, in the present case, V1 /Φ(V therefore conclude:Theorem 17.2. Suppose V and W are faithful 2-tame N -groups ((N with DCCR) and S a subset of tnr(N ). If V1 is the minimal cover of V1 ), M1 /Φ(M M1 ) V (S) and W1 the minimal cover of W (S), then V1 /Φ(V W1 ) and M2 /Φ(M M2 ) (M (M2 a realisation (M1 a realisation of V1 ), W1 /Φ(W (M of W1 ) are all N -isomorphic. This section concludes with an important application of theorem 16.3. Theorem 17.3. Suppose V and W are faithful 2-tame N -groups ((N with DCCR) and S a subset of tnr(N ). If V1 is the minimal cover of V (S) and W1 the minimal cover of W (S), then (0 : V1 ) = (0 : W1 ).

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Proof: Clearly, if it is shown that (0 : W1 ) ≤ (0 : V1 ), then the symmewM ) try of the situation will give equality. By 16.3, (0 : W1 ) = (0 : (sum over all w in W where M is a realisation of V1 ). Thus (0 : W1 ) ≤ (0 : W M ) ≤ (0 : M ), because if α in N annihilates W M , then M α annihilates W and M α = {0} (W is faithful). However, V1 = v1 M where v1 is in V1 and (0 : M ) ≤ (0 : V1 ). The theorem is completely proved.

18.

Central Factors

This paper is about how Z-constraint (see section 25) implies the N isomorphism of certain N -groups. There are two main features of this undertaking. One relates to relationships between N -groups (the N isomorphism aspect). The other covers Z-constraint. Some N -isomorphism material has already been developed (sections 14 to 17). Such matters will also be addressed later. For the moment we postpone developments of this kind. It is the second aspect of this paper, that takes our attention. Moving in this direction, is what this section initiates. Here, we look at central factors. U2 of A subset S of an N -group V is said to centralise the factor U1 /U V , if (a + b)α ≡ aα + bα mod U2 , for all a in S, b in U1 and α in N . U2 of V is central, if V centralises U1 /U U2 , or equivalently A factor U1 /U U2 ≤ Z(V /U U2 ). U1 /U In this section central factors of tame N -groups are considered. The ﬁrst half looks at tame N -groups. After that considerations are restricted to the 2-tame situation. The question arises as to how central factors of N and V are related. For tame N -groups (no chain conditions) we have:Theorem 18.1. Let V be a tame N -group and U/W a minimal factor of V . If all minimal factors of the N -group N , which are N -isomorphic to U/W are central, then U/W is a central factor of V . Proof: We must show that (v + u)α ≡ vα + uα mod W, for all v in V , u in U and α in N . If vN ∩U ≤ W , then (v+u)α−uα−vα is in W and the desired conclusion holds. When vN ∩U ≤ W , vN +W ≥ U and (vN +W )/W ≥ U/W . Now (vN +W )/W is naturally N -isomorphic (by δ say) to N/(W : v) and (U/W )δ/(W : v) is a minimal factor of N/(W : v), N -isomorphic to U/W . Thus (U/W )δ/(W : v) is central in the N -group N and obviously, U/W is central in vN . This means (v + u)α ≡ vα + uα mod W,

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for all v in V , u in U and α in N . The theorem is proved. A further question concerning tame N -groups is what can be said when all minimal factors are central. In general, it is too much to expect this to imply N -nilpotency, but with DCCR expectations are fulﬁlled. Theorem 18.2. If N is a nearring with DCCR, then a tame N -group V is N -nilpotent if, and only if, all minimal factors of V are central. Proof: By 8.5, V has a tame series {0} = V0 ≤ V1 ≤ · · · ≤ Vk = V, where k ≥ 0 is an integer. Assume k is minimal and proceed by induction V1 is N on k. If k = 0 the result holds. It can be assumed that Vk /V nilpotent. If it is shown that V1 ≤ Z(V ), then the result will hold. Now V1 is either {0} or a sum of minimal submodules of V . It follows that, because minimal factors of V are being assumed central, V1 ≤ Z(V ). It remains to show that if V is N -nilpotent, then all minimal factors of W2 is a minimal factor of V , then take the integer V are central. If W1 /W n ≥ 0 as maximal such that Zn (V ) ∩ W1 ≤ W2 . Now, [Z Zn+1 (V ) ∩ W1 + Zn (V )]/Z Zn (V ) is central in V /Z Zn (V ). By 6.3, (Z Zn+1 (V ) ∩ W1 )/(Z Zn (V ) ∩ W1 ) is central in V /(Z Zn (V ) ∩ W1 ). Under the natural N -homomorphism mapping V /(Z Zn (V ) ∩ W1 ) onto (V /W W2 )/[(Z Zn (V ) ∩ W1 + W2 )/W W2 ] and N -isomorphically onto V /W W2 , (Z Zn+1 (V ) ∩ W1 )/(Z Zn (V ) ∩ W1 ) is mapped to (Z Zn+1 (V ) ∩ W1 + W2 )/W W2 (= W1 /W W2 ). W2 is central in V /W W2 and the theorem is proved. Thus W1 /W With tameness strengthened to 2-tameness more is possible. One of the main reasons for this is, that N -isomorphic factors of a 2-tame N group, are better behaved. U2 and W1 /W W2 are Theorem 18.3. If V is a 2-tame N -group and U1 /U U2 if, N -isomorphic factors of V , then a subset S of V centralises U1 /U and only if, it centralises W1 /W W2 .

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Proof: let δ be an N -isomorphism of U1 /U U2 onto W1 /W W2 and a in S. If w1 is in W1 , take u1 in U1 such that (u1 + U2 )δ = w1 + W2 . For any given α in N , there exists β in N such that (a + u1 )α − u1 α − aα = u1 β and (a + w1 )α − w1 α − aα = w1 β. Applying δ we see (u1 β + U2 )δ = (u1 + U2 )βδ = (u1 + U2 )δβ = w1 β + W2 . However, because u1 β ≡ 0 mod U2 , w1 β is in W2 and (a + w1 )α ≡ aα + w1 α mod W2 , for all a in S, w1 in W1 and α in N . Thus if S centralises U1 /U U2 , it W2 . Clearly, if S centralises W1 /W W2 it centralises U1 /U U2 . centralises W1 /W The theorem is proved. U2 and Corollary 18.4. Suppose V is a 2-tame N -group and U1 /U W2 are N -isomorphic factors of V . If U1 /U U2 is central in V , then W1 /W W2 . so is W1 /W The last theorem of this section is something in the nature of a converse of 18.1. However, for its proof, we require faithfulness, 2-tameness and DCCR. Theorem 18.5. Let V be a faithful 2-tame N -group where N has DCCR. If U/W is a central minimal factor of V , then all minimal factors of N , which are N -isomorphic to U/W , are central. Proof: Suppose the theorem does not hold. There exist minimal factors H2 to of N which are N -isomorphic to U/W but not central. Take H1 /H be such a factor with H1 chosen as minimal for this property. Now, if there exists a right ideal K < H1 such that K + H2 = H1 , then H2 , is central in N . However, K/(K ∩ H2 ), being N -isomorphic to H1 /H N/H H2 is an N -homomorphic image of N/(K ∩ H2 ), with K/(K ∩ H2 ) H2 . This is a contradiction to H1 /H H2 being non-central mapped to H1 /H and all right ideals K < H1 of N are such that K ≤ H2 . Since V is faithful there exists v in V , such that H1 ∩ (0 : v) < H1 . From above H1 ∩ (0 : v) ≤ H2 . It is now shown that [H1 + (0 : v)]/[H H2 + (0 : v)] is N -isomorphic to U/W . Since [H1 + (0 : v)]/[H H2 + (0 : v)] = [H1 + (H H2 + (0 : v))]/[H H2 + (0 : v)],

(1)

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it follows that H2 + (0 : v)], [H1 + (0 : v)]/[H is N -isomorphic to H1 /(H1 ∩ [H H2 + (0 : v)]). Now H1 ∩ [H H2 + (0 : v)] coincides with H2 + (0 : v) ∩ H1 and equals H2 and (1) follows because H2 is N -isomorphic to U/W . N/(0 : v) (N -isomorphic to vN ) is H1 /H mapped N -homomorphically onto N/[H H2 + (0 : v)] and under this N homomorphism the minimal factor H2 + (0 : v))/(0 : v)] (N -isomorphic to U/W ) [(H1 + (0 : v))/(0 : v)]/[(H (2) of N/(0 : v) is mapped onto the minimal factor H2 + (0 : v)] [H1 + (0 : v)]/[H

(3)

of N/[H H2 + (0 : v)]. It follows readily from (3), that if (2) is a central factor of N/(0 : v), then (3) is a central factor of N/[H H2 + (0 : v)]. However, by 18.4, all minimal factors of V (thus of vN ) N -isomorphic to U/W are central. It has been shown (3) is central in N . As above H2 + (0 : v)] = [H1 + (H H2 + (0 : v))]/[H H2 + (0 : v)] [H1 + (0 : v)]/[H H2 + (0 : v)]) is a central factor of N . This is just and, by 6.3, H1 /(H1 ∩ [H H2 and we have arrived at the contradiction that H1 /H H2 is central. H1 /H The theorem is fully proved.

19.

Minimal Faithfuls

Results of the last section ﬁnd application to faithful N -subgroups of a 2-tame N -group V (N with DCCR). The fact that V /H (H a faithful N -subgroup of V ), has limited structure is looked at in section 4 of [12]. However, it is possible to say considerably more than indicated by [12]. For one thing, the compatible assumption (see 4.2 of [12]) is not required. Another matter, that can be dealt with, is that of minimal faithful N -subgroups of V (deﬁned below). A third aspect is that of perfect faithfuls. They allow the development of quite meaningful theory. All this is very satisfying. This section undertakes covering some aspects of these matters. If V is an N -group with a faithful N -subgroup H having no N subgroup H1 < H faithful, then H will be called a minimal faithful N -subgroup of V . Proposition 19.1. A faithful tame N -group V (N ( with DCCI) contains faithful ﬁnitely generated N -subgroups.

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Proof: Since the intersection of (0 : vN ), over all v in V is zero, some ﬁnite intersection (0 : v1 N ) ∩ · · · ∩ (0 : vk N ) (k ≥ 1 an integer and vi , i = 1, . . . , k, in V ), is zero. Because (0 : v1 N + v2 N + · · · + vk N ) ≤ (0 : v1 N ) ∩ · · · ∩ (0 : vk N ), v1 N + · · · + vk N is faithful. The proposition holds. Corollary 19.2. If a tame N -group V (N ( with DCCI) has a minimal faithful N -subgroup H, then H is ﬁnitely generated. Corollary 19.3. A faithful tame N -group V (N ( with DCCR) has minimal faithful N -subgroups. Proof: By 19.1, V has a ﬁnitely generated faithful N -subgroup, H say. By 8.6, H has DCCS and a submodule of H, minimal for being faithful is minimal faithful. The main theorem on faithful N -subgroups (N with DCCR) of a 2-tame N -group is the following:Theorem 19.4. If V is a 2-tame N -group ((N with DCCR) and H a faithful N -subgroup, then V /H is N -nilpotent (ie. there exists an integer n ≥ 0 such that V n ≤ H). Proof: This follows from 18.2, if it is shown that all minimal factors H2 /H) of V /H (H Hi , of V /H are central. A minimal factor (H1 /H)/(H H2 i = 1, 2, containing H), is N -isomorphic to the minimal factor H1 /H of V . By 8.10, H has a minimal factor N -isomorphic to H1 /H H2 . Since U1 /U U2 say) is central in H, by 14.2. H2 ≥ H, this minimal factor (U U2 By 18.5, all minimal factors of N , which are N -isomorphic to U1 /U are central. From 18.1, it follows (because U1 /U U2 is N -isomorphic to H2 /H)), that (H1 /H)/(H H2 /H) is central in V /H. The theo(H1 /H)/(H rem is proved. Corollary 19.5. If V is a 2-tame N -group ((N with DCCR) and H a minimal faithful N -subgroup, then V /H is N -nilpotent (ie. there exists an integer n ≥ 0 such that V n ≤ H). For an N -group V , V ω is taken as the intersection ∩n V n , over all n ≥ 0 and V (ω) is taken as the intersection ∩n V (n) over all n ≥ 0. If V is a 2-tame N -group (N with DCCR) and H a minimal faithful N subgroup then, by 19.5, V (n) ≤ V n ≤ H for some integer n ≥ 0. Since H has DCCS, V n ≥ V n+1 ≥ . . . and V (n) ≥ V (n+1) ≥ . . . , it follows that, in this case, there exist integers r ≥ 0 and s ≥ 0, with V ω = V r and V (ω) = V (s) . V ω and V (ω) are of some importance in developing understanding of minimal faithful N -groups (2-tame and N with DCCR). This is

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particularly true when N has a faithful perfect 2-tame N -group. Before looking into this it should be noted that:Proposition 19.6. If V is a faithful N -group ((N with DCCR), then V ω ≤ H, where H is the intersection of all minimal faithful N -subgroups of V . Proof: From above, there exist integers ni , i ∈ I, such that V ni ≤ Hi , where Hi , i ∈ I, is the collection of all minimal faithful N -subgroups of V . Because each Hi is ﬁnitely generated, it follows from 8.6, that H = Hi1 ∩ · · · ∩ Hik , where {i1 , . . . , ik } (k ≥ 1 an integer) is a ﬁnite subset of I. Thus with n = max{nij : j = 1, . . . , k} it follows that V n ≤ H. The proposition is therefore proved. In the case where there exists a faithful perfect 2-tame N -group, much more is true. Theorem 19.7. If the nearring N (with DCCR) is 2-tame on the perfect N -group V , then in any faithful 2-tame N -group W , the minimal cover of nr(W ) (= ( W (tnr(N ))) is the unique minimal faithful N subgroup of W . Proof: First we prove that V is the minimal cover H of nr(V ) (= V (tnr(N ))). If H < V , then there exists a maximal N -subgroup H1 of V containing H. Because V /H1 an element of nr(V ), it follows that H covers V /H1 . This contradiction can only mean H = V . Now, by 17.3, if W1 is a minimal cover of nr(W ) (= W (tnr(N ))), then (0 : W1 ) = (0 : V ) = {0} and W1 is faithful. An N -subgroup W2 < W1 of W , W4 of nr(W ) (= W (tnr(N ))). Thus, by fails to cover some element W3 /W W4 and, by 8.10, 14.3, W2 has no minimal factors N -isomorphic to W3 /W cannot be faithful. It has been shown that W1 is a minimal faithful N subgroup of W . To show it is the unique minimal faithful N -subgroup observe that, by 8.10 and 14.3, any faithful N -subgroup L of W must cover any element of nr(W ). Thus L is a cover of nr(W ) and, by the uniqueness of minimal covers of subsets of nr(W ), W1 ≤ L. Hence W1 = L and the theorem is established. We are now ready to make use of V ω and V (ω) . Theorem 19.8. Suppose the nearring N (with DCCR) is 2-tame on a perfect N -group. If W is a faithful 2-tame N -group, then W ω = W (ω) and W (ω) is the unique minimal faithful N -subgroup of W . Proof: Let H be the minimal faithful N -subgroup of W . If for some integer n ≥ 0, W (n) ≥ H but H + W (n+1) > W (n+1) , then [H + W (n+1) ]/W (n+1) is a non-zero ring module. This implies H/(H∩W (n+1) ) is a non-zero ring module and H is not perfect. This is in contradiction

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to 19.7 and 15.1. Thus, for all integers n ≥ 0, W (n) ≥ H. However, by 19.5, we can ﬁnd an integer r ≥ 0, such that H ≥ W s ≥ W (s) , for all integers s ≥ r. This means H = W s = W (s) , for all integers s ≥ r, and W (ω) = W ω = H. The theorem is proved.

20.

Conjugacy

This section introduces the concept of conjugacy. In the next we look at weak conjugacy. Conjugacy and weak conjugacy occur quite often in certain nearrings. These are nearrings with DCCN . The author has observed such behaviour in a number of situations and it is certainly time, for some formality to be introduced. The concept is not unlike that occuring for ﬁnite groups. In ﬁnite groups systems of subgroups (eg. sylow subgroups), which are conjugate (in the group sense), are common enough. It seems possible that, a similar notion for nearrings, could have something like the same impact on development of nearring theory. A collection Γ, of right N -subgroups of a nearring N (with DCCN ), will be called a conjugacy class if, whenever Mi , i = 1, 2, are in Γ (not M2 = M1 and if necessarily distinct), there exists α in M1 such that αM any right N -subgroup N -isomorphic to an element of Γ, is in Γ. The ﬁrst thing to note is that if M is in a conjugacy class Γ of N (with DCCN ), then M is self monogenic. The next thing to note, is that all elements of Γ are N -isomorphic. Indeed, if Mi , i = 1, 2, are in Γ, then there exists α1 in M1 , such that α1 M2 = M1 . However, there exists α2 in M2 such that α2 α1 M2 = M2 . Because α2 α1 is in M2 it follows, by [8], that (0 : α2 α1 ) ∩ M2 = {0}. Thus (0 : α1 ) ∩ M2 = {0} and the natural map taking M2 onto α1 M2 (= M1 ) is an N -isomorphism. In the deﬁnition of the conjugacy class Γ, the condition that any right N -subgroup N -isomorphic to an element of Γ is in Γ, is not that frequently fulﬁlled. However, it occurs often enough to warrant the deﬁnition. From the statement and proof of 6.1 of [15], it follows that the set of all right N -subgroups of N , minimal for not annihilating a particular minimal ideal of N from the left, is a conjugacy class. This conjugacy class has, in the tame case, a valuable extension. In order to establish this we ﬁrst prove an important lemma. Lemma 20.1. Suppose N is a tame nearring with DCCR and Mi , i = M1 ) 1, 2, are self monogenic right N -subgroups with property q. If M1 /Φ(M M2 ) are N -isomorphic, then there exists γ1 in M1 such that and M2 /Φ(M γ1 M 2 = M1 . Proof: By 7.4 of [15], we have Mi ∩ J(N ) = Φ(M Mi ), for i = 1 and 2. Thus (M M1 + J(N ))/J(N ) is N -isomorphic to (M M2 + J(N ))/J(N ). By

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8.2 of [15], M2 + J(N ))/J(N )] = (M M1 + J(N ))/J(N ). (4) [(M M1 + J(N ))/J(N )][(M By [8], M1 has a left identity e1 . Thus, by (4), there exists γi + J(N ) in γi in Mi ) such that, as elements of N/J(N ) Mi + J(N ) (γ (γ1 + J(N ))(γ2 + J(N )) = e1 + J(N ). It follows that γ1 γ2 ≡ e1 mod J(N ) and, for any element γ of N , e1 γ is of the form γ1 γ2 β + λ, where β is in N and λ in J(N ). Since e1 N = M1 and γ1 γ2 N ≤ γ1 M2 , we have shown M1 ≤ γ1 M2 + J(N ). Now γ1 M2 ≤ M1 and M1 = γ1 M2 + M1 ∩ J(N ). It follows from 7.4 of [15], M1 ). If γ1 M2 < M1 , then ACCN ensures M1 that M1 ∩ J(N ) = Φ(M has a maximal right N -subgroup K containing γ1 M2 . However, since M1 ) ≤ K we have a contradiction. Thus γ1 M2 = M1 γ1 M2 ≤ K and Φ(M and the lemma is proved. If N is a tame nearring (with DCCR) and S a subset of the collection of all minimal ideals of N , then mr(S) is taken as consisting of every right N -subgroup M of N , which is minimal for the property that M B = {0}, for all B in S. By 7.1 of [15], the elements of mr(S) all have property q and, by 7.6 of [15], they are self monogenic. Also, by 8.6 of [15], the M1 ) elements Mi , i = 1, 2, of mr(S) are N -isomorphic and clearly M1 /Φ(M M2 ). It follows, by 20.1, that there exists γ1 is N -isomorphic to M2 /Φ(M in M1 such that γ1 M2 = M1 . However, if H is a right N -subgroup of N , N -isomorphic to the right N -subgroup M of mr(S), then H is in mr(S). This is because HA = {0}, for all A in S and, if H1 < H is a right N subgroup of H, then H1 is N -isomorphic to a proper right N -subgroup K of M and KB = {0}, for some B in S, implying H1 B = {0}. It has been shown that:Theorem 20.2. If N is a tame nearring with DCCR and S a collection of minimal ideals of N , then mr(S) is a conjugacy class.

21.

Weak Conjugacy

The notion of a conjugacy class tends to be a bit overpowering. We therefore deﬁne a weak conjugacy class. If N is a nearring with DCCN and Γ a collection of right N -subgroups of N , then Γ will be called a weak conjugacy class if, for Mi , i = 1, 2, in Γ (not necessarily distinct) there exists α1 in M1 such that α1 M2 = M1 and any self monogenic right N -subgroup, N -isomorphic to an element of Γ, is in Γ. Clearly, as with a conjugacy class, all elements of Γ are self monogenic and N -isomorphic. The ﬁrst example of weak conjugacy that we give requires other developments.

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If N is a nearring with DCCN , then min(N ) will denote the collection of all right N -subgroups of N , which are minimal for being nonnilpotent. If M is an element of min(N ), then a proper submodule of M is nilpotent. The sum of all such submodules of M will be denoted by st(M ). The submodule st(M ) of M is nil because an element of st(M ) is contained in a ﬁnite sum of nilpotent submodules of the nearring M . Since N has DCCN , classical results ensure st(M ) must be nilpotent and st(M ) < M . Thus st(M ) is the unique maximal submodule of M and contains all proper submodules of M . Now an element γ of M which is non-nilpotent is contained in γN (≤ M ) and γN = M . Thus M/st(M ) has no proper submodules and is cyclic. It is an N -group of type 0. For type 0 N -groups we deﬁne classes (distinct from conjugacy or weak conjugacy classes). A type 0 class ∆, is simply the class of all N -groups N -isomorphic to a given type 0 N -group. Now min(∆) is deﬁned as the set of all M in min(N ), where M/st(M ) is in ∆. Lemma 21.1. If N is a nearring with DCCN and ∆ a type 0 class, then min(∆) is non-empty. Proof: If V is in ∆, then V = vN , for some v in V and (0 : v) is a maximal right ideal of N . Take M as a right N -subgroup of N minimal such that M + (0 : v) = N . Now 1 = e + ρ, where e is in M and ρ in (0 : v) and e ≡ e2 mod (0 : v). Because e is not in (0 : v) ((0 : v) = N ), it follows that M is non-nilpotent. Also M/(M ∩ (0 : v)) is N -isomorphic to N/(0 : v) and M ∩ (0 : v) is a maximal submodule of M . Take a right N -subgroup M1 ≤ M of N , minimal for being non-nilpotent. It contains a non-nil element γ and γM M1 = M1 (because γ 2 is in M1 ). By [8], M1 has a left identity e1 and M = M1 + (0 : e1 ) ∩ M , where M1 ∩ ((0 : e1 ) ∩ M ) = {0}. Now (0 : e1 ) ∩ M is a proper submodule of M and cannot be such that (0 : e1 ) ∩ M + (0 : v) ∩ M = M, since otherwise (0 : e1 ) ∩ M + (0 : v) ≥ M + (0 : v) = N, contrary to the nature of M . Thus (0 : e1 ) ∩ M ≤ (0 : v) and N = M1 + (0 : v). The minimality of M forces M1 to equal M . Now M1 ∩ (0 : M1 ). Clearly, since v) is the maximal submodule of M1 and equals st(M M1 ) is N -isomorphic to N/(0 : v). The lemma is M1 = M , M1 /st(M proved. An example of a weak conjugacy class is now given. Theorem 21.2. If N is a nearring with DCCN and ∆ a type 0 class, then min(∆) is a weak conjugacy class.

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Proof: Suppose Mi , i = 1, 2, are in min(∆). Now a non-nil element β M2 = M2 , because β 2 is in βM M2 . Thus M2 has a of M2 is such that βM left identity e and M22 = M2 . This means M2 )].M M2 = M2 /st(M M2 ) = (e + st(M M2 )).M M2 . [M M2 /st(M Take δ as an N -isomorphism of M2 /st(M M2 ) onto M1 /st(M M1 ) and let M1 ), where γ is in M1 . It follows that (e + st(M M2 ))δ = γ + st(M M1 ) = ((e + st(M M2 )).M M2 )δ = (γ + st(M M1 )).M M2 . M1 /st(M It is now clear that, γM M2 + st(M M1 ) = M1 . Because st(M M1 ) is nilpotent, it follows from p.104 of [2], that γM M2 cannot be nilpotent and the M2 to equal M1 . minimality of M1 , forces γM It must be shown that a self monogenic right N -subgroup M of N , N isomorphic (by λ say) to H in min(∆), is in min(∆). Now (st(H))λ−1 is the unique maximal right N -subgroup of M and M/(st(H))λ−1 is N -isomorphic to H/st(H). If it is shown that M is in min(N ), then (st(H))λ−1 will equal st(M ) and the result will follow. A right N subgroup K of M minimal for being non-nilpotent has a left identity e1 and M = K + (0 : e1 ) ∩ M , where K ∩ ((0 : e1 ) ∩ M ) = {0}. If K < M , then Kλ < H and is nilpotent, while ((0 : e1 ) ∩ M )λ is a proper submodule of H and is nilpotent. By p.104 of [2], this would mean Kλ + [(0 : e1 ) ∩ M ]λ is nilpotent, contrary to H being non-nilpotent. The only possibility is that K = M and M is minimal for being nonnilpotent. The theorem is established. In the remainder of this paper, the conjugacy and weak conjugacy classes of 20.2 and 21.2, will not ﬁgure again. The reason they have been presented is to show how important these concepts are. We ﬁnish this section with an example that will be used in what follows. Suppose N is a 2-tame nearring with DCCR, S a subset of tnr(N ) and U any faithful 2-tame N -group. The minimal cover of U (S) is cyclic and has a realisation. We take re(S) as the set of all such realisations (over all faithful 2-tame N -groups U and realisations of minimal covers of U (S)). Theorem 21.3. If N is a 2-tame nearring with DCCR and S a subset of tnr(N ), then re(S) is a weak conjugacy class. Proof: Let Vi , i = 1, 2, be faithful 2-tame N -groups, with minimal covers of Vi (S), respectively W1 and W2 . Let Mi , i = 1, 2, be realisations of Wi . One thing to be shown is that, there exists α1 in M1 , such that α1 M2 = M1 . By 11.3 and 11.2, both Mi have property q and are self M1 ) is N -isomorphic to M2 /Φ(M M2 ) and α1 monogenic. By 17.2, M1 /Φ(M exists by 20.1.

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Now suppose H is a self monogenic right N -subgroup of N , N -isomorphic to an element of re(S) (M M1 say). Clearly H has property q and M1 ) is N -isomorphic to H/Φ(H). Thus, by 20.1, there exists α M1 /Φ(M in M1 such that αH = M1 . With w1 in W1 taken such that w1 M1 = W1 , we see that w1 αH = W1 and H contains a realisation H1 of W1 . From above, there exists β1 in M1 , such that β1 H1 = M1 and β2 in H1 , such that β2 M1 = H1 . Because (0 : β2 β1 ) ∩ H1 = {0}, (0 : β1 ) ∩ H1 = {0} and H1 is N -isomorphic to M1 . Thus H1 is N -isomorphic to H and ≤ H. The DCCN implies H1 = H and H is a realisation of W1 . The theorem is completely proved.

22.

Centrality in the Socle

If V is a tame N -group (N with DCCR), then the socle soc V of V is of particular interest. A result that ensures this is that, when V is faithful, (soc V : V ) = soc N and N/soc N is therefore tame on V /soc V . It is important to have information about Z(V ) ∩ soc V . If Z(V ) ∩ soc V = {0}, then how large can Z(V ) ∩ soc V become? For example, does it contain every minimal N -subgroup of V , N -isomorphic to a minimal N -subgroup of Z(V ) ∩ soc V ? If V is a tame N -group, then a class ∆ of type 2 consists of all N groups N -isomorphic to some minimal N -group and soc(V, ∆) is the sum of all minimal N -subgroups of V which are in ∆. It is an elementary matter that soc(V, ∆) can be expressed as a direct sum of minimal N groups. Thus, for any such direct summand H, soc(V, ∆) = K ⊕ H, where K is a submodule of V . The nature of soc(V, ∆) ensures that some minimal N -subgroup of V in ∆ is not in K and is therefore N isomorphic to H. It has been shown that in any expression of soc(V, ∆) as a direct sum of minimal N -groups, the minimal N -groups involved are in ∆. This also implies that, because any minimal N -subgroup of soc(V, ∆), is N -isomorphic to such a direct summand, all minimal N subgroups of soc(V, ∆) are in ∆. Theorem 22.1. If V is a tame N -group and ∆ a type 2 class such that soc(V, ∆) ∩ Z(V ) = {0}, then soc(V, ∆) ≤ Z(V ). Proof: Since soc(V, ∆) = K ⊕ (soc(V, ∆) ∩ Z(V )), where K is an N -subgroup of V , some minimal N -subgroup of soc(V, ∆) is N -isomorphic to a minimal N -subgroup of soc(V, ∆) ∩ Z(V ). It has been shown that soc(V, ∆) contains a central minimal N -subgroup W in ∆. Let H be any other minimal N -subgroup of ∆ in V . Now (W ⊕H)/H is central in V /H. However, if L is a diagonal N -subgroup of W ⊕ H,

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then L ⊕ H = W ⊕ H and (L ⊕ H)/H ≤ Z(V /H). By 6.3, it follows that L ≤ Z(V ). Thus L ⊕ W ≤ Z(V ) and since H ≤ L ⊕ W , H ≤ Z(V ). Clearly, the theorem is proved. Theorem 22.2. If V is a tame N -group and ∆ is in tnr(N ), then either soc(V, ∆) is {0} or a minimal N -group. Proof: If soc(V, ∆) = {0}, then soc(V, ∆) is expressible as a direct sum of minimal N -groups (all in ∆). In the situation where soc(V, ∆) is not a minimal N -group it contains a direct sum H1 ⊕ H2 of two minimal N -subgroups of V in ∆. A diagonal N -subgroup K of H1 ⊕ H2 is a submodule with projections onto H1 and H2 being H1 and H2 respectively. Also K ∩ H1 = K ∩ H2 = {0}. By 6.4, K is in Z(H1 ⊕ H2 ) and by projecting K onto H1 we see that, H1 is a ring module. Since H1 is in ∆ we have a contradiction. The theorem holds. The ﬁnal theorem of this section requires the DCCR. What can be said about soc(V, ∆) (V a tame N -group and ∆ a type 2 class), when soc(V, ∆) is an inﬁnite direct sum of minimal N -groups? Here we have the following rather beautiful result:Theorem 22.3. Suppose V is a tame N -group ((N with DCCR) and ∆ a type 2 class. If soc(V, ∆) is an inﬁnite direct sum of minimal N groups, then soc(V, ∆) ≤ Z(V ). Proof: It must be shown that, for v in V , w in soc(V, ∆) and α in N , (v + w)α = vα + wα. This will hold if soc(V, ∆) ≤ Z(vN + soc(V, ∆)). Now, vN has a composition series and vN ∩ soc(V, ∆) has a composition series. Clearly soc(V, ∆) does not have a composition series and soc(V, ∆) ≤ vN . Thus soc(V, ∆) = vN ∩ soc(V, ∆) ⊕ H, where H ≤ soc(V, ∆) is a non-zero N -subgroup of V and is expressible as a direct sum of minimal N -groups (of type ∆). Clearly vN +soc(V, ∆) = vN ⊕ H. By 22.2, a minimal N -subgroup of soc(V, ∆) is a ring module, soc(V, ∆) is a ring module and H is a ring module. A minimal N subgroup H1 of H is central in vN + soc(V, ∆). Indeed, for h1 in H1 , h in H and u in vN , we have (u + h + h1 )β − h1 β − (u + h)β = uβ + (h + h1 )β − h1 β − hβ − uβ = uβ − uβ = 0,

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for all β in N . It follows, from 22.1, that soc(V, ∆) ≤ Z(vN + soc(V, ∆)), and the theorem is completely proved.

23.

Complete Reducibility and the Socle of N

Suppose N (with DCCR) is tame on V . Results of the previous section apply to the socle soc V of V . As seen there, centrality in soc V , can be treated with ease. Things there, go very smoothly. It would be nice if the socle of N also shared such behaviour. However, this does not happen. Some change of emphasis is required. This section will have a diﬀerent purpose from the last. Here we look at soc N from an elementary viewpoint. The material covered is, in its own way, of use later. We have previously dealt with some aspects of complete reducibility. At this stage, an extension of this notion is required. If V is an N -group, then a submodule U of V is said to be completely reducible in V , if it is a direct sum of minimal submodules of V which are minimal N -groups. The following result is not diﬃcult to prove:Theorem 23.1. Let V be an N -group and U a submodule of V . The following are equivalent:(i) U is completely reducible in V , (ii) U is a sum of minimal submodules of V which are minimal N groups and (iii) for every N -subgroup U1 ≤ U of V , there exists a submodule W of V such that W + U1 = U and W ∩ U1 = {0}. One of the more obvious examples of complete reducibility in the N group N , occurs with tame nearrings. Here either soc N = {0} (is completely reducible in N by default) or soc N is the sum of minimal right ideals of N . Because minimal right ideals, can be easily shown to be minimal N -groups, theorem 23.1 is applicable and soc N is completely reducible in N . Also, it follows readily that, any right ideal of N contained in soc N , is completely reducible in N . In the case where N has DCCR (N tame), soc N = {0} for N = {0} and soc N is a ﬁnite direct sum of minimal right ideals. If ∆ is a type 2 class, then soc(N, ∆) (N a nearring) will denote the sum of all minimal right ideals of N , which as N -groups, are of type ∆. The notation soc(N, ∆ ), will be used to denote the sum of all minimal right ideals, which are minimal N -groups of some type other than ∆. In the case of a tame N -group V , soc(V, ∆) was central if, and only if, one

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minimal N -subgroup of V of type ∆ was central. For a tame nearring N , we can only conclude that soc(N, ∆) is central (in N ), if all minimal right ideals of N (in ∆) are central. Proposition 23.2. If N is a tame nearring with DCCR and ∆ a type 2 class, then soc N = soc(N, ∆) ⊕ soc(N, ∆ ). This is proved by expressing soc N as a direct sum of minimal right ideals and letting H be the component consisting of all such right ideals which are in ∆ and K be the sum of the remainder. Thus soc N = H ⊕ K and projecting any minimal right ideal R in ∆ onto these direct summands, we see R ≤ H. Thus H = soc(N, ∆) and a similar argument shows K = soc(N, ∆ ). Just when is soc(N, ∆) = {0}? It turns out that, this is the case precisely when some minimal factor R2 /R1 of N (R2 ≤ soc N ) is in ∆. Indeed, if soc(N, ∆) = {0}, then there exists a minimal right ideal R in ∆ and R/{0} is such a factor. When R2 /R1 exists, R2 = R1 ⊕ H, where H is a minimal right ideal of N and, since R2 /R1 is in ∆, H is in ∆ and soc(N, ∆) = {0}. Thus:Proposition 23.3. If N is a tame nearring with DCCR and ∆ a type 2 class, then soc(N, ∆) = {0} if, and only if, there exists a minimal ( 2 ≤ soc N ), with R2 /R1 in ∆. factor R2 /R1 of N (R Because soc N = H ⊕ K (as above), where H is a direct sum of minimal N -groups in ∆ and K a direct sum of minimal N -groups in ∆ , it follows that, for N tame with DCCR, soc(N, ∆) is an ideal of N . Indeed, any minimal right N -subgroup of N is contained in soc N (see 5.4 of [16]) and, if of type ∆, is contained in H. Thus αH1 (α in N and H1 a direct summand of H) is contained in H. It follows that:Proposition 23.4. If N is a tame nearring with DCCR and ∆ a type 2 class, then soc(N, ∆) is an ideal of N .

24.

Minimal N -group Direct Summands

In this section we shall be looking at a tame N -group, which can be expressed as a direct sum U ⊕ W , where U is a minimal N -subgroup of U ⊕ W and W a submodule. An elementary fact concerning such an N -group is that:Proposition 24.1. U ⊕ W has central factors N -isomorphic to U if, and only if, U is a ring module. A deeper result follows:-

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Lemma 24.2. All minimal factors of U ⊕ W , N -isomorphic to U are central if, and only if, U is a ring module. Proof: If all minimal factors of U ⊕ W , N -isomorphic to U are central then, because U/{0} is such a factor, U is a ring module. V2 a minimal factor of U ⊕ W , Suppose U is a ring module and V1 /V N -isomorphic to U . The projection of V1 into U either has kernel V1 (V V1 ≤ W ) or has kernel W1 , where W1 is a maximal N -subgroup of V1 contained in W . Suppose V1 ≤ W . Clearly, in this situation, V2 ≤ W V2 ⊕ W/V V2 has (U + V2 )/V V2 in its center and, by 22.1, has and (U + V2 )/V V2 in its center. In the second case, either V2 = W1 or V2 + W1 = V1 . V1 /V W1 = V1 , then by 6.3, V1 /V V2 is central if, and only if, W1 /(V V2 ∩W W1 ) If V2 +W is central. Since this follows from the case where V1 ≤ W , we are left W1 ⊕ W/W W1 has with the situation where V2 = W1 . Here, (U + W1 )/W W1 in its center and, by 22.1, V1 /W W1 (= V1 /V V2 ) is also in its (U ⊕ W1 )/W center. The lemma is proved. If the U of 24.2 is a ring module, then all minimal factors of U ⊕ W , N -isomorphic to U , are central, but what can be said about factors of N , N -isomorphic to U ? Here we have:Theorem 24.3. Suppose N (with DCCR) is tame on V and V is expressible as a direct sum U ⊕ W of submodules U and W , where U is a minimal N -group. If U is a ring module, then all minimal factors of N , N -isomorphic to U , are central. Proof The proof of this proceeds in a manner similar to that of 18.5. We assume there exists minimal factors of N which are N -isomorphic to U but not central (here we replace U/W of 18.5 by U ). With this replacement being made throughout that proof, our present proof proceeds. The only diﬀerence that needs to be noted is that, in the proof of 18.5 it is stated (six lines from the end) that ‘by 18.4, all minimal factors of V (thus of vN ) N -isomorphic to U/W (ie. to U ) are central’. With this replaced by ‘by 24.2, all . . . ’, the theorem is established. So far, results of this section, have dealt with the situation where U is central (ie. U is a ring module). What can be said when U is not a ring module? Here we have:Proposition 24.4. Suppose N (with DCCR) is tame on V and V is expressible as a direct sum U ⊕ W of submodules U and W , where U is a minimal N -group. If U is not a ring module, then N = (0 : U )⊕(0 : W ). Proof: This will follow from 13.3, if it is shown that W has no minimal W2 is a minimal factor of W , N factors N -isomorphic to U . If W1 /W isomorphic to U , then in V /W W2 , we have the submodule (U + W2 )/W W2 ⊕

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W1 /W W2 , where both components are N -isomorphic to U . Taking a diagonal N -subgroup ∆ of this submodule, we see that ∆ intersects both components trivially. By 6.3, this means ∆ is central in this submodule and W2 projecting ∆ onto the ﬁrst component, yields the fact that (U + W2 )/W (thus U ) is a ring module. This contradiction establishes the proposition. The V of 24.4 is cyclic if, and only if, W is cyclic. Proposition 24.5. Suppose N , V , U and W are as in 24.4. If U is not a ring module, then V is cyclic if, and only if, W is cyclic. Proof: If V is cyclic, then V /U is N -isomorphic to W and W is cyclic. Suppose W is cyclic. There exists w in W such that wN = W and u in U such that uN = U . By 24.4, u(0 : W ) = U and w(0 : U ) = W . Also (u + w)N = (u + w)(0 : W ) + (u + w)(0 : U ) = u(0 : W ) + w(0 : U ) = V and V is cyclic. The result holds.

25.

Z-constraint

The study of central factors of a tame (2-tame) N -group V was undertaken in 18. This material was used in 19 to gain understanding on a faithful N -subgroup H of V . It was seen there, that V /H is N -nilpotent. The minimal factors of V /H already appear in H. In some sense V /H is redundant. Indeed, N is fully represented by its action on H. It is the purpose of this section, to introduce and study, N -groups without redundancy of this kind. The condition introduced is one on which our major results depend. It is hard to overemphasise its value. As indicated above, a very important assumption that can be made on N -groups is that of Z-constraint. An N -group V , will be called Zconstrained, if there are no minimal factors U/W of V with U/W ≤ Z(V /W ). This deﬁnition relates to N -groups and, in particular, tame N -groups. It will be seen shortly, that the notion of Z-constraint for a tame nearring, is diﬀerent. The ﬁrst thing to note, about a Z-constrained N -group, is that it is perfect. Proposition 25.1. If V is a Z-constrained N -group, then it is perfect. Proof: If V (1) < V , then take U1 as the submodule of V generated by an element u1 of V \V (1) and W1 as a submodule of V maximal for containing V (1) and excluding u1 (possible by Zorn’s lemma). Now, W1 is a minimal factor of V and, since W1 ≥ V (1) , V /W W1 (U U1 + W1 )/W is a ring module. This means (U U1 + W1 )/W W1 ≤ Z(V /W W1 ) and V is not Z-constrained. Thus V (1) = V and V is perfect. Corollary 25.2. If V is a tame Z-constrained N -group ((N with DCCR), then V is cyclic.

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Proof: This follows from 10.5. What we are now looking for, is a sensible deﬁnition of Z-constraint in a tame nearring (with DCCR). The fact that, if N is tame on V it does not imply that homomorphic images of N , are tame on N -homomorphic images of V , is an observation that means we must be cautious. We want our deﬁnition of Z-constrained tame nearrings (with DCCR) to relate well to the faithful tame N -group involved. The manner in which N (with DCCR) relates to a faithful tame N -group V , is by the socle (see the statements of section 22). If in any N -group V (possibly N ), soc0 V is taken as {0} and soc1 V as soc V and if for an integer r ≥ 1, socr V is deﬁned, then socr+1 V ≥ socr V and socr+1 V /socr V is taken as the sum of all minimal factors (if such exist) of V of the form U/socr V and as {0} (= {socr V }) if there are no such minimal factors. One nice thing about a tame nearring N with DCCR (tame on V say) is that socr N = N , for some integer r ≥ 1. Also, for this integer socr V = V . However, the socle series of V and N are even more closely related. It is a relatively easy matter (quite classical) to show that (socr V : V ) = socr N , for all integers r ≥ 0. In particular (soc2 V /soc V : V /soc V ) in N/soc N , coincides with soc2 N/soc N and many other combinations can be devised. We are now ready to deﬁne Z-constraint in a tame nearring N (with DCCR). Such a nearring is Z-constrained, if it has no minimal two sided factors of the form A/socr N (r ≥ 0 any integer) with (A/socr N ) ∩ Z(N/socr N ) = {0}. By 12.5, this implies and is implied by, the non-existence of minimal two sided factors A/socr N (r ≥ 0 any integer) with A/socr N ≤ Z(N/socr N ). A point worth noticing about this deﬁnition of Z-constraint, is that it does not follow that homomorphic images of N are Z-constrained (they may not even be tame). However, certain select homomorphic images will be Z-constrained. It can be readily seen that those nearrings of the form N/socr N (r ≥ 0 an integer) are Z-constrained. This is enough to allow the possibility of inductive arguments. The ﬁnal result of this section is an example of this. Theorem 25.3. If the nearring N (with DCCR) is tame on V and is Z-constrained, then V is cyclic. Proof: Let r ≥ 0 be the smallest integer such that socr N = N . Clearly, if r = 0, N = {0} and V = {0} so that the theorem holds. If r ≥ 1 then,

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because N/soc N is Z-constrained and the induction hypothesis applies, we may suppose V /soc V is a cyclic N/soc N -group. Thus, V /soc V is a cyclic N -group and there exists v + soc V (v in V ) in V /soc V such that (v + soc V )N = V /soc V and vN + soc V = V . Since soc V is completely reducible soc V = H ⊕(vN ∩soc V ), where H is a completely reducible submodule of V . It follows readily that V = vN ⊕ H. This means H is a direct sum of minimal N -subgroups of V . If H = {0}, V = vN and the result follows. Now if two of the direct summands of H are N -isomorphic, then considering diagonal N -subgroups, we can show they are ring modules and therefore central in V . Thus, by 8.2, either H = H1 ⊕ · · · ⊕ Hk (k ≥ 1 an integer), where Hi , i = 1, . . . , k, are minimal N -groups which are not ring modules, or V has a minimal N -subgroup K which is a ring module and a direct summand. In the ﬁrst case continued application of 24.5, yields the cyclic nature of V . Consider the second situation. Here, we have from 24.3, that all factors of N , N -isomorphic to K (such exist) are central. Let R1 /R2 be such a factor with R1 chosen minimal for this property. It follows, from the minimal nature of R1 , that R2 is a right ideal of N uniquely maximal for being contained in R1 (see previous arguments). Take s ≥ 0, such that socs N is maximal for R1 ∩ socs N < R1 . Now socs+1 N ≥ R1 and, from the nature of R1 , R1 ∩ socs N ≤ R2 . We have {0} < (R1 + socs N )/socs N ≤ (socs+1 N )/(socs N ). Since (R1 +socs N )/socs N is N -isomorphic to R1 /(R1 ∩socs N ) which has a minimal factor (of N ) N -isomorphic to R1 /R2 (R2 ≥ R1 ∩ socs N )), it follows that there exists minimal factors of N/socs N contained in (socs+1 N )/(socs N ), N -isomorphic to R1 /R2 (type ∆ say). All such minimal factors are central and, by 23.3, soc(N/socs N, ∆) = {0}. Now soc(N/socs N, ∆) is the sum of such central minimal factors. Thus (see 23.4), soc(N/socs N, ∆) contains a non-zero central minimal ideal of N/socs N . Clearly, this means N/socs N has central minimal ideals contrary to Z-constraint. The case of H having a minimal N -subgroup direct summand, which is a ring module, cannot occur. The theorem is proved.

26.

A Theorem on Z -constraint

Suppose N (with DCCR) is tame on V . The assumption that V is Zconstrained is straightforward. One might expect this condition to be of interest. On the other hand, why we should assume N is Z-constrained, is not so clear. This condition, of the last section, was developed because of the close relationship with that of Z-constraint in V . Indeed, in the case of N being 2-tame on V , they are equivalent. In view of the fact

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that, the deﬁnition is for tame N , it would be nice if only tameness were required. However, it looks unlikely that equivalence holds in this situation. What we will prove is the following:Theorem 26.1. If N (with DCCR) is 2-tame on V , then the following are equivalent (i) V is Z-constrained (ii) N is Z-constrained and (iii) all faithful 2-tame N -groups are Z-constrained. Proof: First it is proved that (i) implies (ii). By 25.2, V is cyclic and we can ﬁnd v in V , such that vN = V . If N is not Z-constrained, then there exists a minimal two sided factor of N of the form A/socr N (r ≥ 0 an integer), which is central. Now, vN/socr vN is a faithful N/socr N -group (it is relatvely well known that (socr vN : vN ) coincides with socr N ). With V1 = vN/socr vN , v1 = v + socr vN , N1 = N/socr N and A1 = A/socr N , we see that v1 N1 = V1 and, since A1 is a nonzero central ideal of N1 and N1 is faithful on V1 , v1 A1 is a non-zero central submodule of V1 . Because V1 is an N -homomorphic image of V , V cannot be Z-constrained. This contradiction can only mean N is Z-constrained and (i) implies (ii). It is now proved that (ii) implies (iii). The proof of this result is somewhat similar to some parts of the proof of 25.3. However, the full proof will be included here. Suppose N is Z-constrained and a faithful 2-tame N -group W , is not Z-constrained. This means W has a minimal factor K, which is central. By 18.5, all minimal factors of N , N -isomorphic to K (such exist), are central. Let R1 /R2 be such a factor with R1 chosen minimal for this property. It follows, from the minimal nature of R1 , that R2 is a right ideal of N uniquely maximal for being contained in R1 (see previous arguments). Take r ≥ 0, such that socr N is maximal for R1 ∩ socr N < R1 . Now socr+1 N ≥ R1 and, from the nature of R1 , R1 ∩ socr N ≤ R2 . We have {0} < (R1 + socr N )/socr N ≤ (socr+1 N )/(socr N ). Since (R1 +socr N )/socr N is N -isomorphic to R1 /(R1 ∩socr N ) which has a minimal factor (of N ) N -isomorphic to R1 /R2 (R2 ≥ R1 ∩ socr N )), it follows that there exists minimal factors of N/socr N contained in (socr+1 N )/(socr N ), N -isomorphic to R1 /R2 (type ∆ say). All such minimal factors are central and, by 23.3, soc(N/socr N, ∆) = {0}. Now soc(N/socr N, ∆) is the sum of such central minimal factors. Thus (see 23.4), soc(N/socr N, ∆) contains a non-zero central minimal ideal of N/socr N . Clearly, this means N/socr N has central minimal ideals con-

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trary to Z-constraint. The case of W having a central minimal factor cannot occur. It has been proved that (iii) holds. Clearly, (iii) implies (i) and the theorem holds.

27.

Another Theorem

The result of the last section is a satisfying statement as to what Zconstraint implies. However, in the compatible case this can be taken very much further. In order to do this a lemma is required. To motivate this we ﬁrst look at the tame case. V2 a factor of V . The question Suppose V is a tame N -group and V1 /V arises as to whether it is possible to extend V , by a copy U of the V2 . In particular, is the direct sum U ⊕ V , again a tame N -group V1 /V tame N -group? In order for this to be the case, all N -subgroups of V2 ⊕ V /V V2 , would have to be submodules. Thus a diagonal N (U + V2 )/V V2 ⊕ V1 /V V2 , would be a submodule and central subgroup ∆ of (U + V2 )/V V2 ⊕ V1 /V V2 . This implies (U + V2 )/V V2 is a ring module and in (U + V2 )/V V2 is central in (U + V2 )/V V2 ⊕ V /V V2 . It also follows readily that (U + V2 )/V (U + V2 )/V V2 ⊕ V /V V2 = ∆ ⊕ V /V V2 and ∆ is central in (U +V V2 )/V V2 ⊕V /V V2 . Thus (U +V V2 )/V V2 +∆ is central in V2 ⊕ V /V V2 . Since (U + V2 )/V V2 + ∆ coincides with (U + V2 )/V V2 ⊕ (U + V2 )/V V2 , V1 /V V2 is a central factor of U ⊕V . We have arrived at a condition, V1 /V V2 . that must be satisﬁed in order to extend V directly by a copy of V1 /V V2 is a central factor of V . However, in tame situations, This is that V1 /V V2 this may not be enough to ensure such an extension is possible. V1 /V could be a central minimal factor of V and V2 have a minimal factor, N -isomorphic to V1 /V V2 , but not central. All the indications are that such situations can arise. However, if compatibility is assumed such extensions can be made. The lemma hinted at above, is now given. V2 a central factor of Lemma 27.1. If V is a compatible N -group, V1 /V V2 , then U ⊕ V is a compatible V and U a direct sum of copies of V1 /V N -group. Proof: We have U is a direct sum ⊕U Ui , of copies Ui , i ∈ I of the N V2 . If x + y is any element of U ⊕ V , with x in U and y in V , group V1 /V then for any given α in N (x + y + u + v)α − (x + y)α = (x + u)α − xα + (y + v)α − yα = uα + (y + v)α − yα, for all u in U and v in V . Furthermore, there exists β in N such that (y + v)α − yα = vβ, for all v in V . It is clearly true that, for h in V1 hβ = (y + h)α − yα ≡ hα mod V2 .

The Z -Constrained Conjecture

131

Now, since for each i in I, Ui is a copy of V1 /V V2 , we see ui β = ui α, for all ui in Ui and it follows easily that uβ = uα, for all u in U . Thus (x + y + u + v)α − (x + y)α = uβ + vβ = (u + v)β, for all u in U and v in V . It has been shown that U ⊕ V is a compatible N -group and the lemma is proved. Lemma 27.1 appears to break down if it is only assumed that V is 2-tame. With uj + vj , j = 1, 2, taken in U ⊕ V , there exists β such that (y + vj )α − yα = vj β (see the proof) but when it comes to requiring hβ ≡ hα mod V2 , for all h in V1 , things break down. The reader can convince himself or herself, that the diﬃculty is real and compatibility deﬁnitely appears to be required. Lemma 27.1 allows us to prove the main result of this section. Theorem 27.2. If N (with DCCR) is compatible on V , then the following are equivalent:(i) V is Z-constrained, (ii) N is Z-constrained, (iii) all faithful compatible N -groups are Z-constrained, (iv) all faithful compatible N -groups are cyclic, (v) all faithful compatible N -groups are ﬁnitely generated, (vi) all faithful compatible N -groups have a composition series, (vii) all faithful compatible N -groups have DCCS and (viii) all faithful compatible N -groups have ACCS. Proof: From 26.1, it follows readily that (i) to (iii) are equivalent. From (iii) and 25.2, (iv) follows. Clearly (iv) implies (v). Also (v) to (viii) are, by 8.6, equivalent. It remains to show that (viii) implies (i). Suppose V is not Z-constrained. If this is the case, then there exists V2 of V . Let U1 , U2 , . . . , be copies of V1 /V V2 . a central minimal factor V1 /V By 27.1, V ⊕ U1 ⊕ U2 ⊕ . . . , is a compatible N -group. Clearly, it is faithful (V is faithful) and does not have ACCS. This contradiction to (viii) can only mean V is Z-constrained. The theorem is proved.

28.

Decomposition of Realisations

Suppose the nearring N has DCCR and is 2-tame on V . Take S as a subset of tnr(N ) and W as the minimal cover of V (S). By previous results W is cyclic and has a realisation, M say. We shall be interested in the situation where M decomposes into a sum H + R of a right N subgroup H and a submodule R (of M ), where H ∩R = {0}. Here a quite

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impressive deduction can be made. It can be shown that HR = {0}. In order to prove this, we ﬁrst need a lemma. Lemma 28.1. Adopting the notation outlined above, we have vH (v ( in V ) centralises a minimal factor W/L of W , where W/L is not N isomorphic to a minimal factor H/K of H. Proof: Now H is an N -homomorphic image of M and has property q. It follows that (see 9.1 and 11.4), all N -isomorphism types of possible H/K, are N -isomorphism types of minimal factors W/W W1 . By 10.4, it can be seen that W/Φ(W ) = W2 /Φ(W ) ⊕ W3 /Φ(W ), where W2 > Φ(W ) is a submodule of W , such that W2 /Φ(W ) is a direct sum of minimal N -groups, whose N -isomorphism types contain those of all possible H/K and where W3 > Φ(W ) is a submodule of W , such that W3 /Φ(W ) is N -isomorphic to W/L. Clearly, W2 is a cover of all the minimal factors W/P of W , not N -isomorphic to W3 /Φ(W ). It also centralises W3 /Φ(W ). By 18.3, W2 centralises W/L. Now, as is easily checked, vH is a minimal cover of the minimal factors vH/K1 of vH. vH (an N -homomorphic image of H) is therefore a minimal cover for some of the minimal factors W/P of W , which are not N -isomorphic to W3 /Φ(W ). Thus vH ≤ W2 and vH centralises W/L. The lemma is proved. We are now ready to prove the theorem stated above. Theorem 28.2. With the notation outlined above, we have HR = {0}. Proof: Suppose there exists α in H such that αR = {0}. If this is the case, then (0 : α) ∩ R < R. Take X as a submodule of M containing (0 : α) ∩ R and maximal in R. Since R > X ≥ (0 : α) ∩ R, it follows that αR > αX. Now H + X is contained in a maximal N subgroup X1 of M which, by property q, is a submodule of M . Thus the factor (X1 + R)/(X1 + X) of M is either {0} or N -isomorphic to R/X. Since X ≤ X1 and X1 + R = M , (X1 + R)/(X1 + X) is N -isomorphic to R/X. This means αR/αX is N -isomorphic to the minimal factor M/X1 of M . However, a minimal factor H/H1 of H is N -isomorphic to (H + R)/(H1 + R) = M/(H1 + R). Since H1 + R does not contain H, X1 = H1 + R. Thus αR/αX is not N -isomorphic to a minimal factor H/H1 of H (two minimal factors M/R1 and M/R2 of M , with R1 = R2 cannot be N -isomorphic, by 15.4 and 15.5). Now αR ≤ H (α is in H) and H contains a right N -subgroup H2 minimal for having a minimal H3 , N -isomorphic to αR/αX. This means H3 is the unique factor H2 /H maximal submodule of H2 . Take v in V such that (0 : v) ∩ H2 < H2 .

The Z -Constrained Conjecture

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Clearly, (0 : v) ∩ H2 ≤ H3 and vH H2 /vH H3 is N -isomorphic to αR/αX. H2 /vH H3 Thus vH contains vH H2 = {0}, which has a minimal factor vH (N -isomorphic to αR/αX). By 28.1, vH centralises vH H2 /vH H3 . Since H3 ≤ vH/vH H3 , this can only mean vH H2 /vH H3 is a ring module. vH H2 /vH This contradiction to the perfect nature of M , yields the fact that R ≤ (0 : α), for all α in H and HR = {0}. The theorem is proved.

29.

Left Units of Realisations

If N is a nearring and M a right N -subgroup of N , then a left unit of M , is an element γ1 of M such that γ1 M = M . Such elements of M occur precisely when M is self monogenic. They often play a crucial role in developing theory. When N has DCCN , left units of certain right N subgroups arise. In this area, their value becomes evident. For example, if M is minimal such that M A = {0} (A a minimal ideal of N ), then a left unit γ1 of M is such that γ1 A = M ∩ A. All other α in M satisfy αA = {0}. However, this section looks at left units from a diﬀerent viewpoint. If N is a nearring and M1 ≤ M2 are both self monogenic right N -subgroups of N , then with DCCN , M2 must split into a sum M1 + R (R a submodule of M2 ) in such a way that M1 ∩ R = {0}. The question arises, when a left unit of M1 (γ1 say) can be lifted to one of M2 . In other words, when is it possible to ﬁnd α in R, such that γ1 + α is a left unit of M2 . This appears to be a deep question. It would be of considerable value to have some answers to such a question. In general it appears that this is not possible. However, as indicated in the next paragraph, there are certain situations where it can be done. Since this is all that later developments require, it is suﬃcient for the present paper. How this happens is now outlined. If W is a cyclic N -group (N with DCCN ) and M a realisation of W then, by 11.2, M has a left identity. In this section we shall be interested in ﬁnding conditions ensuring an element of M , derived from an element of a smaller right N -subgroup, is a left unit of M . We shall do this, in the special case of N being 2-tame on V and W being a minimal cover of V (S) (S a subset of tnr(N )). It will be assumed that M has a decomposition H + R, where H is a N -subgroup of M and R a submodule of M such that H ∩ R = {0}. With e a left identity of M , we have e = e1 + α, where e1 is in H and α in R. What we are interested in showing is that γ + α (γ a left unit of H) is a left unit of M . Lemma 29.1. With the notation outlined above, we have (H + Φ(M ))/Φ(M ) is a submodule of M/Φ(M ) and M/Φ(M ) = (H + Φ(M ))/Φ(M ) ⊕ (R + Φ(M ))/Φ(M ).

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Proof: By 15.5, (H + Φ(M ))/Φ(M ) is a submodule of M/Φ(M ). Clearly M/Φ(M ) = (H + Φ(M ))/Φ(M ) + (R + Φ(M ))/Φ(M ). Suppose [(H + Φ(M ))/Φ(M )] ∩ [(R + Φ(M ))/Φ(M )] = {0} and K/Φ(M ), is a minimal factor of M contained in this submodule. Thus M/Φ(M ) = K1 /Φ(M ) ⊕ K/Φ(M ), where K1 > Φ(M ) is a maximal submodule of M . Because [K1 /Φ(M )] ∩ [(H + Φ(M ))/Φ(M )] < (H + Φ(M ))/Φ(M ) and [K1 /Φ(M )] ∩ [(R + Φ(M ))/Φ(M )] < (R + Φ(M ))/Φ(M ), it follows that K1 ∩ H < H and K1 ∩ R < R. Now M = H + K1 , and M/K1 is N -isomorphic to H/(H ∩ K1 ). However, (H + R)/(H ∩ K1 + R) is an N -homomorphic image of this minimal N -group. It cannot be zero as H + R = M and H ∩ K1 + R < M . Thus K1 and H ∩ K1 + R are both maximal submodules of M with M/K1 and M/(H ∩ K1 + R), N -isomorphic. The only possibility (see the proof of 28.2), is that K1 = H ∩ K1 + R and R ≤ K1 . This contradiction completes the proof. From section 11 and 7.4 of [15], we have that:Proposition 29.2. With the notation outlined above, Φ(M ) = J(N ) ∩ M and Φ(M ) is an ideal of M . Proposition 29.3. With the notation outlined above, we have that (H +Φ(M ))/Φ(M ) and (R+Φ(M ))/Φ(M ) are both ideals of the nearring M/Φ(M ). Proof: Now (H + Φ(M ))/Φ(M ) (= X1 say) is a direct sum of minimal N -groups. The same is true for (R + Φ(M ))/Φ(M ) (= X2 say). No such summands of X1 are N -isomorphic to those of X2 (see 15.5 and 15.4). It follows readily that X1 X2 = X2 X1 = {0} and Xi are clearly ideals of M/Φ(M ). The proposition holds. We now prove the theorem stated above. Theorem 29.4. The notation outlined above is employed. If e is a left identity of M , γ a left unit of H and e = e1 + α, where e1 is in H and α in R, then γ + α is a left unit of M .

The Z -Constrained Conjecture

135

Proof: Now e + Φ(M ) is a left identity of M/Φ(M ) and e + Φ(M ) = e1 + Φ(M ) + α + Φ(M ), where e1 + Φ(M ) is in (H + Φ(M ))/Φ(M ) and α + Φ(M ) is in (R + Φ(M ))/Φ(M ). Thus α + Φ(M ) is a left identity of (R + Φ(M ))/Φ(M ) (see 29.1 and 29.3). Also, γ + Φ(M ) is a left unit of (H + Φ(M ))/Φ(M ). Thus (γ + α + Φ(M ))[(H + Φ(M ))/Φ(M ) ⊕ (R + Φ(M ))/Φ(M )] (see 29.1) must, by 29.3, coincide with (γ + Φ(M ))[(H + Φ(M ))/Φ(M )] ⊕ (α + Φ(M ))[(R + Φ(M ))/Φ(M )] which, by the indicated properties of γ + Φ(M ) and α + Φ(M ), coincides with (H + Φ(M ))/Φ(M ) ⊕ (R + Φ(M ))/Φ(M ). An easy deduction of this, is that (γ + α)(H + R) + Φ(M ) = H + R

(5)

and, if (γ + α)(H + R) < M , then a maximal N -subgroup K of H + R containing (γ + α)(H + R) cannot exist ((5) implies K = H + R). Thus (γ + α)(H + R) = H + R and the theorem is proved.

30.

Minimal Ideals and Realisations

When N is a nearring with DCCN , then the relationship between a self monogenic right N -subgroup of N and an ideal of N , is particularly pleasing. A special case of this is exhibited in 12.1. It is easy to state and prove a fuller result. Proposition 30.1. If N is a nearring with DCCN , M a self monogenic right N -subgroup of N and A an ideal of N , then eA = M ∩ A = M A, where e is a left identity of M . Proof: This follows because M ∩ A = e.M ∩ A ≤ eA ⊆ M A ⊆ M ∩ A. If A is a minimal ideal of N and M is in mr(A), then M ∩ A contains minimal submodules of M (N -submodules) contained in A. If R is such a minimal submodule then, for any γ in M , γM < M or γM = M . In the ﬁrst case γM A = {0} and γ.M ∩ A = {0} so that γR = {0}. In the second case if γR = {0} then, because γM = M , γR is a minimal

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submodule of M . This means the sum γR (= X say) over all γ in M is a non-zero submodule of M (it is non-zero because M has a left identity). Clearly, each γR ≤ M ∩ A and X ≤ M ∩ A. However, it is clearly the case that, M X ⊆ X and M.Id(X) (Id(X) the ideal of N generated by X) is contained in X. Because Id(X) = A, we see M ∩ A ≤ X. Thus, we have:Lemma 30.2. If N is a nearring with DCCN , A a minimal ideal of N , M is in mr(A) and R a minimal submodule of M contained in M ∩ A, then M ∩ A = γR (the sum over all submodules γR of M with γ in M ). Because the γR of 30.2 can only be non-zero when γ is a left unit of M (see above explanation) we have: Corollary 30.3. If N , A, M and R are as in 30.2, then M ∩A = γR, where the sum is over all left units γ of M . What is really required is a certain type of extension of 30.3. It can happen that, when N is a nearring with DCCN , A a minimal ideal of N , M a self monogenic right N -subgroup of N and R a minimal submodule of M contained in M ∩ A (necessarily non-zero), then M ∩ A = γR where the sum is over all left units of M . This possibility arises in 2-tame situations, when dealing with realisations of perfect 2-tame N -groups. Theorem 30.4. Suppose N (with DCCR) is 2-tame on the perfect N -group V , M is a realisation of V and A a minimal ideal of N . We have (i) M has a left identity e and eA = M ∩ A = M A, (ii) a minimal submodule R of M contained in M ∩ A is a minimal N -group, (iii) there exists a right N -subgroup H of mr(A) contained in M , (iv) H ∩ A = M ∩ A, (v) R is a minimal submodule of H contained in H ∩ A and (vi) M ∩ A = βR, where the sum is over all left units β of M . Proof: By 11.2 and 30.1, (i) holds. Clearly M ∩A is a submodule of M and, if non-zero (see (iii)) contains a minimal submodule. By [10], A is a direct sum of minimal right ideals, which are minimal N -groups. Thus eA (= M ∩A) is contained in M and a direct sum of minimal submodules, which are minimal N -subgroups of M . It follows readily that R is necessarily a minimal N -group and (ii) holds. Now there exists w in V such that wM = V and, by faithfulness, M A = {0}. Thus (iii) holds (ie. H exists).

The Z -Constrained Conjecture

137

Since H is, by 6.1 of [15], self monogenic it has a left identity e1 . Now e1 M = H and M = H + (0 : e1 ) ∩ M , where (0 : e1 ) ∩ M ∩ H = {0}. Thus M = H + X, where X (taken as (0 : e1 ) ∩ M ) is a submodule of M such that X ∩ H = {0}. Now an element of M ∩ A is of the form α1 +α2 , where α1 is in H and α2 in X. Thus e1 (α1 +α2 ) = α1 is in H ∩A (≤ M ∩ A) and α2 is in M ∩ A. It follows that M ∩ A ≤ H ∩ A + X ∩ A. Clearly the reverse inclusion holds and we have equality. If it is shown that X ∩ A = {0}, then (iv) will hold. Now M.X = (H + X).X ⊆ X, by 28.2. Also M A ⊆ A, and it therefore follows that M.(X ∩A) ⊆ X ∩A. Thus M.Id(X ∩ A) ⊆ X ∩ A and, if X ∩ A = {0}, then M.A ⊆ X ∩ A. This is clearly impossible (H ∩ A = {0}). Thus (iv) holds. Now R is a submodule of H, by (iv). It is therefore a minimal submodule of H by (ii). It remains to show that (vi) holds. Suppose e = e2 + α, where e2 is in H and α in X. If it is shown that H ∩A=

(γ + α)R, where γ is a left unit of H

(6)

then, by (iv) and 29.4, (vi) will hold. Now (γ + α)ρ − γρ (ρ in R) is in A (because ρ is in A). It is also in X (α is in X). Thus (γ + α)ρ − γρ is in X ∩ A = {0} and (γ + α)ρ = γρ, for all ρ in R. It follows that the sum (6) is just γR over all left units γ of H. By (v) and 30.3, γR coincides with H ∩ A. It has been shown that (vi) holds and the theorem is proved. Before leaving this section, we brieﬂy recap what has been shown. For a nearring N (with DCCN ), minimal ideal A of N and right N subgroup M of mr(A), many special properties hold. One of these is that the orbit (it should be reasonably clear how this word is being used) of a minimal submodule of M contained in M ∩ A, under the left units of M , is just M ∩ A. In general, it is not possible to draw such a conclusion for a self monogenic right N -subgroup of N , not annihilating A from the left. In particular A may not be the orbit of a minimal right ideal, under the units of N . However, theorem 30.4 gives us an example of a situation where M can be enlarged. This is covered in (vi) of 30.4. It is the most important aspect of that theorem. What it shows is that M1 ∩ A, where M1 is a realisation of a perfect 2-tame N -group, has the desired property. Such a conclusion, is a quite real success, in our attempts to prove the main results of this paper. It depends quite heavily on previous theorems. One is 28.2. Another is 29.4.

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Centralisers

A big success, that has helped the advance of tame theory, has been the recognition that centralisers are well behaved. Centralisers always exist but in order that they be useful, we often require that they be N -subgroups. This is the case for 3-tame N -groups (therefore for ntame, n ≥ 3, and compatible N -groups). These results are now quite well known. In this section we shall give an alternative proof of this fact. However, this section is about more than just centralisers in 3tame N -groups. It also covers centralisers in 3-tame nearrings. This is a development that is used in a special way in the next section. If V is an N -group and S a subset of V , then an N -subgroup U of V , will be said to centralise S if (h + u)α = hα + uα, for all h in S, u in U and α in N . Since N has an identity and V is unitary, this deﬁning relationship can be replaced by any one of three others. These are those obtained by interchanging h and u on the left and/or the right. The centraliser of S is simply taken as the union of all N -subgroups of V that centralise S. This N -subset of V is denoted by CV (S). In the case of W , being an N -subgroup of V , we denote W ∩ CV (S) by CW (S). In 3-tame situations CW (S) is always an N -subgroup of V . Obviously, this follows if it is established that CV (S) is an N -subgroup of V . Theorem 31.1. If V is a 3-tame N -group and S a subset of V , then CV (S) is an N -subgroup of V . Proof: Since CV (S) is a union of N -subgroups of V , CV (S)N ⊆ CV (S). Also, if a is in CV (S), then −a is in CV (S). It remains to show that if ai , i = 1, 2, are in CV (S), then so is a1 + a2 . Thus we are required to show that, for u in S and α in N (a1 + a2 + u)α = (a1 + a2 )α + uα. Now, there exists β in N such that (a1 + wi )α − a1 α = wi β, for any ‘three’ given elements wi , i = 1, 2, 3, of V . With w1 = a2 + u, w2 = a2 and w3 = u, we can ﬁnd the appropriate β. Thus (a1 + a2 + u)α − a1 α = (a2 + u)β, (a1 + a2 )α − a1 α = a2 β, and (a1 + u)α − a1 α = uβ. However, (a2 + u)β = a2 β + uβ and (a1 + a2 + u)α − a1 α = (a1 + a2 )α − a1 α + (a1 + u)α − a1 α. This yields the fact that (a1 + a2 + u)α = (a1 + a2 )α − a1 α + (a1 + u)α

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and, because a1 is in CV (S) (a1 + a2 + u)α = (a1 + a2 )α − a1 α + a1 α + uα = (a1 + a2 )α + uα. The theorem is proved. Theorem 31.1 tells us that in 3-tame N -groups, centralisers exist in a more real way than in arbitrary N -groups. Also the notation has a U2 a factor of V , valuable extension. If V is a 3-tame N -group and U1 /U U1 /U U2 ) exists and is an N -subgroup of V /U U2 . The then clearly CV /U U2 (U U1 /U U2 ) will be used to denote the N -subgroup H ≥ U2 of notation CV (U U1 /U U2 ). By 18.3, we have that:V that is such that H/U U2 = CV /U U2 (U U2 and W1 /W W2 are Theorem 31.2. If V is a 3-tame N -group and U1 /U U1 /U U2 ) = CV (W W1 /W W2 ). N -isomorphic factors of V , then CV (U If V is a 3-tame N -group and H a submodule of V , then the notation CV (H) is well deﬁned. It may happen that there exists a nearring N1 such that V is a 3-tame N1 -group and H is an N1 -submodule of V . In order to make the distinction, the centraliser of H in V with respect to N1 will sometimes be denoted by CV (H)N1 . Proposition 31.3. If V is a 3-tame N -group, U ≤ H submodules of V and N1 = N/(U : V ), then CV /U (H/U ) = CV /U (H/U )N1 . The centraliser of a subset S of a nearring N exists. With the notation outlined at the beginning of this section it is denoted by CN (S). Also, if CN (S). A particularly M is a right N -subgroup of N , then CM (S) = M ∩C pleasing feature of 3-tame nearrings is that the centralisers CN (S) are right ideals of N . Theorem 31.4. If N is a 3-tame nearring and S a subset of N , then CN (S) is a right ideal of N . In order to prove this result, we establish two preliminaries. Lemma 31.5. If the nearring N is 3-tame on V and S a subset of N , then a right N -subgroup M of N centralises S if, and only if, vM ≤ CV (vS), for all v in V . Proof: If vM ≤ CV (vS), for all v in V , then for m in M , β in S and α in N , we have (vm + vβ)α = vmα + vβα, for all v in V . Thus v(m + β)α = v(mα + βα), for all v in V and (m + β)α = mα + βα, showing M ≤ CN (S). On the other hand, if M ≤ CN (S) then, (m + β)α = mα + βα, for all m in M , β in S and α in N . This means (vm + vβ)α = vmα + vβα, for all m in M , β in S and α in N . Clearly this implies vM ≤ CV (vS), for all v in V . The lemma is proved.

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Proposition 31.6. If N is a 3-tame nearring, S a subset of N and M a right N -subgroup of N such that M ≤ CN (S), then R(M ) (the right ideal of N generated by M ) is contained in CN (S). Proof: Let V be a faithful 3-tame N -group. For each v in V , M +(0 : v) is a right ideal of N (see 8.9). Since M ≤ R(M ) ≤ M + (0 : v), it is clear that vM = vR(M ). Thus vR(M ) ≤ CV (vS), for all v in V , by 31.5. Again, by 31.5, R(M ) ≤ CN (S). The proposition is proved. We are now ready for:Proof of 31.4: Let Mi , i ∈ I, be the set of all right N -subgroups of N , that centralise S. It will follow that 31.4 holds, if it is shown that R(M Mi ) (the sum over all i in I), is contained in CN (S). By 31.6, each R(M Mi ) is contained in CN (S). Let V be a faithful 3-tame N -group. Thus, by 31.5, vR(M Mi ) ≤ CV (vS) for each v in V . By 31.1, we have that v R(M Mi ) ≤ CV (vS), for each v in V . Now, from 31.5, we conclude that R(M Mi ) ≤ CN (S). The theorem is proved. This section opened with statements to the eﬀect that, in many tame situations, centralisers are well behaved. As stated there, this has been a big help to the advance of tame theory. However, we should also look at the reverse side of the coin. Are centralisers as well behaved as we would like them to be? Certainly, in 3-tame N -groups centralisers are as well behaved as can be expected. For example, they exist in N -homomorphic images of such N -groups. When it comes to centralisers of subsets of a 3-tame nearring N , things are not so pleasant. If N is such a nearring, R a right ideal of N and S a subset of N/R, then it is not clear as to what the centraliser of S in N/R, should be taken as. Even when R is an ideal this is so. Certainly, when R is an ideal of the form socr N , all is well, but in general, this is not the case. This is unfortunate, but it may not be that bad. The need to use centralisers in N/R does not appear to arise all that often. In any case, in d.g. situations, centralisers will always exist. This section ﬁnishes with an examination of d.g. N -groups (N d.g.). For such an N -group V , with subset S, we have the additive centraliser CV+ (S) of V and the maximal N -subgroup of CV+ (S) exist. The N -subgroup in question is simply that generated by all N -subgroups contained in CV+ (S). It has all the properties one would expect of a centraliser. Thus, in d.g. situations, centralisers are available in N homomorphic images also. This means they are not only available in N but in all N/R (R a right ideal of N ). The d.g. situation illustrates something further. If in a d.g. N -group V , we take an N -subgroup V1 , then the N -subgroup of V generated by all N -subgroups that normalise V1 again normalises V1 . This means, in these situations, not only do we

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have centralisers, but normalisers are present. They are also present in all N -homomorphic images. Do normalisers occur in tame (eg. 3-tame) situations? For a 3-tame N -group the notion is vacuous but for 3-tame nearrings the question is valid. However, it appears that in general they do not exist. In certain interesting special cases we have existence, but for arbitrary 3-tame nearrings things appear to be more complex. More work remains to be done. Normalisers might turn out to be a valuable tool.

32.

A Uniqueness Result

The question arises as to when a faithful perfect 3-tame N -group V (N with DCCR) is unique (up to N -isomorphism). It appears that it is much too much to hope that uniqueness always holds. What is more reasonable is that certain factors V /W of V are unique. What are the factors that we want this to hold for? They are those of the form U1 /U U2 ), where U1 /U U2 is a minimal factor of V . It is within the V /C CV (U realms of possibility that if Vi , i = 1, 2, are two faithful perfect 3-tame CV1 (H1 /H H2 ) is N -isomorphic to V2 /C CV2 (W W1 /W W2 ), N -groups, then V1 /C H2 and W1 /W W2 are N -isomorphic minimal factors of V1 whenever H1 /H and V2 respectively. It is not only within the realms of possibility but, as seen in this section, it is also true. The result outlined above, will be a key ingredient in the proof of the main result of this paper. In order to establish, this very important preliminary, two lemmas are required. Lemma 32.1. Suppose N (with DCCR) is 3-tame on the perfect N group V , M is a realisation of V (V ( is cyclic) and U is a minimal submodule of V of type ∆ such that soc(N, ∆) = {0}. If A is a minimal ideal of N , contained in soc(N, ∆) (see 23.4), then (i) M ∩ A = {0}, (ii) for v in V such that vM = V , vR = {0} for some minimal submodule R of M contained in M ∩ A, (iii) for R as in (ii), vR is N -isomorphic to U and CM (R). (iv) for R as in (ii), V /C CV (U ) is N -isomorphic to M/C Proof: Since there exists w in V such that wM = V , we see M A = {0} and M ∩ A = {0}. Thus (i) holds. Let v in V , be such that, vM = V . Let e be a left identity of M (see 11.2). Now A is a direct sum of minimal right ideals N -isomorphic to U (see section 23). Thus eA (= M ∩ A) is a sum of minimal submodules of eN (= M ), N -isomorphic to U . One such submodule R must be such

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that vR = {0}, because otherwise veA = {0} and vM A = V A = {0}. Thus (ii) holds. From (ii), R is N -isomorphic to U and therefore so is vR. Thus (iii) holds. The main content of the lemma is contained in (iv). We now proceed to prove this. Since vR = {0}, (0 : v) ∩ R = {0} (R is a minimal right N -subgroup of N , from the proof of (ii)). Thus (0 : v) ∩ M ∩ R = {0} and (0 : v) ∩ M is a submodule of M centralising R. It follows that (0 : v) ∩ M ≤ CM (R). Now, for all m1 in CM (R), ρ in R and α in N , (m1 + ρ)α = m1 α + ρα and (vm1 + vρ)α = vm1 α + vρα. This means vC CM (R) ≤ CV (vR) = CV (U ). However, if u is in CV (U ), then u is in CV (vR), and with m2 in M such that vm2 = u, we see that (vm2 + vρ)α = vm2 α + vρα, for all ρ in R and α in N . This means (m2 + ρ)α − ρα − m2 α is in (0 : v) ∩ M . Clearly, it is in R, and because (0 : v) ∩ M ∩ R = {0}, (m2 + ρ)α = m2 α + ρα and CM (R). It has been shown that m2 is in CM (R). Thus vm2 (= u) is in vC CV (U ) coincides with vM/vC CM (R), which vC CM (R) = CV (U ). Thus V /C is N -isomorphic to [M/(M ∩ (0 : v))]/[C CM (R)/(M ∩ (0 : v))] and N -isomorphic to M/C CM (R). The lemma is proved. Another substantial step, in proving the result discussed above follows. Lemma 32.2. Suppose H is a minimal N -group of type ∆ and N (with DCCR) is 3-tame on the perfect N -groups Vi , i = 1, 2. If soc(N, ∆) = {0}, then Vi , i = 1, 2, contain minimal N -subgroups Ui , i = 1, 2, of type CV1 (U U1 ) is N -isomorphic to V2 /C CV2 (U U2 ). ∆ and V1 /C Proof: Because soc(N, ∆) contains a minimal right N -subgroup K of type ∆ and Vi K = {0}, for i = 1, 2, there exists ai in Vi , such that ai K = {0} and taking Ui as ai K (N -isomorphic to K) we have established the existence of Ui as in the statement of the lemma. The Vi , i = 1, 2, are both minimal covers of V (tnr(N )). They are cyclic and have realisations Mi , i = 1, 2. Thus there exists wi in Vi such that wi Mi = Vi , i = 1, 2. However, by 21.3, M1 and M2 belong to the same weak conjugacy class. Thus there exists α2 in M2 such that α2 M1 = M2 and w2 α2 M1 = V2 . This means M1 contains a realisation M1 = H and m1 in H of V2 . By 21.3, there exists h in H such that hM M1 = M1 . Since m1 h is in M1 , from [8] M1 , such that m1 H = m1 hM (0 : m1 h) ∩ M1 = {0} and (0 : h) ∩ M1 = {0}. Thus M1 is N -isomorphic to hM M1 (= H) and DCCN ensures H = M1 . It follows that, M1 is

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not only a realisation of V1 , but also of V2 . We have v1 M1 = V1 and v2 M1 = V2 , for some vi , i = 1, 2, in Vi . The next step in the proof of 32.2, is to use 32.1 to show both CV1 (U U1 ) and V2 /C CV2 (U U2 ) are N -isomorphic. Take A as a minimal V1 /C ideal of N contained in soc(N, ∆) (see 23.4). Now, by (ii) and (iii) of 32.1, M1 ∩A contains a minimal submodule R of M1 , N -isomorphic to U1 and such that v1 R = {0}. By (iv) of 32.1, V1 /C CV1 (U U1 ) is N -isomorphic to CM1 (R). From 30.4, M1 ∩ A is the sum βR over all left units β of M1 /C M1 ∩A) = {0}. Thus, for M1 . Because V2 A = {0}, v2 M1 A = {0} and v2 (M some left unit β1 of M1 , v2 β1 R = {0}. Clearly v2 β1 R is N -isomorphic to CV2 (U U2 ) U2 . Also v2 β1 M1 = V2 and so, again by (iv) of 32.1, we have V2 /C CM1 (R). It has been shown that V1 /C CV1 (U U1 ) is N -isomorphic to M1 /C CV2 (U U2 ) are N -isomorphic. The lemma is entirely proved. and V2 /C The theorem outlined at the outset of this section can now be proved. Theorem 32.3. Suppose N has DCCR and is 3-tame on the perfect N groups Vi , i = 1, 2. If U1 /U U2 is a minimal factor of V1 , N -isomorphic to W2 of V2 , then V1 /C CV1 (U U1 /U U2 ) is N -isomorphic the minimal factor W1 /W CV2 (W W1 /W W2 ). to V2 /C U2 of V1 (type ∆ say) is N -isomorphic Proof: The minimal factor U1 /U to a minimal factor of N . Now N is such that {0} = soc0 N < soc1 N < · · · < socr N = N, for some integer r ≥ 1. Also, N has a composition series and there exists an integer s ≥ 0, with (socs+1 N )/(socs N ) containing a minimal right ideal of N/socs N of type ∆. Thus soc(N/socs N, ∆) = {0}. Also N/socs N is faithful on the perfect N -groups Vi /socs Vi , i = 1, 2. By 32.2, both V1 /socs V1 and V2 /socs V2 contain minimal N/socs N -subgroups (thus minimal N -groups) of type ∆ (H1 and H2 say). Again, by 32.2, (V V1 /socs V1 )/K1 is N/socs N -isomorphic (thus N -isomorphic) to K2 , where (V V2 /socs V2 )/K K1 = CV1 /socs V1 (H1 )N/socs N and K2 = CV2 /socs V2 (H H2 )N/socs N . Hi ). Now, there exists By 31.3, Ki , i = 1, 2, coincide with CVi /socs Vi (H minimal factors Li /socs Vi , i = 1, 2, of Vi such that Li /socs Vi = Hi and Ki = CVi (Li /socs Vi )/socs Vi . Because (V V1 /socs V1 )/K1 is N -isomorphic to (V V2 /socs V2 )/K K2 , it follows that CV1 (L1 /socs V1 ) is N -isomorphic to V2 /C CV2 (L2 /socs V2 ). V1 /C

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Both L1 /socs V1 and L2 /socs V2 are of type ∆ and N -isomorphic to U1 /U U2 and W1 /W W2 . By 31.2, V1 /C CV1 (U U1 /U U2 ) is N -isomorphic to CV2 (W W1 /W W2 ). The theorem is proved. V2 /C Theorem 32.3 can be applied in a number of meaningful ways. What can be said about faithful perfect 3-tame N -groups, with a unique minimal non-ring submodule? If N has DCCR, all such N -groups are N isomorphic. This can be modiﬁed to include the case where the unique minimal submodule is self centralising. Here, the factors (with respect to the minimal submodule) must all be N -isomorphic. These two instances of applications of 32.3 can be generalised in a number of intriguing ways. It is not within the goals of this paper to do this here, but it is an endeavour that proves rewarding.

33.

Local N -nilpotency

In [12] local N -nilpotency is deﬁned for tame N -groups. This deﬁnition is insuﬃcient for our present needs. We require a deﬁnition of local N -nilpotency, that encompasses N -groups other than tame ones. The reason is that, we have to apply the deﬁnition to tame nearrings. Since tame nearrings are, as N -groups, not necessarily tame, a more embracing deﬁnition is required. This is not hard to come by. It is an obvious extension of the tame N -group deﬁnition. An N -group V is said to be locally N -nilpotent, if every ﬁnitely generated N -subgroup of V is N -nilpotent (cf. section three of [12]). By [12], if V is N -nilpotent then it is locally N -nilpotent. An easily deduced consequence of this deﬁnition is as follows:Proposition 33.1. Any N -subgroup or N -homomorphic image of a locally N -nilpotent N -group is locally N -nilpotent. The 3-tame situation is of special interest. According to 3.3 of [12], a ﬁnite sum of N -nilpotent N -subgroups (ie. submodules) of a 3-tame N -group is N -nilpotent. This gives (see 3.4 of [12]):Proposition 33.2. If V is a 3-tame N -group, then any sum of locally N -nilpotent submodules of V is locally N -nilpotent, any sum of N -nilpotent submodules of V is locally N -nilpotent, and any locally N nilpotent submodule of V is a sum of N -nilpotent submodules of V . If V is a 3-tame N -group, then the sum of all locally N -nilpotent submodules of V is locally N -nilpotent. This submodule of V is denoted by L(V ). It is the unique maximal locally N -nilpotent submodule of V . From 33.1, any N -subgroup of L(V ) is locally N -nilpotent. Also from 33.1, if U is a submodule of V , then (L(V ) + U )/U ≤ L(V /U ).

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We now seek to ﬁnd a similar N -subgroup (it will turn out to be an ideal) of a 3-tame nearring. For a 3-tame nearring N , L(N ) is taken as the right N -subgroup of N generated by all locally N -nilpotent right N -subgroups. Our goal is to show L(N ) is particularly well behaved. It will be seen to be a locally N -nilpotent ideal that coincides with the sum of all N -nilpotent right ideals. Proposition 33.3. If N is a 3-tame nearring and M an N -nilpotent right N -subgroup of N , then R(M ) is N -nilpotent. Proof: Let V be a faithful 3-tame N -group. By 8.9, each M + (0 : v), v in V , is a right ideal of N and M ≤ R(M ) ≤ M + (0 : v). Thus, for all v in V , vR(M ) = vM . Now R(M ) can be embedded in the direct product vR(M ), taken over all v in V . Since each vR(M ) = vM and is of N -nilpotency class not greater than that of M , it is clear that vR(M ) is N -nilpotent. Thus R(M ) is N -nilpotent. Proposition 33.4. A ﬁnite sum of N -nilpotent right ideals of a 3-tame nearring N is N -nilpotent. Proof: Clearly this will follow if it is shown that, whenever Ri , i = 1, 2, are N -nilpotent right ideals of N , then so is R1 + R2 . Now R1 + R2 can be embedded in the direct product v(R1 + R2 ), taken over all v in V . Since vRi has N -nilpotency class ≤ mi (the class of Ri ), v(R1 + R2 ) is, by 3.3 of [12], N -nilpotent of class ≤ m1 + m2 . Thus v(R1 + R2 ) is N -nilpotent and so is R1 + R2 . The proposition is proved. Now L(N ) is generated by all Mi , i ∈ I, which are locally N -nilpotent right N -subgroups of N . Thus, for γ in some Mj , j in I, γN is N nilpotent and,by 33.3, R(γ) is N -nilpotent. This implies R(γ) ≤ L(N ) and, the sum R(γ) over all such γ, is contained in L(N ). Clearly each Mi , i ∈ I, is contained in this sum and L(N ) is a sum of N -nilpotent right ideals of N . Any N -nilpotent right ideal of N is contained in L(N ). Thus we have:Proposition 33.5. If N is a 3-tame nearring, then L(N ) is the sum of all N -nilpotent right ideals of N . Corollary 33.6. L(N) is locally N -nilpotent. Proof: A ﬁnitely generated right N -subgroup of L(N ) is, by 33.5, contained in a ﬁnite sum of N -nilpotent right ideals. By 33.4, such a right N -subgroup is N -nilpotent. Proposition 33.7. If N is a 3-tame nearring, then L(N ) is an ideal of N . right ideals of N . Proof: By 33.5, L(N ) = Ri over all N -nilpotent Thus αL(N ), α in N , coincides with the sum αRi of all submodules

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αRi of αL(N ). Each αRi , is N -nilpotent and, by 33.3, so is R(αRi ). Thus αL(N ) ≤ R(αRi ) ≤ L(N ), by 33.5, and the proposition is proved. If N is 3-tame on V , then what is the relationship between L(V ) and L(N )? One feature of this is presented in the next proposition. Propsition 33.8. If N is 3-tame on V , then L(N ) ≤ (L(V ) : V ). Proof: If R is any N -nilpotent right ideal of N , then V R⊆ vR (the sum over all v in V ). Each vR, v in V , is N -nilpotent and vR ≤ L(V ). Thus R ≤ (L(V ) : V ) and, by 33.5, L(N ) ≤ (L(V ) : V ). Proposition 33.8 is all very well, but what is really required is conditions that ensure the reverse inclusion also holds. This will be the case if L(V ) is N -nilpotent. There are a number of important situations where this holds. Illustrations are as follows. If N , with DCCR, is 3-tame on V then, by 6.4 of [12], L(V ) is N -nilpotent. If N , with DCCI or ACCI, is compatible on V then, by 7.3 or 9.2 of [12], L(V ) is N -nilpotent. In these situations the next result is applicable. Proposition 33.9. If N is 3-tame on V and L(V ) is N -nilpotent, then L(N ) = (L(V ) : V ). Proof: This will follow from 33.8, if it is shown that (L(V ) : V ) is N -nilpotent. For each v in V , v(L(V ) : V ) ≤ L(V ) and is N -nilpotent of class ≤ thatof L(V ) (m say). Thus (L(V ) : V ) embedds into the direct product v(L(V ) : V ) (over all v in V ) of N -nilpotent N -groups of class ≤ m. Consequently (L(V ) : V ) is N -nilpotent (class ≤ m) and the proposition is proved. In the above case V /L(V ) is a faithful 3-tame N/L(N )-group. It makes sense, in the case of N having DCCR, to talk about N/L(N ) being Z-constrained. This will happen in section 39. Before leaving this section we brieﬂy look at another way of specifying L(V ). In 3-tame situations centralisers are available. If V is a 3-tame N U1 /U U2 ) (U U1 /U U2 minimal factors group, then the intersection of all CV (U of V ) is called the Fitting submodule of V and denoted by F (V ). If N has DCCR then, as is reasonably well known, L(V ) = F (V ). The factor V /F (V ) of V turns out to be quite important. It will be called the Fitting factor. A big result of [12] is that, with compatibility, the Fitting factor is ﬁnite.

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34.

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Local N -solubility

This section is, in many regards, similar to § 33. We are requiring a deﬁnition of local N -solubility that covers any N -group. That of [12] is insuﬃcient for our present needs, as it only covers the tame case. Because tame nearrings are, as N -groups, not necessarily tame, a more general notion is required. As in the case of local N -nilpotency, this is not hard to come by. It is an obvious extension of the tame N -group deﬁnition. An N -group V is said to be locally N -soluble, if every ﬁnitely generated N -subgroup of V is N -soluble (cf. section three of [12]). Rather trivial examples of local N -solubility have already been encountered. By 7.1, if V is N -soluble, then it is locally N -soluble. An easily deduced consequence of this deﬁnition is as follows:Proposition 34.1. Any N -subgroup or N -homomorphic image of a locally N -soluble N -group is locally N -soluble. How does N -solubility behave as far as sums of submodules are concerned? In the case of N -nilpotency the 3-tame assumption is required. However, for N -solubility things are more general. If an N -group V is a sum W + U of an N -soluble N -subgroup W of N -solubility length l1 and an N -soluble submodule U of V (length l2 ), then by considering U and V /U , it can be seen that V is N -soluble (length ≤ l1 +l2 ). Thus, in tame situations, a ﬁnite sum of N -soluble N -subgroups (ie. submodules) is again N -soluble. This gives us the following:Proposition 34.2. If V is a tame N -group, then any sum of locally N -soluble submodules of V is locally N -soluble, any sum of N -soluble submodules of V is locally N -soluble, and any locally N -soluble submodule of V is a sum of N -soluble submodules of V . If V is a tame N -group, then the sum of all locally N -soluble submodules of V is locally N -soluble. This submodule of V is denoted by S(V ). It is the unique maximal locally N -soluble submodule of V . From 34.1, any N -subgroup of S(V ) is locally N -soluble. Also, from 34.1, if U is a submodule of V , then (S(V ) + U )/U ≤ S(V /U ). In the case of V being a 3-tame N -group and L(V ) being N -nilpotent, it cannot be assumed that L(V /L(V )) = {0}. However, for N -solubility things are diﬀerent. If V is tame and S(V ) is N -soluble, then necessarily S(V /S(V )) = {0}. Thus, S(V ) has more of the properties of a radical, than does L(V ) (V 3-tame). For a tame N -group, S(V ) is by 34.1 and 34.2, particularly well behaved. We now seek to ﬁnd a similar N -subgroup (it will turn out to be an ideal) of a tame nearring. For a tame nearring N , S(N ) is taken as the

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right N -subgroup of N generated by all locally N -soluble N -subgroups. It will be shown that S(N ) is also particularly well behaved. Indeed, it turns out to be a locally N -soluble ideal that coincides with the sum of all N -soluble right ideals. Proposition 34.3. If N is a tame nearring and M an N -soluble right N -subgroup of N , then R(M ) is N -soluble. Proof: Let V be a faithful tame N -group. It is easily proved (see 33.3) that, for v in V, vR(M ) = vM . Now R(M ) can be embedded into the direct product vR(M ), taken over all v in V . Since each vR(M ) = vM and is of N -soluble length not greater than that of M , it is clear that vR(M ) is N -soluble. Thus R(M ) is N -soluble. Because the sum U1 + U2 of N -soluble submodules Ui , i = 1, 2, of U1 + U2 is N -soluble, it is rather obvious that:Proposition 34.4. A ﬁnite sum of N -soluble right ideals of a tame nearring is N -soluble. Now S(N ) is generated by all Mi , i ∈ I, which are locally N -soluble right N -subgroups of N . Thus, for γ in some Mj , j in I, γN is N -soluble and, by 34.3, R(γ) is N -soluble. This implies R(γ) ≤ S(N ) and, the sum R(γ) over all such γ, is contained in S(N ). Clearly each Mi , i ∈ I, is contained in this sum and S(N ) is a sum of N -soluble right ideals of N . Any N -soluble right ideal of N is contained in S(N ). Thus we have:Proposition 34.5. If N is a tame nearring, then S(N ) is the sum of all N -soluble right ideals of N . Corollary 34.6. S(N ) is locally N -soluble. Proof: A ﬁnitely generated right N -subgroup of S(N ) is, by 34.5, contained in a ﬁnite sum of N -soluble right ideals. Such sums are N -soluble. Thus, the ﬁnitely generated right N -subgroup is N -soluble. Proposition 34.7. If N is a tame nearring, then S(N ) is an ideal of N. right ideals of N . Proof: By 34.5, S(N ) = Ri over all N -soluble Thus αS(N ), α in N , coincides with the sum αRi of all submodules αRi of αS(N ). Each αRi , is N -soluble and, by 34.3, so is R(αRi ). Thus αS(N ) ≤ R(αRi ) ≤ S(N ), by 34.5, and the proposition is proved. If N is tame on V , then what can be said about the relationship between S(V ) and S(N ). As in the case of 3-tame V , where L(N ) ≤ (L(V ) : V ), we have:-

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Proposition 34.8. If N is tame on V , then S(N ) ≤ (S(V ) : V ). Proof: If R is any N -soluble right ideal of N , then V R⊆ vR (the sum over all v in V ). Each vR, v in V , is N -soluble and vR ≤ S(V ). Thus R ≤ (S(V ) : V ) and, by 34.5, S(N ) ≤ (S(V ) : V ). In more restricted situations, proposition 34.8 fails to go far enough. Often we have S(V ) is N -soluble. This happens when N has DCCR or (see 8.3 and 9.3 of [12]), when N has DCCI or ACCI and is compatible on V . In all such cases, the inclusion of 34.8, is equality. Proposition 34.9. If N is tame on V and S(V ) is N -soluble, then S(N ) = (S(V ) : V ). Proof: This will follow from 34.8, if it is shown that (S(V ) : V ) is N soluble. For each v in V , v(S(V ) : V ) ≤ S(V ) and is N -soluble of length ≤ that of S(V ) (l say). Thus (S(V ) : V ) embedds into the direct product v(S(V ) : V ) (over all v in V ) of N -soluble N -groups of length ≤ l. Consequently (S(V ) : V ) is N -soluble (length ≤ l) and the proposition is proved. In the above case V /S(V ) is a faithful tame N/S(N )-group. It makes sense, in the case of N having DCCR, to talk about N/S(N ) being Z-constrained. This will happen in section 40. The factor V /S(V ) of V turns out to be quite important. As will be seen in the next two sections, where the semi-Fitting submodule is looked at, this is just (in cases of importance to us) the semi-Fitting factor (deﬁned in the section now following).

35.

The Semi-Fitting Submodule

At the end of section 33, there followed a brief discussion on the Fitting submodule of a 3-tame N -group V . It was deﬁned in terms of centralisers of minimal factors of V . In the case where N has DCCR, the well known fact that L(V ) = F (V ), was mentioned. This section addresses a somewhat analogous concept. Can we, in terms of centralisers, deﬁne something like a Fitting submodule, doing the same for N -solubility? What is wanted is some canonical intersection of centralisers that, in the case where DCCR holds, in fact coincides with S(V ). A pleasing development of this section and the next, is to show that such a project can be carried out. Not only is such a submodule presented, but it is found under more general conditions. Suppose V is an N -group and U a minimal submodule of V which is not a ring module. The set S of submodules of V intersecting U trivially is, by 3.22 of [2], additively closed. By Zorn’s lemma S has a maximal element H and, because H + H1 is in S, whenever H1 is in S,

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H is the unique maximal submodule of V intersecting U trivially. Thus, in the tame case, H is the union of all N -subgroups that centralise U and is an N -subgroup. Therefore, in this case, the notation CV (U ) is U2 a minimal available for this submodule. Now, for tame V , with U1 /U U1 /U U2 ) factor of V , which is not a ring module, the submodule CV /U U2 (U U1 /U U2 ) be that submodule H ≥ U2 of V such that exists. Let CV (U H/U U2 = CV /U U1 /U U2 ). It is reasonably clear that, CV (U U1 /U U2 ) is the U2 (U unique maximal submodule K of V such that U1 ∩ K ≤ U2 . We are ready to deﬁne the semi-Fitting submodule of V . For tame U1 /U U2 ), where U1 /U U2 are V , this is taken as the intersection of all CV (U minimal factors of V , which are not ring modules. It is denoted by sF (V ). Theorem 35.1. If V is a tame N -group and H a submodule of V , then (sF (V ) + H)/H ≤ sF (V /H). Proof: Let X1 /X2 be a minimal factor of V /H which is not a ring module. Now Xi = Ui /H, i = 1, 2, where the Ui ≥ H are submodules U2 and U1 /U U2 is a minimal of V . Clearly X1 /X2 is N -isomorphic to U1 /U factor of V , which is not a ring module. If U1 /U U2 )/H] ∩ [U U1 /H] ≤ U2 /H [C CV (U then, it follows readily that CV (U U1 /U U2 )∩U U1 ≤ U2 , contrary to the nature U1 /U U2 ). Thus it can be concluded that of CV (U U1 /U U2 )/H ≤ CV /H ((U U1 /H)/(U U2 /H)) = CV /H (X1 /X2 ). CV (U Hence (sF (V ) + H)/H ≤ CV /H (X1 /X2 ), for all X1 /X2 minimal factors of V /H, which are not ring modules. The theorem follows. Proposition 35.2. If V is a tame N -group, then sF (sF (V )) = sF (V ). U2 is a minimal Proof: Clearly sF (sF (V )) ≤ sF (V ). Also, if U1 /U factor of sF (V ), which is not a ring module then, with straightforward notation, it can be seen that U1 /U U2 ) = sF (V ) ∩ CV (U U1 /U U2 ). CsF (V ) (U Thus CsF (V ) (U U1 /U U2 ) ≥ sF (V ) and sF (sF (V )) ≥ sF (V ). The proof is complete. The ﬁnal result of this section shows that sF (V ) contains S(V ). Theorem 35.3. If V is a tame N -group, then S(V ) ≤ sF (V ).

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Proof: By 34.2, S(V ) is a sum of N -soluble N -subgroups of V . Thus the result will follow if it is shown that any N -soluble N -subgroup U of U2 is a minimal factor of V , which is V is contained in sF (V ). If U1 /U not a ring module, then either (U U1 /U U2 ) ∩ ((U + U2 )/U U2 ) = {0} or U1 /U U2 ≤ (U + U2 )/U U2 . U2 and the second case cannot occur Since U is N -soluble so is (U + U2 )/U U2 is N -soluble). Thus (it would imply U1 /U U1 /U U2 ) ≥ (U + U2 )/U U2 CV /U U2 (U and CV (U U1 /U U2 ) ≥ U + U2 ≥ U . This implies sF (V ) ≥ U and the theorem holds. If V is a tame N -group, then the factor V /sF (V ) of V (called the semi-Fitting factor of V ) is of considerable importance. As will be seen in the next section, in situations of interest to us, it in fact coincides with the factor V /S(V ) of V .

36.

When sF (V ) and S(V ) Coincide

In 35.3, it was seen that for V a tame N -group, S(V ) ≤ sF (V ). In general it cannot be expected that S(V ) = sF (V ). However, with reasonable chain conditions this conclusion holds. Theorem 36.1. If V is a tame N -group ((N with DCCR), then sF (V ) is N -soluble and sF (V ) = S(V ). Proof: Since S(V ) ≤ sF (V ) (see 35.3), the theorem will follow if it is shown that sF (V ) is N -soluble. By 8.5, sF (V ) has a tame series {0} = V0 < V1 < · · · < Vr = sF (V ), where the integer r ≥ 0 is minimal. The proof will be by induction on r. If r = 0, sF (V ) = {0} and clearly sF (V ) is N -soluble. Assume r ≥ 1. Now, V1 ) ≥ (sF (sF (V )) + V1 )/V V1 ≥ sF (V )/V V1 , sF (V )/V V1 ≥ sF (sF (V )/V V1 ) and, because the by 35.1 and 35.2. Thus sF (V )/V V1 = sF (sF (V )/V N -group sF (V )/V V1 has a tame series of length ≤ r − 1, the theorem will follow if it is shown that V1 is a ring module. Now V1 is a direct sum of minimal submodules of sF (V ). It will therefore follow that sF (V ) is N -soluble, if it is shown that a minimal submodule U of sF (V ) is a ring module. If U is not a ring module, then CV (U ) ∩ U = {0} and

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CV (U ) ≥ sF (V ). This contradiction to the nature of sF (V ), completes the proof. The conclusion of 36.1 holds when V is a compatible N -group (N with DCCI). In order to establish this result, we need a result on annihilators. This is just 8.2 of [12]. Lemma 36.2. Let V be a non-zero compatible N -group, n ≥ 1 an integer and {0} = U0 < U1 < · · · < Un ≤ V a sequence of submodules of V . If each Ui+1 /U Ui , i = 0, . . . , n − 1, is a maximal ring submodule of V /U Ui , then (0 : U0 ) > (0 : U1 ) > · · · > (0 : Un ). It is not possible, to prove the theorem mentioned above, without some further knowledge. According to section two of [12] (see also [17]), a quasi-minimal submodule U of an N -group V , is one which is minimal (amongst submodules) such that (U : V ) > (0 : V ). If N has DCCI and V = {0}, such submodules exist. In the compatible case, such a submodule U is either central, or has U ∩Z(V ) as its maximal submodule (see 2.3 of [12]). The stage is now set to state and prove the theorem indicated above. Theorem 36.3. If V is a compatible N -group ((N with DCCI), then sF (V ) is N -soluble and S(V ) = sF (V ). Proof: Since S(V ) ≤ sF (V ) (see 35.3), the theorem will follow if it is shown that sF (V ) is N -soluble. In order to do this, we ﬁrst show that if U < sF (V ) is a submodule of V , then sF (V )/U has a non-zero ring submodule. Take H as a quasi-minimal submodule of sF (V )/U . As indicated above, either H contains a non-zero central submodule of sF (V )/U , or H is minimal. In the ﬁrst case sF (V )/U has a non-zero ring submodule. It may be assumed that H is minimal. Now H is a minimal factor of V of the form H1 /U . If it is not a ring module, then CV (H1 /U ) ∩ H1 ≤ U and CV (H1 /U ) ≥ sF (V ). This contradiction to the nature of sF (V ), establishes that sF (V )/U always has non-zero ring submodules. If sF (V ) = {0} take U1 > {0} as a maximal ring submodule of sF (V ) (possible by Zorn’s lemma). When sF (V )/U U1 = {0}, take U2 > U1 as U1 is a maximal ring submodule a submodule of sF (V ) such that U2 /U of sF (V )/U U1 (again possible by Zorn’s lemma). This process can be continued as long as Un (n ≥ 1 an integer) is properly contained in sF (V ). However, the process must terminate. This is because, by lemma

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36.2, the ideals (0 : U0 ) (here U0 = {0}), (0 : U1 ), (0 : U2 ), etc, form a strictly descending chain. It follows that, for some integer r ≥ 0, Ur = sF (V ) and, as is now easily seen, sF (V ) is N -soluble. The theorem is proved.

37.

Certain Subdirect Sums

It is well known that, an N -group V with a ﬁnite collection of subUk = {0}, can be embedded modules Ui , i = 1, . . . , k, such that U1 ∩· · ·∩U Hi copies of V /U Ui ), in such (by δ say) into the direct sum H1 ⊕ · · · ⊕ Hk (H Hk onto Hi , have the property a way that the projections πi of H1 ⊕· · ·⊕H that V δπi = Hi . This is usually expressed by saying, V is a subdirect sum of the Hi , i = 1, . . . , k. Often there are numerous subdirect sums of Hi , i = 1, . . . , k. H1 ⊕ · · · ⊕ Hk is one of them. However, in this and the next section, we shall be looking at situations where the subdirect sum is, in some sense, unique. Indeed, in the next section it will be seen that with Z-constraint (and compatibility) uniqueness holds. This section is, essentially a preliminary to the next. Here we look at subdirect sums of two N -groups. The results contained here can be found in [13]. Because all of what follows in this and the next section is so important, full coverage is given. Lemma 37.1. Suppose the N -group V is a direct sum H1 ⊕ H2 of submodules Hi , i = 1, 2, and Tj , j = 1, 2, are 2-tame N -subgroups of V such that Tj + Hi = V and Tj ∩ Hi = {0}, for j = 1, 2, and i = 1, 2. If T1 (or T2 ) is either centerless or perfect, then T1 = T2 . Proof: Let πi , i = 1, 2, be the projections of V onto Hi . Since T1 ∩ H2 = {0} the restriction π11 of π1 to T1 is an N -isomorphism. Because T1 + H2 = V , π1 maps T1 onto H1 and T1 π11 = H1 . Similarly, the restriction π12 of π1 to T2 is an N -isomorphism of T2 onto H1 . Two further N -isomorphisms arise from π2 . These are the restrictions of π2 to T1 and T2 (respectively denoted here by π21 and π22 ). They are respectively N -isomorphisms of H2 onto T1 and T2 . −1 = T2 and T1 is N -isomorphic to T2 . Now T1 π11 = H1 and T1 π11 π12 It follows that, if T1 (or T2 ) is centerless or perfect then so is T2 (or T1 ). Thus it can be assumed that T1 is centerless or perfect. −1 −1 = T2 (where π11 π12 is an N Now, we have T1 π11 = H1 , T1 π11 π12 −1 −1 isomorphism), T1 π11 π12 π22 = H2 (π11 π112 π22 an N -isomorphism) and −1 −1 −1 −1 π22 π21 = T1 and π11 π12 π22 π21 (= δ say) is an N -automorT1 π11 π12 phism on T1 . Since T1 is a 2-tame N -group, T1 is a central sum of T1 δ and T1 (1 − δ) (see 6.6). However T1 δ = T1 and, the N -endomorphism 1 − δ of T1 maps T1 into Z(T T1 ). If T1 is centerless this can only mean

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1−δ = 0. When T1 is perfect, we see that T1 /ker(1−δ) is a ring module, −1 −1 π22 π21 = 1. ker(1−δ) = T1 and 1−δ = 0. It has been shown that π11 π12 Returning to the goal of showing T1 = T2 , take x2 in T2 . This element −1 of T2 can be expressed in the form x2 π12 + x2 π22 . Because π11 π12 is an N -isomorphism of T1 onto T2 , there exists x1 in T1 such that −1 −1 −1 = x2 . Thus x2 = x1 π11 π12 π12 + x1 π11 π12 π22 and, from the x1 π11 π12 −1 −1 fact that π11 π12 π22 π21 = 1, we conclude that x2 = x1 π11 + x1 π21 = x1 . It has been shown that x2 is in T1 and T2 ≤ T1 . Similarly T1 ≤ T2 and the lemma holds. The next and ﬁnal result of this section, is an extension of 37.1. Lemma 37.2. Suppose the N -group V is a direct sum H1 ⊕ H2 of submodules Hi , i = 1, 2, and Tj , j = 1, 2, are 2-tame N -subgroups of V , such that T1 ∩ Hi = T2 ∩ Hi , for i = 1, 2, and Tj + Hi = V for j = 1, 2, i = 1, 2. If T1 or T2 is perfect, then T1 = T2 . Proof: Let π1 be the projection of V onto H1 . Now kerπ1 = H2 and, because T1 + H2 = V , T1 π1 = V π1 = H1 . Thus H1 , being an N homomorphic image of T1 , is 2-tame. Similarly H2 is 2-tame. It follows readily, that any N -subgroup of H1 (or H2 ) is a submodule of V . At this stage deﬁne the submodule X of V as T1 ∩ H1 ⊕ T1 ∩ H2 . From the statement of the lemma, this is just T2 ∩ H1 ⊕ T2 ∩ H2 . Now deﬁne V as V /X, H i as (H Hi + X)/X and T j = Tj /X for j = 1, 2. It will be shown that H i and T j satisfy the conditions of lemma 37.1. First of all V = H 1 + H 2 . Also, (H1 + X) ∩ (H H2 + X) = (H1 + T1 ∩ H2 ) ∩ (H H 2 + T1 ∩ H 1 ) H 2 + T 1 ∩ H 1 ) + T1 ∩ H 2 = H1 ∩ (H = T1 ∩ H1 + T1 ∩ H2 = X. Thus V is a direct sum of H i , i = 1, 2. Now T j ∩ H i = {0}, for j = 1, 2 and i = 1, 2. This is because Tj ∩ (H1 + Tj ∩ H2 ) = X, from which it follows that T j ∩ H 1 = {0} and similarly, T j ∩ H 2 = {0}. Using the natural N -homomorphism of V onto V and material of the statement of the lemma, we conclude also that V = T j + H i for j = 1, 2, and i = 1, 2. Finally, because the T j are respective N -homomorphic images of the Tj , j = 1, 2, one of the T j is perfect. Thus all the conditions of 37.1 are fulﬁlled for V , H i and T j . From that lemma it must follow that T 1 = T 2 . It has been shown that T1 /X and T2 /X coincide and clearly T1 and T2 must also. The lemma is proved.

38.

Uniqueness of Subdirect Sums

According to the two results of the previous section, a 2-tame perfect subdirect sum of two N -groups, has a restricted nature. It is in fact

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speciﬁed by its intersection with the direct sum components (see 37.2). So the question arises as to what condition, on this 2-tame N -group, will also ensure these intersections are determined. It is possible that all N groups of suitable type have a unique representation. What is to be taken as a suitable type of N -group? It must at least be perfect and 2-tame. Thus we can look amongst such compatible N -groups for a condition. A diﬃculty of compatible N -groups that sometimes intrudes is as follows. A compatible N -group V might possess submodules U < W with (U : V ) = (W : V ). This possibility is excluded if Z-constraint is assumed. Moreover, with Z-constraint V is perfect. With this ‘pathology’ out of the way (assuming Z-constraint) things go through in a remarkably straightforward manner. These results are not unknown. They have appeared in a manuscript of the author (see [7]). In that manuscript it was suggested that the uniqueness of subdirect sums was likely to have bearing on the Zconstrained conjecture. It is not just pleasing that this is so, but also pleasing that they enter into arguments in a straightforward manner. This will be seen in the next section. As far as this section goes it is simply a matter of establishing uniqueness of subdirect sums. Considerations start with a fairly easy consequence of Z-constraint. Proposition 38.1. Suppose the Z-constrained compatible N -group V is cyclic. If v in V is a generator of V and H a submodule of V , then v(H : V ) = H. Proof: Clearly v(H : V ) ≤ H. Now, V (H : V ) = vN (H : V ) = v(H : V ) and (H : V ) ≤ (v(H : V ) : V ). Thus, by 6.8 H/v(H : V ) ≤ Z(V /v(H : V )), so that, by Z-constraint, H = v(H : V ). The proposition is proved. It is now proved that the intersection property, brieﬂy outlined above, holds true. Lemma 38.2. Suppose the N -group U is a direct sum U1 ⊕ U2 of submodules Ui , i = 1, 2. If Tj , j = 1, 2, are cyclic Z-constrained compatible N -subgroups of U such that Tj + Ui = U , for j = 1, 2, i = 1, 2, then Ui ∩ T1 = Ui ∩ T2 , for i = 1, 2. Proof: Now the projections πi , i = 1, 2, of U onto Ui are clearly such that T1 πi = T2 πi = Ui . The kernel of π1 is U2 , so that U1 is N -isomorphic to Tj /(T Tj ∩ U2 ), for j = 1 and 2 and a similar argument applies to U2 .

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This means the annihilators of U1 and Tj /(T Tj ∩ U2 ) are the same. It Tj ∩ U2 : Tj ), for j = 1, 2. Take aj as has been shown that (0 : U1 ) = (T generators of Tj . By proposition 38.1, aj (0 : U1 ) = Tj ∩ U2 . However, aj = aj π1 + aj π2 and clearly aj π2 (0 : U1 ) = Tj ∩ U2 , where aj π2 are generators of U2 . Now, a1 π2 (0 : U1 ) = a1 π2 N (0 : U1 ) = U2 .(0 : U1 ) and similarly a2 π2 (0 : U1 ) = U2 .(0 : U1 ). It has been shown that T1 ∩ U2 = T2 ∩ U2 . An entirely similar argument shows T1 ∩ U1 = T2 ∩ U1 . The proof is complete. Corollary 38.3. If U , Ui , i = 1, 2, and Tj , j = 1, 2, are as in the statement of 38.2, then T1 = T2 . Proof: By 38.2, Ui ∩ T1 = Ui ∩ T2 for i = 1, 2, and therefore by 37.2, T1 = T2 . Theorem 38.4. Suppose n ≥ 1 is an integer and the N -group V is a direct sum V1 ⊕ · · · ⊕ Vn , of submodules Vi , i = 1, . . . , n. If there exists a compatible Z-constrained N -subgroup W of V , which is cyclic ( i , i = 1, . . . , n, the obvious projections), then and such that W πi = Vi (π it is unique. Proof: Suppose H is a cyclic Z-constrained N -subgroup of V distinct from W and having the property that Hπi = Vi , for i = 1, . . . , n. We proceed by induction on n. The result is clearly true if n = 1 and we may assume n ≥ 2. Let U1 = V1 ⊕ · · · ⊕ Vn−1 and U2 = Vn . With δ the projection of V onto U1 , we have that the restriction of the projections πi , i = 1, . . . , n − 1, of U1 onto Vi , i = 1, . . . , n − 1, are such that Hδπi = Vi . The same applies to W δ and, by the induction assumption and the cyclic compatible Z-constrained nature of these N -groups, we have Hδ = W δ. Thus H and W are both contained in Hδ ⊕ Vn , and have the property that the projections onto Hδ and Vn are respectively Hδ and Vn . From corollary 38.3, we conclude that H = W and the theorem is completely proved. A ﬁnal point about 38.4 is worth noting. The cyclic nature of W is often implied by Z-constraint. As has been seen, this is the case if N has DCCR.

39.

Main Results

Much of this paper is now behind us. What has been covered is extensive. It consists of some powerful results and many of less signiﬁcance. A large number of the smaller results have been necessary to prove the

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substantial ones. However, most of the material, whether basic or deep, has been developed with one goal in view. This has been that of providing the proof of the main results of this paper. These results consist of a theorem and a number of corollaries. The theorem is what this paper is all about. Essentially, the corollaries are ways of reformulating the theorem. The fact that so much work has been necessary to reach this position is evidence of the signiﬁcance of the result. For a number of years it has been apparent that tame theory throws up many very real possibilities. With results like this at hand, tame theory almost seems to have now taken on a new dimension. Let us hope that, it will continue to open up much more in the way of new horizons. What does 39.2 (the theorem) rest on? As indicated above most of the material of this paper goes into it. However, two results stick out. These are 32.3 and 38.4. 32.3 means factors of certain compatible N groups are N -isomorphic, while 38.4 allows us to piece together these factors. How this is done will be seen in the proof. This material is not all that is required. Before stating and proving the theorem, we must cover something else. It will be helpful to look at a certain submodule λ(V ), of a 2-tame N -group (N with DCCR). This is just the minimal cover of all elements of nr(V ) (or alternatively V (tnr(N ))). Lemma 39.1. If V is a 2-tame N -group ((N with DCCR) and U a submodule of V such that V /U is perfect, then U + λ(V ) = V . Proof: Suppose U + λ(V ) < V . Since V /(U + λ(V )) has maximal N -subgroups 8.7, we can ﬁnd a maximal N -subgroup K of V such that K ≥ U +λ(V ). Now U ≤ K and K/U is a maximal N -subgroup of V /U . Since V /U is perfect the minimal factor (V /U )/(K/U ) of V /U is not a ring module and consequently V /K is a minimal factor of V which is not a ring module. However, λ(V ) covers all minimal factors of V which are not ring modules. This means K + λ(V ) = V , contrary to the fact that λ(V ) ≤ K. This contradiction completes the proof. Theorem 39.2. If N is a compatible nearring with DCCR and N/L(N ) being Z-constrained (see section 33), then all Fitting factors of faithful compatible N -groups are N -isomorphic. Proof: Let Vj , j = 1, 2, be two faithful compatible N -groups. We are V1 ) and V2 /F (V V2 ) are N -isomorphic. In required to show that V1 /F (V order to do this we ﬁrst take two preliminary steps. Step 1. Here it is proved that Vj /F (V Vj ) is, for j = 1, 2, N -isomorphic Vj )). Because N/L(N ) is Z-constrained, it follows from to λ(V Vj )/F (λ(V Vj ) are Z-constrained. By 25.1, they are 26.1 and 33.9, that the Vj /F (V perfect and from lemma 39.1, F (V Vj ) + λ(V Vj ) = Vj , for j = 1 and 2. This

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means Vj /F (V Vj ) is N -isomorphic to λ(V Vj )/(F (V Vj ) ∩ λ(V Vj )), for j = 1 Vj ) is an N -nilpotent N -subgroup of λ(V Vj ) and 2. However, F (V Vj ) ∩ λ(V Vj ) ≤ F (λ(V Vj )) (see [12]). On the other hand and therefore F (V Vj ) ∩ λ(V Vj ) (clearly F (λ(V Vj )) is an N -nilpotent N -subgroup of Vj contained in F (V Vj )) = F (V Vj ) ∩ λ(V Vj ) it is contained in λ(V Vj )). It has been shown F (λ(V and step 1. is proved. V1 ) and V2 /F (V V2 ) Step 2. Here it is proved that, in order to show V1 /F (V are N -isomorphic, it suﬃces to assume Vj , j = 1, 2, are perfect. The Vj ))-groups. λ(V Vj ), j = 1, 2, are perfect faithful compatible N/(0 : λ(V V2 )), they are perfect faithful N1 Since, by 17.3 (0 : λ(V V1 )) = (0 : λ(V V1 )). Because N/L(N ) is Z-constrained, groups, where N1 = N/(0 : λ(V Vj ) are Z-constrained. By it follows from 26.1 and 33.9, that Vj /F (V Vj )/F (λ(V Vj )) are necessarily Zstep 1. (see the deﬁnition of N1 ) λ(V N1 ) is constrained with respect to N1 . Thus (see 26.1 and 33.9) N1 /L(N Z-constrained. If our result were to hold for perfect V1 and V2 , it could V1 )) is N1 -isomorphic (hence N therefore be concluded that λ(V V1 )/F (λ(V V2 )). This would, by step 1., give the result isomorphic) to λ(V V2 )/F (λ(V for general Vj , j = 1, 2. It can, by step 2., be assumed that Vj , j = 1, 2, are perfect. Let ∆1 , . . . , ∆n , be the set of all N -isomorphism types of minimal N -groups Wi , i = 1, . . . , n, as minimal factors of (n ≥ 1 is ﬁnite by 8.2). Take Ui /W Ki , i = 1, . . . , n, as V1 of type ∆i (they exist by 8.10). Also, take Hi /K minimal factors of V2 of type ∆i . By 31.2, Ui /W Wi ) and F (V V2 ) = ∩ni=1 CV2 (H Hi /K Ki ). F (V V1 ) = ∩ni=1 CV1 (U CV1 (U Ui /W Wi ) and Pi = V2 /C CV2 (H Hi /K Ki ). By 32.3, Li is Let Li = V1 /C Vj ), j = N -isomorphic, by δi say, to Pi , for i = 1, . . . , n. Now Vj /F (V Vj ) (see [12]) and, by 26.1 and 33.9, are Z1, 2, coincide with Vj /L(V constrained. It follows that the Z-constrained (necessarily cyclic) N V1 ) and V2 /F (V V2 ) are respectively embedded (as subdirect groups V1 /F (V sums) into L1 ⊕ · · · ⊕ Ln and P1 ⊕ · · · ⊕ Pn . Thus L1 ⊕ · · · ⊕ Ln contains V1 ), where the projection of L1 ⊕ · · · ⊕ Ln onto each a copy Q1 of V1 /F (V Li , maps Q1 onto Li . Similarly, P1 ⊕ · · · ⊕ Pn contains a copy Q2 of V2 ), where the projection of P1 ⊕ · · · ⊕ Pn onto each Pi , maps Q2 V2 /F (V onto Pi . Each Li is N -isomorphic by δi to Pi and it follows, that the map δ of L1 ⊕ · · · ⊕ Ln into P1 ⊕ · · · ⊕ Pn taking u1 + u2 + · · · + un (ui in Li ) to u1 δ1 + · · · + un δn is an N -isomorphism of L1 ⊕ · · · ⊕ Ln onto P1 ⊕ · · · ⊕ Pn . It can be shown that Q1 δ is such that, the projection of P1 ⊕ · · · ⊕ Pn onto each Pi , maps Q1 δ onto Pi . Indeed, for any a in some Pi , there exists v1 + v2 + · · · + vn in Q1 , where v1 , . . . , vn are respectively

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in L1 to Ln , with vi = aδi−1 . Thus Q1 δ contains (v1 + · · · + vn )δ = v1 δ1 + · · · + vi−1 δi−1 + a + vi+1 δi+1 + · · · + vn δn and Q1 δ is contained in P1 ⊕ · · · ⊕ Pn in the manner indicated. Now Q2 and Q1 δ are both cyclic and Z-constrained so that, by 38.4, Q1 δ = Q2 . δ is an N -isomorphism and it has been shown that Q1 is N -isomorphic to Vj ), it follows that V1 /F (V V1 ) Q2 . Since Qj , j = 1, 2, are copies of Vj /F (V V2 ). The theorem is completely proved. is N -isomorphic to V2 /F (V Corollary 39.3. If N is a compatible nearring with DCCR and N/L(N ) is Z-constrained, then all U/L(U ) (U ( a faithful compatible N -group) are N -isomorphic. Proof: By [12]. Corollary 39.4. If N (with DCCR) is compatible on V and V /F (V ) is Z-constrained, then all U/F (U ) (U ( a faithful compatible N -group) are N -isomorphic. Proof: By [12] and 26.1. Corollary 39.5. If N (with DCCR) is compatible on V and V /L(V ) is Z-constrained, then all U/L(U ) (U ( a faithful compatible N -group) are N -isomorphic. Proof: By [12]. Corollary 39.6. If N (with DCCR) is compatible on the Z-constrained N -group V , then all U/F (U ) (U ( a faithful compatible N -group) are N isomorphic. Proof: By 39.4. Corollary 39.7. If N (with DCCR) is compatible on the Z-constrained N -group V , then all U/L(U ) (U ( a faithful compatible N -group) are N isomorphic. Proof: This follows from 39.5.

40.

Other Important Results

The last section contains the main results of this paper. However, a collection of similar results can be obtained with N -solubility replacing N -nilpotency and Fitting submodules replaced by semi-Fitting submodules. The statement and proof of this material parallels, to some extent, that of the last section. Because this material is somewhat diﬀerent at key points, this section (not all that diﬀerent from the last), is included to cover it.

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Theorem 40.1. If N is a compatible nearring with DCCR and N/S(N ) being Z-constrained (see section 34), then all semi-Fitting factors of faithful compatible N -groups are N -isomorphic. Proof: Let Vj , j = 1, 2, be two faithful compatible N -groups. We are V1 ) and V2 /sF (V V2 ) are N -isomorphic. In required to show that V1 /sF (V order to do this we ﬁrst take two preliminary steps. Vj ) is, for j = 1, 2, N -isomorphic Step 1. Here it is proved that Vj /sF (V Vj )). Because N/S(N ) is Z-constrained, it follows from to λ(V Vj )/sF (λ(V Vj ) are Z-constrained. By 25.1, they 26.1, 34.9 and 36.1, that the Vj /sF (V Vj ) = Vj , for j = 1 and are perfect and from lemma 39.1, sF (V Vj ) + λ(V Vj ) is N -isomorphic to λ(V Vj )/(sF (V Vj ) ∩ λ(V Vj )), 2. This means Vj /sF (V Vj ) is an N -soluble N -subgroup for j = 1 and 2. However, sF (V Vj ) ∩ λ(V Vj ) ∩ λ(V Vj ) ≤ sF (λ(V Vj )) (see 36.1). On the of λ(V Vj ) and therefore sF (V other hand sF (λ(V Vj )) is an N -soluble N -subgroup of Vj contained in Vj )). It has been shown sF (λ(V Vj )) = sF (V Vj ) (clearly it is contained in λ(V Vj ) and step 1. is proved. sF (V Vj ) ∩ λ(V Step 2. Here it is proved that, in order to show V1 /sF (V V1 ) and V2 /sF (V V2 ) are N -isomorphic, it suﬃces to assume Vj , j = 1, 2, are perfect. The Vj ))-groups. λ(V Vj ), j = 1, 2, are perfect faithful compatible N/(0 : λ(V V2 )), they are perfect faithful N1 Since, by 17.3 (0 : λ(V V1 )) = (0 : λ(V groups, where N1 = N/(0 : λ(V V1 )). Because N/S(N ) is Z-constrained, Vj ) are Z-constrained. it follows from 26.1, 34.9 and 36.1, that Vj /sF (V Vj )/sF (λ(V Vj )) are necessarily ZBy step 1. (see the deﬁnition of N1 ) λ(V N1 ) is constrained with respect to N1 . Thus (see 26.1 and 34.9), N1 /S(N Z-constrained. If our result were to hold for perfect V1 and V2 , it could V1 )) is N1 -isomorphic (hence therefore be concluded that λ(V V1 )/sF (λ(V V2 )). This would, by step 1., give the N -isomorphic) to λ(V V2 )/sF (λ(V result for general Vj , j = 1, 2. It can, by step 2., be assumed that Vj , j = 1, 2, are perfect. Let ∆1 , . . . , ∆n , be the set of all N -isomorphism types of minimal N -groups Wi , i = which are not ring modules (n ≥ 1 is ﬁnite by 8.2). Take Ui /W 1, . . . , n, as minimal factors of V1 of type ∆i (they exist by 8.10). Also, Ki , i = 1, . . . , n, as minimal factors of V2 of type ∆i . By 31.2, take Hi /K Ui /W Wi ) and sF (V V2 ) = ∩ni=1 CV2 (H Hi /K Ki ). sF (V V1 ) = ∩ni=1 CV1 (U CV1 (U Ui /W Wi ) and Pi = V2 /C CV2 (H Hi /K Ki ). By 32.3, Li is Let Li = V1 /C Vj ), N -isomorphic, by δi say, to Pi , for i = 1, . . . , n. Now Vj /sF (V Vj ) (see 36.1) and, by 26.1 and 34.9, are j = 1, 2, coincide with Vj /S(V Z-constrained. It follows that the Z-constrained (necessarily cyclic) N V1 ) and V2 /sF (V V2 ) are respectively embedded (as subdigroups V1 /sF (V rect sums) into L1 ⊕ · · · ⊕ Ln and P1 ⊕ · · · ⊕ Pn . Thus L1 ⊕ · · · ⊕ Ln

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contains a copy Q1 of V1 /sF (V V1 ), where the projection of L1 ⊕ · · · ⊕ Ln onto each Li , maps Q1 onto Li . Similarly, P1 ⊕ · · · ⊕ Pn contains a copy V2 ), where the projection of P1 ⊕ · · · ⊕ Pn onto each Pi , Q2 of V2 /sF (V maps Q2 onto Pi . Each Li is N -isomorphic by δi to Pi and it follows, Pn taking u1 +u2 +· · ·+un that the map δ of L1 ⊕· · ·⊕Ln into P1 ⊕· · ·⊕P (ui in Li ) to u1 δ1 + · · · + un δn is an N -isomorphism of L1 ⊕ · · · ⊕ Ln onto P1 ⊕ · · · ⊕ Pn . It can be shown that Q1 δ is such that, the projection of P1 ⊕ · · · ⊕ Pn onto each Pi , maps Q1 δ onto Pi . Indeed, for any a in some Pi , there exists v1 + v2 + · · · + vn in Q1 , where v1 , . . . , vn are respectively in L1 to Ln , with vi = aδi−1 . Thus Q1 δ contains (v1 + · · · + vn )δ = v1 δ1 + · · · + vi−1 δi−1 + a + vi+1 δi+1 + · · · + vn δn and Q1 δ is contained in P1 ⊕ · · · ⊕ Pn in the manner indicated. Now Q2 and Q1 δ are both cyclic and Z-constrained so that, by 38.4, Q1 δ = Q2 . δ is an N -isomorphism and it has been shown that Q1 is N -isomorphic to Vj ), it follows that V1 /sF (V V1 ) Q2 . Since Qj , j = 1, 2, are copies of Vj /sF (V V2 ). The theorem is completely proved. is N -isomorphic to V2 /sF (V Corollary 40.2. If N is a compatible nearring with DCCR and N/S(N ) is Z-constrained, then all U/S(U ) (U ( a faithful compatible N -group) are N -isomorphic. Proof: By 36.1. Corollary 40.3. If N (with DCCR) is compatible on V and V /sF (V ) is Z-constrained, then all U/sF (U ) (U ( a faithful compatible N -group) are N -isomorphic. Proof: By 26.1 and 36.1. Corollary 40.4. If N (with DCCR) is compatible on V and V /S(V ) is Z-constrained, then all U/S(U ) (U ( a faithful compatible N -group) are N -isomorphic. Proof: By 36.1. Corollary 40.5. If N (with DCCR) is compatible on the Z-constrained N -group V , then all U/sF (U ) (U ( a faithful compatible N -group) are N isomorphic. Proof: By 40.3. Corollary 40.6. If N (with DCCR) is compatible on the Z-constrained N -group V , then all U/S(U ) (U ( a faithful compatible N -group) are N isomorphic. Proof: By 40.4.

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A Special Case

Theorems 39.2 and 40.1 are not two way implications. However, they may not be so far removed, from being such. In this section we look at a special case of these results, where the N -isomorphism of certain N -groups, is implied and implied by Z-constraint. The main result of this section (theorem 41.4) was proved about ten years ago (see [19]). It provided motivation for further investigations. Originally, I had thought that Z-constraint, on a faithful compatible N -group (N with DCCR), implied the N -isomorphism of all such N groups. The results of [19] at ﬁrst looked as though they provided weight to this possibility. Also, some of the contents of theorem 27.2 were known at that time. This appeared to suggest, the possibility just mentioned, had some foundation. Unsuccessfully trying to prove it has not been futile. Much of the material of [15] has been supplied by such attempts. In view of 39.2 these attempts may have been somewhat misdirected. They did however, furnish many new insights. Indeed, quite a number of results of this paper have come about in this way. We now look at some of the contents of [19] more closely. A semiprimary N -group is one without non-zero submodules which are ring modules. Studying such N -groups is rewarding. Such a study allows interesting theoretical developments. This is pleasing because they arise frequently. Indeed, if we take any N -group and keep factoring out non-zero submodules which are ring modules, then a semiprimary N -group is obtained. A meaningful result about semiprimary N -groups is that they can be expressed as a subdirect product of much more restricted N -groups. These are the primary N -groups. They are those semiprimary N -groups which are also uniform (ie. have no two non-zero submodules with zero intersection). Extensive theory of compatible primary N -groups has been developed (see [20]). Such N -groups can be studied by very real use of topological initiatives (see also [6]). Although the above deﬁnitions relate to N -groups, it is an easy matter to give a deﬁnition of a semiprimary nearring, which relates well to the N -group deﬁnition. A semiprimary nearring will be taken as one without non-zero ideals which are ring modules. Proposition 41.1. If N is compatible on the semiprimary N -group V , then N is semiprimary. Proof: If N has a non-zero ideal A, which is a ring module and v in V is such that vA = {0}, then vA is a ring submodule of V . The proposition holds.

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The question of some type of converse to 41.1, is slightly more subtle. Proposition 41.2. If the semiprimary nearring N is compatible on V , then V /Z(V ) is a faithful semiprimary N -group. Proof: It is easily seen that (Z(V ) : V ) is an ideal of N which is a ring module and, therefore (Z(V ) : V ) = {0} and N is faithful on the compatible N -group V /Z(V ). However, if there exists a submodule U > Z(V ) of V with U/Z(V ) a ring module, then (U : V ) = {0}. This is because otherwise, 6.8 implies U ≤ Z(V ). It is easily seen that the ideal (U/Z(V ) : V /Z(V )) of N/(Z(V ) : V ), is a ring submodule of N/(Z(V ) : V ) (= N/{0}) and, since this coincides with the non-zero ideal (U : V )/{0}, the proposition holds. Proposition 41.2 has the following corollary:Corollary 41.3. Suppose N is a semiprimary compatible nearring with DCCR. If N is Z-constrained, then all faithful compatible N -groups are semiprimary. Proof: If V is a faithful compatible N -group, then by 26.1, V is Zconstrained and Z(V ) = {0}. The conclusion follows from 41.2. We are now ready to state and prove the theorem mentioned above. Theorem 41.4. Let N be a compatible semiprimary nearring. If N has DCCR, then all faithful compatible N -groups are N -isomorphic if, and only if, N is Z-constrained. Proof: First it will be shown that if N is Z-constrained, then all faithful compatible N -groups are N -isomorphic. If Vi , i = 1, 2, are ‘two’ faithful compatible N -groups then, by 41.3, they are semiprimary. This means that the only N -soluble N -subgroup of Vi , i = 1, 2, is {0}. It also means that the only N -nilpotent N -subgroup of Vi is {0}. Consequently (see Vi ) = {0}. It now follows from, either 36.1 or [12]) sF (V Vi ) = {0} and F (V theorem 40.1 or 39.2, that the Vi , i = 1, 2, are N -isomorphic. Suppose all faithful compatible N -groups are N -isomorphic but N is not Z-constrained. By 26.1, a faithful compatible N -group V must have a minimal factor H/W with H/W ≤ Z(V /W ). Let n be a cardinal > |V | and take U as a direct sum of n copies of H/W . By 27.1, U ⊕ V is a compatible N -group. Clearly |U ⊕V | ≥ n, and U ⊕V and V cannot be N isomorphic. However, U ⊕V is faithful, since (0 : U ⊕V ) ≤ (0 : V ) = {0}. This contradiction yields the result. The theorem is proved.

42.

A Loose End

This paper has used quite a number of diverse concepts. Some have been used more than others. One concept that has played a substantial

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role is realisations. These right N -subgroups ﬁrst appeared in section 11. Considerable theory relating to them was developed in subsequent sections. Sections 28, 29 and 30 brought these developments to completion. In the key result 32.3 it is seen just how essential they are. Nearly all realisations considered have been restricted. They have not just been realisations of arbitrary cyclic N -groups but have arisen from perfect N -groups. It has been minimal covers of subsets of nr(V ) (V a 2-tame N -group and N with DCCR), that have given rise to these right N -subgroups. What is the reason that such objects have proved to be so signiﬁcant? An answer can be given by considering a concept which arose in [15]. When the Z-constrained conjecture ﬁrst received serious attention, it became apparent that faithful hulls might have some real input into answering it. Faithful hulls (see [15] where they are called hulls) are just right N -subgroups which are minimal for being faithful. In situations that are important to us, we have the following two facts about them. They are self monogenic and N -isomorphic (see theorem 8.7 of [15]). These facts and other considerations, suggest they are likely to be of use in attacking the Z-constrained conjecture. The Z-constrained conjecture is about uniqueness (up to N -isomorphism), so faithful hulls might just turn out to play a key part. This is certainly the case. Although faithful hulls have not been used explicitly in this paper, they have occured often. It is in a somewhat disguised form, in which they have arisen. Realisations (in the perfect case outlined above) are just faithful hulls. However, there has been good reason to approach our main theorems through realisations, rather than faithful hulls. In many respects this approach is more natural. It allows development to proceed at a steady pace toward theorems such as 32.3. However, the fact that most of our realisations are faithful hulls does, to some extent, underpin many of the arguments encountered. The relationship is important. This section is included to establish it. Theorem 42.1. If N with DCCR is 2-tame on the perfect N -group V (necessarily cyclic) and M is a realisation of V , then M is a faithful hull. Proof: Take S as tnr(N ). The minimal cover W of V (S) covers all minimal factors V /K of V . From this it follows readily that W = V and M is a realisation of the minimal cover V of V (S). It now follows, by 28.2, that if H is a right N -subgroup of M , such that there exists a submodule R of M with H + R = M and H ∩ R = {0}, then HR = {0}. It can be shown that we can take H ≤ M as a faithful hull. Indeed, M is a faithful N -group because, vM = V for some v in V and, if

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M α = {0}, α = 0 in N then V α = {0}, contrary to faithfulness. Thus M contains a faithful hull which, by [15], has a left identity e1 . Thus, with R = (0 : e1 ) ∩ M , we see M = H + R where H ∩ R = {0}. Clearly, if it is shown that R = {0}, then it will follow that H = M and M is a faithful hull. As outlined above, we have HR = {0}. This implies R ≤ (0 : H). However, the N -group H is faithful and R must necessarily be zero. The theorem is therefore proved.

43.

Discussion

In [5] it began to appear Z-constraint might be important. It was shown there, that if a faithful 2-tame N -group (N with DCCR) was Z-constrained, then so were all faithful 2-tame N -groups. This was not unlike what happened in a more restricted context. This was the situation where an N -group V was r-constrained (had no minimal factors which were ring modules). If a faithful 2-tame N -group (N with DCCR) was r-constrained, then so were all faithful 2-tame N -groups. Indeed, this was implied and implied by N being ring-free (no non-zero homomorphic image of N being a ring). Also, in this case, the DCCR (only necessary on N/J(N )) was enough to imply such faithfuls were ﬁnite (see [13]). However, in this restricted situation much more was true. In [13], it was shown that all faithful 2-tame N -groups were N -isomorphic. Now r-constraint is certainly much stronger than Z-constraint, but it still made sense in the weaker situation, to ask if the faithfuls were N isomorphic. At ﬁrst this possibility looked very promising. It appeared to me as a really deep question, where any insight would probably have spin oﬀ. A partial answer in this direction came about in [19]. The result was established for semiprimary N (or V ). The semiprimary result used compatibility. There was good reason for this. Compatibility behaves well when a type of redundancy is present. For submodules U < W of a faithful compatible N -group V , it can happen that (U : V ) = (W : V ). Compatibility can exclude this. With Z-constraint this type of redundancy cannot occur. Thus the semiprimary case is enough to reinforce the need for compatibility. Was it true that all faithful compatible N -groups (N with DCCR) were N isomorphic when one was Z-constrained? Again answers tended to be provided by the semiprimary case. It looked as though faithful hulls (see section 42) would be of help with discarding the semiprimary assumption. Thus a whole new area of tame theory was introduced. Tame fusion began to give indications of possibilities (see [15]). However, in working with faithful hulls many diﬃculties arise. These tended to make my eﬀorts at proving the Z-constrained conjecture look rather futile. It

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was time to step back. Maybe looking for some weaker result might make sense. I began to think of the possibility of the factors V /F (V ) all being N -isomorphic. One reason that this possibility is so promising is that V /F (V ) can be expressed in a certain way as a subdirect sum. The subdirect sum ﬁts in well with understanding provided by [7]. It looked like this weaker conjecture was pertinent. In this manner the Z-constrained conjecture was replaced. The real question became; if a faithful compatible N -group (N with DCCR) is Z-constrained, are all W/F (W ) (W faithful compatible) N -isomorphic? This paper has established that (see 39.6). It is very pleasing that with modiﬁcation something signiﬁcant holds. I always felt there was something deep to be had here. More can be said about Z-constraint. How natural is this assumption? Is it reasonably general? Examples indicate something in the nature of answers. The nearring P0 (V ) of zero-ﬁxing polynomial maps of an Ω-group V into V is, as far as such questions go, worth looking at. Approaching Ω-groups through such nearrings is very rewarding (see [18]). Here V is a compatible P0 (V )-group. What does it mean for V to be Z-constrained? Let us look at the case of V an associative ring. When V has an identity it is very close (as a P0 (V )-group) to being Z-constrained. Indeed, apart from the possibility of certain exceptional minimal factors of V of order two, Z-constraint holds. This example can be generalised. In [18] full Ω-groups are deﬁned. They are simply Ω-groups which, with respect to the product (see § 9 of [18]), have no redundancy. Again these Ω-groups are very close to being Z-constrained (as P0 (V )-groups). Again, apart from the possibility of really quite exceptional minimal factors of order two, Z-constraint will hold. What does our main theorem tell us about Ω-groups? If V and W are two Ω-groups with P0 (V ) isomorphic to P0 (W ) (having DCCR), then it appears V and W become compatible P0 (V )-groups. There is very little diﬀerence between the Fitting submodules F (V ) and F (W ) and their respective counterparts Fm (V ) and Fm (W ) (see [18]). Does fullFm (W ) are P0 (V )ness (very like Z-constraint) imply V /F Fm (V ) and W/F isomorphic? This might prove to be an interesting question. Certainly it deserves looking into. Just what some of the major results of this paper tell us, in an Ω-group context, is not immediately clear. The results appear to give tantalising leads. The above question, as to what our main theorem tells us about Ωgroups, can be considered in another way. In the paragraph above we looked at ways of adapting the theorem to an Ω-group setting. However, even without adaption it still has bearing on such P0 (V )-groups. The question arises as to how much bearing it has. When V /F (V ) is

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Z-constrained, we know a lot about the uniqueness of this factor. So the question is one of how small can F (V ) be expected to be. Unfortunately in most Ω-groups it appears that it rarely reduces to zero. Although in the semiprimary case F (V ) is zero this seems to be somewhat exceptional. However, there is good reason to believe that in reasonably complex Ω-groups, F (V ) is small. This last statement needs some qualiﬁcation. Certainly for an associative ring V , F (V ) tends to be large. In this case it is the radical. However, for Lie rings and more complex algebras, F (V ) is restricted. In these cases it is likely to be much smaller than S(V ) (S(V ) = F (V ) in the case of associative rings). Thus our main theorem appears to say a considerable amount about V . Simpler cases can also show this behaviour. Certainly this is so for I(V ), A(V ) and E(V ) (V a group). In many situations a large factor of V is uniquely speciﬁed. As mentioned in § 41, theorems 39.2 and 40.1 are not two way implications. Indeed, a closer inspection of these theorems make converses seem somewhat unlikely. The qualiﬁcation of their unlikelyness is intentional. Theorem 41.4 suggests they have some tendancy to be two way implications. It could be rewarding to look at faithful compatible N -groups (N with DCCR), where all Fitting factors are N -isomorphic. Attempts at ﬁnding some type of converse to 39.2 might yield insights. This could be even more the case for theorem 40.1. It was developed with a converse in mind. What can be said about the DCCR assumption? Can it be weakened? A chain condition, that works well in a number of situations, is DCCI. However, replacing DCCR by DCCI is a big step. In the present context it leaves us very much in the dark. What holds along the lines of 39.2 or 40.1, with DCCR replaced by DCCI, is not known. There are reasons that DCCI introduces diﬃculties. One is that, the number of minimal factors of V of distinct N -isomorphism type, may no longer be ﬁnite. Although with DCCI, F (V ) = L(V ), problems like that just mentioned, makes this appear as thorny territory. Compatibility is an important condition. Our main theorems show that this assumption allows the proof of deep results. It is a condition that encompasses a wide range of naturally occuring nearrings. Certainly, in an abstract context, compatible nearrings are of the form P0 (V ) (V an Ω-group). However, to always cast them in this framework is misleading. It is rather like studying Lie rings in terms of commutators of associative rings, or nearrings as subnearrings of transformation nearrings. The material of this paper illustrates this fact. How this paper evolved is more or less covered in the ﬁrst two paragraphs of this section. However, motivational input into this endeavour

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has also come from other sources. Gary Peterson and Carter Lyons digested the material of [19] and discussed with me questions surrounding it. The 1999 Edinburgh conference involved us in stimulating discussion concerning the Z-constrained conjecture. I appreciate this stimulation very much. It is such input that has provided me with the will to write what you have read.

References [1] E. Aichinger, On Near-ring Idempotents and Polynomials on Direct Products of Ω-groups, Proc. Edin. Math. Soc. 44 (2001), 379-388. [2] J. D. P. Meldrum, Nearrings and their Links with Groups, Pitman Publishing, Lon. (1985). [3] G. L. Peterson, On an Isomorphism Problem for Endomorphism Near-rings, Proc. Amer. Math. Soc. 126 (1998), no.7. 1897-1900. [4] G. Pilz, Nearrings, North-Holland Pub (1983). Amsterdam. [5] S. D. Scott, Central Factors of 2-tame N -groups, 10 page manuscript (1996). [6] S. D. Scott, Compatible Nearrings, 146 page unpublished book (2000). [7] S. D. Scott, Compatible Z-constrained N -groups, 4 page manuscript (1997). [8] S. D. Scott, Idempotents in Near-rings with Minimal Condition, J. Lon. Math. Soc. (2) 6 (1973), 464-466. [9] S. D. Scott, Linear Ω-Groups, Polynomial Maps, Contr. Gen. Alg. 8 (1991), 239-293. [10] S. D. Scott, Minimal Ideals of Near-rings with Minimal Condition, J. Lon. Math. Soc. (2) 8 (1974), 8-12. [11] S. D. Scott, Near-rings and near-ring modules, Doct. diss. (Austr. Nat. Uni. 1970). [12] S. D. Scott, N -solubility and N -nilpotency in Tame N -groups, Alg. Col. 5:4 (1998), 425-448. [13] S. D. Scott, On the Finiteness and Uniqueness of Certain 2-tame N -groups, Proc. Edin. Math. Soc. 38 (1995), 193-205. [14] S. D. Scott, On the Structure of Certain 2-Tame Near-rings, Klu. Acad. Pub. (1995) (Netherlands), Near-rings and Near-ﬁelds, 239-256. [15] S. D. Scott, Tame Fusion, Alg. Col. 10:4 (2003), 543-566. [16] S. D. Scott, Tame Near-rings and N -Groups, Proc. Edin. Math. Soc. 23 (1980), 275-296. [17] S. D. Scott, Tameness and the Right Ideal Q(N ), Alg. Col. 6:4 (1999), 413-438. [18] S. D. Scott, The Structure of Ω-Groups, Klu. Acad. Pub (1997) (Netherlands), Nearrings, Nearﬁelds and K-loops, 47-137. [19] S. D. Scott, The Uniqueness of Certain Compatible N -groups, 29 page manuscript (1997). [20] S. D. Scott, Topology and Primary N -groups, Klu. Acad. Pub. (2001) (Netherlands), Near-rings and Near-ﬁelds, 151-197.

II

CONTRIBUTED PAPERS

PRIMENESS AND RADICALS IN NEARRINGS OF CONTINUOUS FUNCTIONS Geoﬀrey L. Booth University of Port Elizabeth, Port Elizabeth 6000, South Africa

Abstract

In this note, we summarise some resullts previously obtained by the present author together with P.R. Hall ([1], [2]) and present some new ones. In particular, we characterise the strongly prime radical of the near-ring of continuous, zero-preserving self-maps of R, and that of some related near-rings. 2000 Mathematics Subject Classiﬁcation: 16Y30; 22A05.

Keywords: Topological group, near-ring of continuous functions, prime, 3-prime, equiprime, strongly prime, strongly equiprime, Hoehnke radical.

1.

Preliminaries

In this paper, all near-rings will be right distributive. The notation “A N ” means “A is an ideal of N ”. There are a number of deﬁnitions of primeness for near-rings in the literature. The classical notion is given in Pilz [1]: A near-ring N is called prime (resp. semiprime) if A, B N (resp. A N ), AB = 0 implies A = 0 or B = 0 (resp. A2 = 0 implies A = 0). N is called 3-prime (resp. 3-semiprime) if x, y ∈ N (resp. x ∈ N ) xN y = 0 implies x = 0 or y = 0 (resp. xN x = 0 implies x = 0). N is called equiprime (cf Booth, Groenewald and Veldsman [3]) if a, x, y ∈ N , anx = any for all n ∈ N implies a = 0 or x = y. All of these deﬁnitions of primeness generalise the usual notion of primeness for associative rings. It is well known that equiprime = =⇒ 3-prime =⇒ = prime = semiprime for near-rings, and that these implications are strict. =⇒ There are two notions of strongly prime for near-rings. A near-ring N is strongly prime [5] if 0 = a ∈ N implies that there exists a ﬁnite subset F of N such that aF x = 0 implies x = 0, for all x ∈ N . N is strongly equiprime [4] if 0 = a ∈ N implies that there exists a ﬁnite subset F of N such that x, y ∈ N, af x = af y for all f ∈ F implies x = y. Clearly strongly equiprime = =⇒ strongly prime = =⇒ 3-prime and strongly equiprime = =⇒ equiprime. We refer to Pilz [1] for all undeﬁned 171 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 171–176. c 2005 Springer. Printed in the Neatherlands.

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concepts concerning near-rings. An ideal I of N is called prime (resp. semiprime, 3-prime, equiprime, strongly prime, strongly equiprime) if the factor near-ring N/I is prime (resp. semiprime, 3-prime, equiprime, strongly prime, strongly equiprime). Let U be a universal class of near-rings. Recall that a Hoehnke radical map (or simply radical map ) in U is a mapping ρ that assigns to each element N of U an ideal ρ(N ) of N such that (i) if N, R ∈ U and f : N → R is a surjective near-ring homomorphism, then f (ρ(N )) ⊆ ρ(R) and (ii) ρ(N/ρ(N )) = 0. It is well known that if M is a subclass of U and ρ(N ) := ∩{I N | N/I ∈ M} then ρ is a radical map in U. The prime (resp. 3-prime, equiprime, strongly prime, strongly equiprime) radical of a near-ring is the intersection of the prime (resp. 3-prime, equiprime, strongly prime, strongly equiprime) ideals of N . These radicals will be denoted P(N ), P3 (N ), Pe (N ), S(N ) and Se (N ), respectively. In the sequel G will denote a T0 (and hence completely regular) additive topological group. The set of zero-preserving continuous self-maps of G forms a zerosymmetric near-ring with respect to addition and composition of functions, and is denoted N0 (G). If the topology on G is discrete, N0 (G) is the set of all zero-preserving self-maps of G, and is denoted M0 (G) in this case. In order to avoid trivial cases, all topological groups will be assumed to contain more than one element. Composition of functions will be denoted by juxtaposition, e.g. ab rather than a ◦ b. For all undeﬁned concepts concerning near-rings, we refer to Pilz [1]. For surveys of work done on near-rings of continuous functions, [7] and [8] can be consulted.

2.

Primeness in N0 (G)

It is well known (cf [10]) that M0 (G) is equiprime. Primeness and prime radicals of N0 (G) were considered in [1] and [2]. In this Section we summarise the main results presented there and prove some new ones. Proposition 2.1. [2, Proposition 2.2] Let G be a 0-dimensional, T0 topological group. Then (a) N0 (G) is equiprime. (b) N0 (G) is strongly prime if and only if the topology on G is discrete. (c) N0 (G) is strongly equiprime if and only if G is ﬁnite. Proposition 2.2. [2, Proposition 3.2] Let G be a T0 , arcwise connected topological group. Then N0 (G) is equiprime. The next two results show that, unlike M0 (G), N0 (G) can be very far from being equiprime.

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Proposition 2.3. [2, Proposition 1.1] Let G be a disconnected topological group, with open components which contains more than one element. Then N0 (G) is not semiprime. Proposition 2.4. [1, Proposition 2.2] Let G be a disconnected topological group, with open components which are arcwise connected and which contain more than one element. Let H be the component of G which contains 0, I := {a ∈ N0 (G) | a(G) ⊆ H} and J := {a ∈ N0 (G) | N0 (G)) = I ∩ J. a(H) = 0}. Then P(N N0 (G)) = Pe (N We now consider strongly prime in the case that G is T0 and arcwise connected. We will need the following Lemma. Lemma 2.5. Let G be arcwise connected and let U be an open set in G such that 0 ∈ U . Then there exists 0 = a ∈ N0 (G) such that a(G) ⊆ U . Proof. Let 0 = g ∈ U . Let α : G → [0, 1] be a continuous function such that α(0) = 0 and α(g) = 1. Since G is arcwise connected, there exists a continuous function β : [0, 1] → G such that β(0) = 0 and β(1) = g. We may assume that β(t) = 0 for 0 < t ≤ 1. For if not, let s := sup{t ∈ [0, 1] | β(t) = 0}. Then s ∈ [0, 1] and β(s) = 0. Note that this implies s < 1. Let γ(t) := (1−s)t+s for t ∈ [0, 1]. Let β (t) = βγ(t) for t ∈ [0, 1]. Then β (0) = β(s) = 0 and β (1) = β(1) = g. Moreover β (t) = 0 for 0 < t ≤ 1, and hence we may replace β with β . Since U is open in G, β −1 (U ) is open in [0, 1]. Hence there exists 0 < u < 1 such that [0, u) ⊆ β −1 (U ). Let 0 < v < u. Let δ(t) := vt for t ∈ [0, 1]. Then δ maps [0, 1] onto [0, v]. Hence βδ maps [0, 1] into U . Let a := βδα. Then a maps G into U , and a(0) = βδα(0) = βδ(0) = β(0) = 0. Hence a ∈ N0 (G). Moreover, a(g) = βδα(g) = βδ(1) = β(v) = 0 since v > 0. This completes the proof. Proposition 2.6. Suppose G is a T0 , arcwise connected topological group. Then N0 (G) is not strongly prime (and hence not strongly equiprime). Proof. Let U be an open set containing 0 whose closure c(U ) is not G. Let g ∈ G\c(U ). Since G is completely regular, there exists a continuous function α : G −→ [0, 1] such that α(c(U )) = 0 and α(g) = 1. Since G is arcwise connected, there exists a continuous function β : [0, 1] → G such that β(0) = 0 and β(1) = g. Let a := βα. Then a ∈ N0 (G) and a(U ) = 0. Now let F := {f1 , ..., fn } be a ﬁnite subset of N0 (G). Let Vi := fi−1 (U ) for 1 ≤ i ≤ n and V := ni=1 Vi . Note that 0 ∈ V . If V = G, affi = 0 for 1 ≤ i ≤ n so aF x = 0 for any x ∈ N0 (G) and we are done. Suppose that V = G. By Lemma 2.5 there exists 0 = b ∈ N0 (G) such that b(G) ⊆ V .

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Then affi b = 0 for 1 ≤ i ≤ n, i.e. aF b = 0. Hence N0 (G) is not strongly prime. We remark that Proposition 2.6 improves [2, Proposition 3.3], which required that the topology on G has a base consisting of arcwise connected open sets. The question arises how we can characterise the strongly prime radicals of N0 (G) when G satisﬁes the conditions of Proposition 2.6. We will do this in the case of N0 (R), where R denotes the additive group of the real numbers, with the usual topology. A more general characterisation awaits further study. Lemma 2.7. Let I be an ideal of N0 (R) which contains all the bounded mappings. Then I = N0 (R). 2 x≥0 x , n(x) := x, a(x) := arctan x. Then Proof. Let m(x) := 2 −x x < 0 a ∈ I. Hence b := m(a + n) − mn ∈ I. It may be shown that b(x) = 2x arctan x + (arctan x)2 x ≥ 0 . Moreover, b is unbounded and −2x arctan x − (arctan x)2 x < 0 strictly monotone increasing and hence is a bijection of R onto itself. Hence b has a continuous inverse from which we deduce that I = N0 (R). Proposition 2.8. Let P := {a ∈ N0 (R) |there exists an open set U in R such that 0 ∈ U ⊆ a−1 (0)}. Then P is a strongly prime ideal of N0 (R) which is contained in every other strongly prime ideal of N0 (R). Proof. It follows from [6, Lemma 2.2 and Theorem 2.3] that P N0 (R). It is easily seen P = {a ∈ N0 (R) |there exists ε > 0 such that (−ε, ε) ⊆ a−1 (0)}. Let a ∈ N0 (R)\P . Then (at least) one of the following is true: (i) for all ε > 0, [0, ε) a−1 (0) or (ii) for all ε > 0, (−ε, 0] a−1 (0). Assume that (i) is true. Let f (x) := x2 . Let b ∈ N0 (R)\P and let ε > 0. Then f b((−ε, ε)) is an interval which contains 0 and which is contained in [0, ∞). It follows from (i) that af b((−ε, ε)) = 0. Since ε was arbitrarily chosen, this implies that af b ∈ / P . Similarly, if a satisﬁes / P implies agb ∈ / P. (ii), there exists g ∈ N0 (R) such that b ∈ We conclude the proof by showing that, if I N0 (R) and P I, then I cannot be strongly prime. Let a ∈ P \I. Then there exists an open set U such that 0 ∈ U and a(U ) = 0. Let F := {f1 , ..., fn } be a ﬁnite subset of N0 (R). Let V := ni=1 fi−1 (U ). Since P I, I = N0 (R). It follows from Lemma 2.7 that there exists a bounded element b of / I. Since V is open, there exists ε > 0 such that N0 (R) such that b ∈ ε(b(x)) ∈ V for all x ∈ R. Hence kε b(R) ⊆ V where kε (x) := εx. Since / I, kε b ∈ / I. But affi kε b(R) ⊆ affi (V ) ⊆ a(U ) = 0. kε is invertible and b ∈ Hence affi kε b = 0 ∈ I for 1 ≤ i ≤ n so I is not strongly prime.

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Corollary 2.9. S(N N0 (R)) = P , where P is as in Proposition 2.8. Proof. Since P is a strongly prime ideal of N0 (R) by Proposition 2.8, S(N N0 (R)) ⊆ P . If Q is a strongly prime ideal of N0 (R), it follows from Proposition 2.8 that P ⊆ Q. Hence P ⊆ ∩{Q | Q is a strongly prime N0 (R)). Equality follows. ideal of N0 (R)}, i.e. P ⊆ S(N Lemma 2.10. Let G be a topological group with open connected components, and let H be the component of G which contains 0. If a ∈ N0 (G), let a denote the restriction of a to N0 (H) and let ϕ(a) := a for all a ∈ N0 (G). Let PG := {a ∈ N0 (G) |there exists an open set U in G such that 0 ∈ U ⊆ a−1 (0)} and PH := {a ∈ N0 (H) |there exists an open set PG ) = PH . U in H such that 0 ∈ U ⊆ a−1 (0)}. Then ϕ(P Proof. Suppose a ∈ G. Then there exists U open in G such that 0 ∈ U ⊆ a−1 (0), whence 0 ∈ U ∩H ⊆ a−1 (0)∩H = a−1 (0). Hence ϕ(a) = a ∈ PH , so ϕ(P PG ) ⊆ PH . Conversely, suppose that b ∈ PH . Deﬁne a : G → G by b(g) g∈H a(g) := . g g ∈ G\H Since H is open, a ∈ N0 (G). Moreover, ϕ(a) = a = b. Since b ∈ N0 (H), there exists an open subset V of H such that 0 ∈ V ⊆ b−1 (0) = a−1 (0). PG ). Since H is open in G, V is open in G. Hence b ∈ PH , so PH ⊆ ϕ(P Equality follows. Remark 2.11. In the proof of [1, Proposition 2.2 ] it is shown that ϕ, as deﬁned in Lemma 2.10, is a near-ring homomorphism of N0 (G) onto N0 (H), and hence N0 (H) ∼ = N0 (G)/J, where J := {a ∈ N0 (G) | a(H) = 0}. Clearly the subset of N0 (G)/J which corresponds to PH under this isomorphism is PG /J. Proposition 2.12. Let G be a topological group with open connected components such that the component containing 0 is isomorphic to R. Let I := {a ∈ N0 (G) | a(G) ⊆ H} and P := {a ∈ N0 (G) |there exists an N0 (G)) = I ∩ P. open set U in G such that 0 ∈ U ⊆ a−1 (0)}. Then S(N Proof. Let J := {a ∈ N0 (G) | a(H) = 0}. Let J := {a ∈ N0 (G) | a(H) = 0} and let Q be a strongly prime ideal of N0 (G). Clearly J(I ∩ P ) = 0. Since Q is a prime ideal, either J ⊆ Q or I ∩ P ⊆ Q. In the former case, we have that (N N0 (G)/J)/(Q/J) ∼ = N0 (G)/Q, which is a strongly prime near-ring, whence Q/J is a strongly prime ideal of N0 (G)/J ∼ = N0 (R). It follows from Proposition 2.8, Lemma 2.10 and Remark 2.11 that P/J ⊆ Q/J, whence P ⊆ Q. Hence I ∩ P ⊆ P ⊆ Q in this case. It follows that I ∩ P ⊆ Q for every strongly prime ideal Q

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of N0 (G). Taking the intersection as Q runs through the strongly prime N0 (G). ideals of N0 (G), we have that I ∩ P ⊆ S(N Since H is open in G, it follows from [1, Lemma 2.1] that the quotient topology on G/H is discrete. From the proof of [1, Proposition 2.2], N0 (G)/I ∼ = N0 (G/H), and the latter is strongly prime by PropoN0 (G)) ⊆ sition 2.1. Hence I is a strongly prime ideal of N0 (G), so S(N ∼ I. Again by the proof of [1, Proposition 2.2], N0 (G)/J = N0 (H) ∼ = N0 (R). It follows from Corollary 2.9, Lemma 2.10 and Remark 2.11 that S(N N0 (G)/J) = P/J. Since S is a radical, map, it follows from the deﬁN0 (G)) ⊆ P . nition of such a that (S(N N0 (G) + J))/J ⊆ P/J, whence S(N Hence S(N N0 (G) ⊆ I ∩ P . This completes the proof. Corollary 2.13. Let G := R × K, where K is any additive group, with the product topology with respect to the usual topology on R and the discrete topology on K. Then S(N N0 (G)) = I ∩ P , where I and P are deﬁned as in Proposition 2.12. Question. Does there exist a non-discrete topological group G such that N0 (G) is strongly prime?

References [1] G.L. Booth: Primeness in near-rings of continuous functions 2, Beitrage ¨ Alg. Geom., to appear. [2] G.L. Booth and P.R. Hall: Primeness in near-rings of continuous functions, Beitrage ¨ Alg. Geom., 45 (2004), No. 1, 21-27. [3] G.L. Booth, N.J. Groenewald and S. Veldsman: A Kurosh-Amitsur prime radical for near-rings, Comm. in Algebra 18 (1990), No. 9, 3111-3122. [4] G.L. Booth, N.J. Groenewald and S. Veldsman: Strongly equiprime near-rings, Quaestiones Math. 14 (1991), No. 4, 483-489. [5] N.J. Groenewald: Strongly prime near-rings, Proc. Edinburgh Math. Soc. 31 (1988), No. 3, 337-343. [6] R.D. Hofer: Near-rings of continuous functions on disconnected groups, J. Austral. Math. Soc. Ser. A 28 (1979), 433-451. [7] K.D. Magill: Near-rings of continuous self-maps: a brief survey and some open problems, Proc. Conf. San Bernadetto del Tronto, 1981, 25-47, 1982. [8] K.D. Magill: A survey of topological nearrings and nearrings of continuous functions, Proc. Tenn. Top. Conf., World Scientiﬁc Pub. Co., Singapore, 1997, 121140. [9] G. Pilz: Near-rings, 2nd ed., North-Holland, Amsterdam, 1983. [10] S. Veldsman: On equiprime near-rings, Comm. in Algebra 20 (1992), No. 9, 2569-2587.

DIFFERENCE METHODS AND FERRERO PAIRS Tim Boykett∗ and Peter Mayr∗ Institut f¨ fur Algebra Johannes Kepler Universit¨t ¨ Linz 4040 Linz, Austria [email protected] [email protected]

Abstract

1.

We present a construction method of BIB-designs from a ﬁnite group G and a group of automorphisms Φ on G such that |Φ(x)| = |Φ| for all x ∈ G, x = 0. By using a generalization of the concept of a diﬀerence family we can so unify several previous constructions of BIB-designs from planar near-rings. 2000 Mathematics Subject Classiﬁcation: 05B05; (16Y30).

Introduction

This paper introduces a general construction that encompasses several known constructions of designs from near-rings. The original work of Ferrero, lying historically between that of Bose and Wilson, introduced the construction, related to diﬀerence families. This paper uses a slightly more general concept, the short diﬀerence family, to generalize these constructions. We proceed ﬁrst by introducing our terminology, then the short diﬀerence family construct. We show that such constructions arise naturally from ﬁxed-point-free automorphism groups acting on groups.

2.

Designs and Diﬀerence Families

Let V be a set of size v with v > 1. Let k, λ be positive integers. For our purposes, a (v, k, λ)-BIB-design (or design for short) is a set B of subsets of V that satisﬁes the following conditions: ∗ This

work has been supported by grant P15691 of the Austrian National Science Foundation (Fonds zur F¨ ¨ orderung der wissenschaftlichen Forschung).

177 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 177–187. c 2005 Springer. Printed in the Neatherlands.

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(1) |B| = k for all B ∈ B; (2) |{B ∈ B | u, v ∈ B}| = λ for all u, v ∈ V, u = v . The elements of V are usually referred to as points; the elements of B are called blocks. By our deﬁnition a design does not have repeated (multiple) blocks. Apart from that our deﬁnition is equivalent to that of a 2-balanced design [BJL99, p. 15] or (v, k, λ)-BIBD [CD96]. A bijection ϕ on V such that ϕ(B) ∈ B for all B ∈ B is called automorphism of the design B. In this note we will use the following concepts: Let (G, +) be a ﬁnite group, not necessarily abelian, of order v > 1. We denote G \ {0} by G∗ . The group G acts on itself by right translation. For B ⊆ G, we let GB := {g ∈ G | B + g = B} denote the stabilizer of the set B under the action of G. Let B be a set of subsets of G. Then devB := {B + g | B ∈ B, g ∈ G} is called the development of B in G. For B, C ∈ B, we deﬁne B ∼ C iﬀ ∃g ∈ G : B = C + g. This relation ∼ is an equivalence relation on B. The equivalence class of B ∈ B modulo ∼ is denoted by B/∼. Deﬁnition 2.1. Let B be a set of subsets of the group (G, +) with |G| > 1. Let v = |G|, and let k, µ, ν, λ be positive integers such that the following are satisﬁed: (1) |B| = k for all B ∈ B; (2) |GB | = µ for all B ∈ B; (3) |B/∼| = ν for all B ∈ B; (4) |{(B, a, b) | B ∈ B, a, b ∈ B, a − b = d}| = λ for all d ∈ G, d = 0.

λ Then B is called a (v, k, µν )-short diﬀerence family (or sdf ) in G.

We note that an sdf with µ = 1 is in fact a (v, k, λ)-diﬀerence family as deﬁned in [BJL99, p. 470]. However, the general deﬁnition of a diﬀerence family allows multiple blocks, which we avoid. The term short diﬀerence family seems appropriate since in the literature a block B ∈ B with |GB | > 1 is said to have a short orbit. In [CD96, p.271] a set of subsets B of G is called a partial diﬀerence family if devB is a BIB-design. The following proposition says that short diﬀerence families are partial diﬀerence families in this sense.

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Proposition 2.2. Let B be a (v, k, λ)-sdf in the group G. Then devB is a (v, k, λ)-design. We note that G acts as a group of design-automorphisms on devB for an sdf B. Hence the automorphism group of devB acts transitively on the point set G of devB. Proof of Proposition 2.2. Let B be a (v, k, λ)-sdf. Then there exist µ, ν such that |GB | = µ and |B/∼| = ν for all B ∈ B. We have |{(B, a, b) | B ∈ B, a, b ∈ B, a − b = d}| = λµν for all d ∈ G, d = 0. If B ∼ C for B, C ∈ B, then dev{B} = dev{C}. Let R be a set of representatives of ∼ on B. Then dev{B}, devB = B∈R

where the union is disjoint. Let u, v ∈ G, u = v. We count the number of blocks in devB that contain {u, v}: |{C ∈ dev{B} | u, v ∈ C}| |{C ∈ devB | u, v ∈ C}| = B∈R

=

|{C ∈ dev{B} | u, v ∈ C}| |B/∼|

B∈B

= =

1 |{g ∈ G | u, v ∈ B + g}| ν |GB | B∈B 1 |{g ∈ G | u − g, v − g ∈ B}| µν B∈B

For B ∈ B ﬁxed, we now consider the map f : {g ∈ G | u − g, v − g ∈ B} → {(a, b) ∈ B × B | a − b = u − v}, g → (u − g, v − g). Then f has an inverse, given by f −1 : (a, b) → −a + u. In particular, f is bijective. Hence we obtain |{C ∈ devB | u, v ∈ C}| =

1 |{(a, b) ∈ B × B | a − b = u − v}| µν B∈B

1 · |{(B, a, b) | B ∈ B, a, b ∈ B, a − b = u − v}| = µν 1 · λµν = µν = λ.

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Thus each set {u, v} with u, v ∈ G, u = v, is contained in λ distinct blocks of devB. This completes the proof that devB is a (v, k, λ)-design. In the proof above we used that a set of representatives R for ∼ on B has the same development as B. Now each equivalence class of ∼ on R has size 1. Thus we could have required |B/∼| = 1 for all B ∈ B instead of condition (3) in our deﬁnition of an sdf, and we would still obtain the same designs. It is just a matter of notational convenience that we allow |B/∼| = ν for an arbitrary, but ﬁxed integer ν. In the following lemma GB is characterized as the unique maximal subgroup of which B ⊆ G can be expressed as union of left cosets. Lemma 2.3. Let G be a group, and let B ⊆ G. Then GB is the unique maximal (with respect to containment) element in {H ≤ G | B = B+H}. Proof. Straightforward. As a corollary of Lemma 2.3, we obtain that GB = {0} if |B| and |G| are relatively prime.

3.

Short diﬀerence families from fpf automorphisms

Let (G, +) be a ﬁnite group. We denote the identity mapping on G by 1, and let 0 : G → G, x → 0. A group Φ of automorphisms on (G, +) is said to be ﬁxed-point-free (fpf ) iﬀ |Φ(x)| = |Φ| for all x ∈ G∗ . Proposition 3.1. Let Φ be a group of fpf automorphisms on (G, +), and let S ⊆ Φ ∪ {0} with |S| > 1. Let µ, ν be positive integers such that the following are satisﬁed: (1) |GS(a) | = µ for all a ∈ G∗ ; (2) |S(a)/∼| = ν for all a ∈ G∗ . Then {S(x) | x ∈ G∗ } is a (|G|, |S|, |S|·(|S|−1) )-sdf. µν Proof. Since Φ is fpf, we have |S(x)| = |S| for all x ∈ G∗ . We note that for the same reason the map α − β : G → G, x → α(x) − β(x) is bijective for all α, β ∈ Φ ∪ {0}, α = β. Thus (α − β)−1 exists. It remains to check condition (4) of Deﬁnition 2.1. Let d ∈ G∗ , and let B := {S(x) | x ∈ G∗ }. Now we have {(B, a, b) | B ∈ B, a, b ∈ B, a − b = d} = {(S(x), α(x), β(x)) | x ∈ G∗ , α, β ∈ S, α(x) − β(x) = d} = {(S(x), α(x), β(x)) | α, β ∈ S, α = β, x = (α − β)−1 (d)}.

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The cardinality of the last set is |S| · (|S| − 1). Thus B is an sdf with parameters as given in the proposition. Several of the designs from planar near-rings as described in [Cla92] arise in the situation of Proposition 3.1 for diﬀerent choices of a set S of endomorphisms on a group G. We will not prove the next two wellknown results. We note that also for the original proofs the diﬃculty lies entirely in verifying the conditions (1) and (2) as given in Proposition 3.1. Corollary 3.2. [Cla92, cf. p. 59, Theorem 5.5] Let Φ be fpf on G. Then dev{Φ(x) | x ∈ G∗ } is a (|G|, |Φ|, |Φ| − 1)-design. Corollary 3.3. Let Φ be fpf on G. Let S := Φ ∪ {0}, and let B := {S(x) | x ∈ G∗ }. (1) [Cla92, cf. p. 118, Theorem 7.9] If B is not a subgroup of G for any B ∈ B, then devB is a (|G|, |Φ| + 1, |Φ| + 1)-design. (2) [Cla92, cf. p. 118, Theorem 7.11] If B is a subgroup of G for all B ∈ B, then devB is a (|G|, |Φ| + 1, 1)-design. Proposition 3.4. Let Φ be fpf on G, and let S ⊆ Φ ∪ {0}. Let Ψ be a group of automorphisms of G such that Ψ normalizes S and Ψ is transitive on G∗ . Then B = {S(x) | x ∈ G∗ } is an sdf, and G Ψ acts as a doubly transitive group of automorphisms on devB. Proof. It suﬃces to check that the assumptions of Proposition 3.1 are satisﬁed. Let a, b ∈ G∗ , c ∈ G be such that S(a) = S(b) + c, and let ψ ∈ Ψ. Since ψ normalizes S, we have S(ψ(a)) = S(ψ(b)) + ψ(c). Thus we obtain GS(ψ(a)) = ψ(GS(a) ) and |S(ψ(a))/∼| = |S(a)/∼|. Since Ψ is transitive on G∗ , the assumptions of Proposition 3.1 are satisﬁed. The result follows. Corollary 3.5. Let (F, +, ·) be a ﬁnite left near-ﬁeld, and let S ⊆ F be such that xS = Sx for all x ∈ F . Then B = {Sx | x ∈ F ∗ } is an sdf. Proof. Straightforward by Proposition 3.4. We note that the assumption xS = Sx for all x ∈ F in the above corollary is trivially fulﬁlled if (F, +, ·) is a ﬁeld.

4.

A generalization of Sun’s segments

We will now describe how the designs of segments as deﬁned in [Sun01] ﬁt into our setting of short diﬀerence families. Our main result is the following.

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Proposition 4.1. Let (G, +) be a group, and let S be a set of endomorphisms on G such that the following conditions are satisﬁed: (1) 0, 1 ∈ S and |S| > 2; (2) {1 − α | α ∈ S} = S; (3) S ∗ is a group of fpf automorphisms on G. We assume that |G| and |S ∗ | are odd. Then {S(x) | x ∈ G∗ } is a )-sdf. (|G|, |S|, |S|·(|S|−1) 2 Here and in the following we write S ∗ for S \ {0}. See Examples 4.7 to 4.10 at the end of this section for various choices of G and S that satisfy the conditions (1), (2), and (3) of Proposition 4.1. For the proof of Proposition 4.1, we need a bit of preparation. Lemma 4.2. Let α be an automorphism on (G, +) such that 1 − α is also an automorphism on (G, +). Then (G, +) is abelian. Proof. Let β = 1 − α, and let x, y ∈ G. Then we have β(x + y) = x + y − α(x + y) = x + y − α(y) − α(x)

(4.1)

β(x + y) = β(x) + β(y) = x − α(x) + y − α(y).

(4.2)

and

By (4.1) and (4.2), we obtain (y − α(y)) − α(x) = −α(x) + (y − α(y)).

(4.3)

Since −α and 1−α are bijections on G, equation (4.3) yields a+b = b+a for all a, b ∈ G. Lemma 4.3. Let Φ be fpf on (G, +), and let α, ϕ ∈ Φ be such that 1 − α ∈ Φ. If {α, 1 − α} ⊆ NΦ (ϕ), then {α, 1 − α} ⊆ CΦ (ϕ). Proof. Let {α, 1 − α} ⊆ NΦ (ϕ). We note that α, 1 − α is an abelian and hence cyclic group of fpf automorphisms. Let β ∈ Φ generate α, 1− α, and let α = β i , 1 − α = β j . Since β normalizes ϕ, there exists an i j integer r such that ϕβ = ϕr . Then ϕα = ϕ(r ) and ϕ1−α = ϕ(r ) . By i j i ϕ(1 − α) = ϕ − ϕα = ϕ − αϕ(r ) , we have (1 − α)ϕ(r ) = ϕ − αϕ(r ) . Thus we obtain i

j

j

α(ϕ(r ) − ϕ(r ) ) = ϕ − ϕ(r ) . i

j

i

j

(4.4)

If ϕ(r ) = ϕ(r ) , then ϕ(r ) − ϕ(r ) is invertible because Φ is fpf. Then j i j α = (ϕ − ϕ(r ) )(ϕ(r ) − ϕ(r ) )−1 , and, in particular, α commutes with

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j

ϕ. But then also 1 − α commutes with ϕ. We obtain ϕ(r ) = ϕ = ϕ(r ) i j in contradiction to our assumption. Thus we have ϕ(r ) = ϕ(r ) and j ϕ = ϕ(r ) by (4.4). Both α and 1 − α commute with ϕ. Corollary 4.4. Let S be a set of endomorphisms of (G, +) such that S = 1 − S and Φ = S ∗ is a group of fpf automorphisms of G. Then Φ/Z(Φ) is trivial or isomorphic to one of the following: A4 , S4 , A5 , S5 . Proof. By [Bro01, Theorem 1.4], we have a unique normal subgroup N of Φ such that all Sylow subgroups of N are cyclic and Φ/N is isomorphic to one of the following groups: 1, V4 , A4 , S4 , A5 , S5 where 1 denotes the trivial group and V4 denotes the Klein group. We note that N is cyclic and normal in Φ. Thus the elements of S ∗ normalize N and, by Lemma 4.3, S ∗ centralizes N . Hence we have N ⊆ Z(Φ). In particular, N is nilpotent. Now N is the direct product of its cyclic Sylow subgroups, and N is cyclic. As a cyclic normal subgroup, N is central in Φ by Lemma 4.3. Thus N ⊆ Z(Φ). First we assume that Φ/N is abelian. Since N is central, this yields that Φ is nilpotent. Let P denote the Sylow 2-subgroup of Φ. Then P is a cyclic group or a generalized quaternion group [Rob82, 10.5.6 (ii)]. In any case, P has a normal cyclic subgroup R of index 2. Since Φ is the direct product of its Sylow subgroups, R is normal in Φ. By Lemma 4.3, R is central in Φ, which implies that P is cyclic. Thus Φ/N is not isomorphic to V4 . It remains that Φ/N is isomorphic to 1, A4 , S4 , A5 , or S5 . Since all these groups have trivial center, we ﬁnally obtain N = Z(Φ). The corollary is proved. By Corollary 4.4, Φ = S ∗ is cyclic if Φ is metacyclic or if |Φ| is odd. Lemma 4.5. Let G and S satisfy the conditions (1), (2), and (3) of Proposition 4.1. Let a, b ∈ G∗ , c ∈ G. If S(a) = S(b) + c, then −a + b + 2c ∈ GS(a) . Proof. We assume that S(a) = S(b) + c. Since (G, +) is abelian and S(x) = x − S(x) for all x ∈ G∗ , we obtain: S(a) − a + b + 2c = = = = =

−S(a) + b + 2c −(S(b) + c) + b + 2c −S(b) + b + c S(b) + c S(a)

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Thus −a + b + 2c ∈ GS(a) . Lemma 4.6. Let G and S satisfy the conditions (1), (2), and (3) of Proposition 4.1. Let Φ = S \ {0}. We assume that |Φ| is odd and that ( = a and c = 0) GS(a) = {0} for all a ∈ G∗ . Then S(a) = S(b) + c iﬀ (b or ((b = −a and c = a). Proof. We assume that S(a) = S(b) + c. By Lemma 4.5, we have −a + b + 2c ∈ GS(a) . Since GS(a) = {0} by assumption, we obtain b = a − 2c and S(a) = S(a − 2c) + c. We note that S(a − 2c) contains the elements 0 and a − 2c. By S(a) = S(a − 2c) + c, we obtain that c and a − c are in S(a). With {0, a} ⊆ S(a) and S(a) − c = S(a − 2c) we ﬁnd {−c, a − c} ⊆ S(a − 2c). Summing up, we have {0, a, c, a − c} ⊆ S(a) and {0, a − 2c, −c, a − c} ⊆ S(a − 2c). Seeking a contradiction, we suppose that c = a and c = 0. Let ∗ S = S \ {0}. Since a − c ∈ S ∗ (a) and a − c ∈ S ∗ (a − 2c), both S ∗ (a) and S ∗ (a − 2c) are contained in Φ(a − c). In particular, c and −c are in Φ(a − c). If c = −c, then 2c = 0 and S(a) = S(a) + c. Now GS(a) = 0 yields c = 0, which contradicts our assumption that c = 0. If c = −c, then there exists a ϕ ∈ Φ, ϕ = 1, such that ϕ(c) = −c. Now c = 0 is a ﬁxed point of ϕ2 . Hence ϕ2 = 1 and |Φ| is even, which contradicts the assumption of the lemma. Thus we have c = a or c = 0. The lemma is proved. Proof of Proposition 4.1. Let G and S satisfy the assumptions, and let a ∈ G∗ . We will prove that |GS(a) | = 1.

(4.5)

Seeking a contradiction, we suppose that there exists a d ∈ GS(a) such that d = 0. Since 0 ∈ S, we then have {d, −d} ∈ S(a). For Φ = S ∗ , we obtain {d, −d} ∈ Φ(a). In particular, there is a ϕ ∈ Φ such that ϕ(d) = −d. We have d = −d and ϕ = 1 by the assumption that |G| is odd. Since ϕ2 ﬁxes d = 0, we then have ordϕ = 2. This contradicts the assumption that |Φ| is odd. Thus we have d = 0, and (4.5) is proved. Next we show that |S(a)/∼| = 2. (4.6) From Lemma 4.6 we obtain S(a)/∼ = {S(a), S(−a)}. Since a = −a by the assumption that |G| is odd, we have S(a) = S(−a). This proves (4.6). The result follows from (4.5) and (4.6) with Proposition 3.1. We give several examples for designs of segments from short diﬀerence families.

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Example 4.7. Let G be an elementary abelian 2-group, and let S be a set of endomorphisms on G that satisﬁes the conditions (1), (2), and (3) of Proposition 4.1. From 1 − S = 1 + S we obtain {0, a} ⊆ GS(a) for all a ∈ G∗ . We assume equality, that is, GS(a) = {0, a} for all a ∈ G∗ . Then Lemma 4.5 yields |S(a)/∼| = 1 for all a ∈ G∗ . By Proposition 3.1, )-sdf for G. {S(a) | a ∈ G∗ } is a (|G|, |S|, |S|·(|S|−1) 2 Example 4.8. Let G and S satisfy the conditions (1), (2), and (3) of Proposition 4.1. We assume that |S| = 3. Then S = {0, 1, 2−1 } where 2−1 is the inverse mapping of 2 : G → G, x → x + x. Suppose there is a ∈ G∗ such that GS(a) = {0}. Then GS(a) = S(a), which yields 2−1 (a) = −a. Hence 2−1 is a ﬁxed-point-free automorphism of order 2, and 2−1 = 2 = −1. Now 2x = −x for all x ∈ G∗ yields 3x = 0; G is an elementary abelian 3-group. Thus S(x) = x = GS(x) for all x ∈ G. By Proposition 3.1, {S(a) | a ∈ G∗ } forms a (|G|, 3, 1)-sdf for G. If |S| = 3 and GS(a) = {0} for all a ∈ G∗ , then S(a)/∼ = {S(a), −S(a)} for all a ∈ G∗ by Lemma 4.5, and {S(a) | a ∈ G∗ } forms a (|G|, 3, 3)-sdf for G. Example 4.9. We consider a group of fpf automorphisms Ψ on G. Let S ⊆ {α ∈ Ψ | ordα = 6} ∪ {0, 1} be such that {0, 1} ∈ S, |S| > 2, and α5 ∈ S for all α ∈ S. We shall show that S = 1−S, |S| is even, |S| ≤ 22, )-sdf for G. and ﬁnally that {S(a) | a ∈ G∗ } is a (|G|, |S|, |S|·(|S|−1) 2 First let α ∈ Ψ with ordα = 6. We show that α5 = 1 − α.

(4.7)

Since α3 is a ﬁxed-point-free automorphism of order 2, we have α3 = −1 and G is abelian [Rob82, Exercise 8.5.6]. Let x ∈ G. By α2 (x + α2 (x) + α4 (x)) = α2 (x) + α4 (x) + x, we ﬁnd that x + α2 (x) + α4 (x) is a ﬁxed point of α2 = 1. Thus x + α2 (x) + α4 (x) = 0 for all x ∈ G. From 1 + α2 + α4 = 0 we obtain 1 + α4 = −α2 . Together with α3 = −1 this yields (4.7). Hence we have S = 1 − S. That |S| is even follows immediately from the deﬁnition of S. Next we show that |S| ≤ 22. Since the Sylow 3-subgroups of Φ := S ∗ are cyclic [Rob82, 10.5.6], either all elements of order 3 in Φ are central or none of them are central in Φ. First we consider the former case. Since −1 is the unique element of order 2 in Φ, we then have that all generators of Φ are central. Hence Φ is cyclic of order 6 and |S| = 4. Next we assume that no element of order 3 is central in Φ. Since −1 ∈ Z(Φ), we have that Φ/Z(Φ) is generated by elements of order 3. From Corollary 4.4 we obtain that Φ/Z(Φ) is isomorphic to A4 or to A5 . Hence there are at most 20 elements of order 3 in Φ and at most 20 elements of order

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6. This proves the assertion that |S| ≤ 22. We note that actually Φ is cyclic of order 6, isomorphic to SL(2, 3) or to SL(2, 5). We proceed to show that GS(a) = {0} for all a ∈ G∗ .

(4.8)

Let a ∈ G∗ , c ∈ GS(a) . Then we have α ∈ S such that c = α(a). Since −c ∈ GS(a) , we also have β ∈ S such that −c = β(a). We note that |G| is odd because S ∗ has even order. Suppose that α = 0. Then c = 0 and βα−1 (c) = −c = c. By assumption βα−1 is ﬁxed-point-free and has order 2. Hence βα−1 = −1 and β = −α. If α = 1, then we obtain β = −1 ∈ S which contradicts the assumption on S. If ordα = 6, then β = α4 ∈ S and ordβ = 3 which yields a contradiction. Thus we have α = 0 and (4.8) is proved. Now we show that S(a)/∼ = {S(a), −S(a)} for all a ∈ G∗ .

(4.9)

Let a, b ∈ G∗ , c ∈ G be such that S(a) = S(b)+c. Seeking a contradiction we suppose that c ∈ {0, a}. Then we have α ∈ S \{0, 1} and β ∈ S ∗ such that c = α(a) and c = −β(b). We note that β = 1 because otherwise S(a) = S(−c)+c = S(c) and Lemma 4.5 yields a = c. Hence both α and β have order 6. From Lemma 4.5 we obtain −α−1 (c) − β −1 (c) + 2c = 0. By (4.7), we have α−1 = 1−α, β −1 = 1−β which yields α(c)+β(c) = 0. Thus β = −α by the assumption that S ∗ is fpf. This contradicts that both α and β have order 6. Hence c ∈ {0, a} and we have (4.9). Proposition 3.1 yields that {S(a) | a ∈ G∗ } is an sdf. Example 4.10. We note that for a left near-ﬁeld (F, +, ·) each element a ∈ F ∗ induces an automorphism λa : x → a · x on (F, +). Let Φ = {λa | a ∈ F ∗ }. Then Φ is a group of fpf automorphisms of (F, +), and Φ is isomorphic to (F ∗ , ·). With 1 denoting the identity of (F, +, ·) we have λ1 = 1. If (F, +, ·) is not a ﬁeld, then we do not have 1 − λa = λ1−a in general. While 1 − λa for a = 1 is an automorphism of (F, +), in general it is not true that 1 − λa ∈ Φ. We note that Φ ∩ (1 − Φ) = {λa | a ∈ F ∗ such that (1 − a)x = x − ax for all x ∈ F }. Let (F, +, ·) be a Dickson near-ﬁeld with Dickson pair (q, n). That is, (F, +, ·) is coupled to the ﬁeld GF(q n ) = (F, +, ∗). Let Φ = {λf | f ∈ F ∗ }. Then there exist a, b ∈ F such that Φ is generated by α : x → a ∗ x and β : x → b ∗ xq . Thus Φ and hence (F ∗ , ·) are metacyclic. Let S ∗ ⊆ Φ and S = 1 − S. By Lemma 4.3, we have S ∗ ⊆ α. In particular

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S ∗ is cyclic. All mappings in S are of the form µf : x → f ∗ x where f ∈ GF(q n ) and ∗ is the multiplication in the ﬁeld GF(q n ). Let T = {f ∈ F | µf ∈ S}. Then S(x) = T ∗ x for all x ∈ F . Since T is central in the multiplicative group of GF(q n ), Corollary 3.5 yields that {T ∗ x | x ∈ F ∗ } is an sdf.

5.

Conclusion and further work

We have tried to explain designs from planar near-rings as pioneered by Ferrero and Clay [Cla92], using a generalization of the well known concept of diﬀerence families for groups. This allows a further analysis of segments, both in the geometric view (the deﬁnition of segments by their endpoints) and the design view (determination of some of the resulting designs). We do not know how concepts like circularity [BFK96, Ke92, KK96] relate to diﬀerence families. Future work may investigate this.

References [BFK96]

K. Beidar, Y. Fong, and W.-F. Ke. On ﬁnite circular planar nearrings. J. Algebra, 185(3):688–709, 1996.

[BJL99]

Thomas Beth, Dieter Jungnickel, and Hanfried Lenz. Design theory. Vol. II. Cambridge University Press, Cambridge, second edition, 1999.

[Bro01]

Ron Brown. Frobenius groups and classical maximal orders. Mem. Amer. Math. Soc., 151(717):viii+110, 2001.

[CD96]

Charles J. Colbourn and Jeﬀrey H. Dinitz, editors. The CRC handbook of combinatorial designs. CRC Press Series on Discrete Mathematics and its Applications. CRC Press, Boca Raton, FL, 1996.

[Cla92]

James R. Clay. Nearrings. The Clarendon Press Oxford University Press, New York, 1992. Geneses and applications.

[Ke92]

W.-F. Ke. Structures of circular planar nearrings. PhD thesis, University of Arizona, 1992.

[KK96]

W.-F. Ke and H. Kiechle. Combinatorial properties of ring generated circular planar nearrings. J. Combin. Theory Ser. A, 73(2):286–301, 1996.

[Rob82]

D. J. S. Robinson. A course in the theory of groups. Springer-Verlag, 1982.

[Sun01]

Hsin-Min Sun. Segments in a planar nearring. Discrete Math., 240(13):205–217, 2001.

ZERO-DIVISOR GRAPHS OF NEARRINGS AND SEMIGROUPS G. Alan Cannon Department of Mathematics, Southeastern Louisiana University, Hammond, LA 70402, U. S. A. [email protected]

Kent M. Neuerburg Department of Mathematics, Southeastern Louisiana University, Hammond, LA 70402, U. S. A. [email protected]

Shane P. Redmond Department of Mathematics, Eastern Kentucky University, Richmond, KY 40475, U. S. A. [email protected]

Abstract

Zero-divisor graphs of rings have been developed and explored by D. F. Anderson and Livingston, Redmond, and others ([4], [5], [15], [17]). Additionally, these ideas have been adapted to semigroups by DeMeyer, McKenzie, and Schneider [10]. Results concerning the properties of graphs of semigroups are presented. All possible zero-divisor graphs of nearrings with identity in which the graph has less than ﬁve vertices are classiﬁed, and the additive group of each nonring is identiﬁed. Following the example of [7], we include a table of nearrings with identity of orders between sixteen and thirty-one. 2000 Mathematics Subject Classiﬁcation: 16Y30, 05C99.

Keywords: zero-divisor graph, nearring, semigroup

1.

Introduction

The idea of a zero-divisor graph was introduced by Beck in [6] and further developed by D. D. Anderson and Naseer in [3]. Let R be a commutative ring and Z(R) be the set of nonzero zero-divisors of R. The zero-divisor graph, as deﬁned in [4], is the graph with vertices Z(R) 189 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 189–200. c 2005 Springer. Printed in the Neatherlands.

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and edges a − b if and only if ab = 0 and a = b. Later these ideas were applied to noncommutative rings by Redmond in [15] and [17] and to commutative semigroups by DeMeyer, McKenzie, and Schneider in [10]. In this paper the concept of the zero-divisor graph is extended to noncommutative semigroups and to nearrings. Let S be any semigroup and Z(S) be the set of nonzero zero-divisors of S. Deﬁne Γ(S) to be the directed graph whose vertices are elements of Z(S) with edges a → b if ¯ ab = 0 and a = b. Further, let Γ(S) be the undirected graph of Z(S) with edges a − b if a = b and either ab = 0 or ba = 0. Properties of the graphs ¯ Γ(S) and Γ(S) are determined. The same ideas are applied to a nearring ¯ ). A complete catalogue of the N to generate the graphs Γ(N ) and Γ(N possible nonempty zero-divisor graphs of zero-symmetric nearrings with identity having four or fewer vertices is given. In particular, examples of graphs that are not achievable on rings are presented. For an introduction to graph theory and nearrings including basic deﬁnitions and properties consult [9], [4], [1] and [18].

2.

General Results

We begin by considering zero-divisor graphs of semigroups. A semigroup S is zero-symmetric if there exists an element 0 ∈ S such that 0a = a0 = 0 for all a ∈ S. A (directed) graph G is connected if there is a path following the (directed) edges of G from any vertex of G to any other vertex of G. A graph G has diameter n if n is the smallest integer such that any two vertices of G are connected by a path having at most n (directed) edges. The proofs of the next two results are omitted as they follow those for the analogous results for rings as found in [15].

Theorem 2.1. Let S be a zero-symmetric semigroup. Then Γ(S) is connected if and only if the set of left zero divisors is equal to the set of right zero divisors. Moreover, if Γ(S) is connected, then diam(Γ(S)) ≤ 3.

¯ Theorem 2.2. Let S be a zero-symmetric semigroup. Then Γ(S) is a ¯ connected graph and diam(Γ(S)) ≤ 3.

Theorem 2.3. Let S be a zero-symmetric semigroup. ¯ ¯ 1. If Γ(S) does not contain a cycle, then Γ(S) is a connected subgraph of the graph G given in Figure 1.

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Figure 1. Graph G for Theorem 2.3. ¯ 2. Every connected subgraph of G is the graph Γ(S) of some semigroup S. Proof. Although the proof of this theorem is just a slight modiﬁcation of that for Theorem 3 given in [10], we include a sketch of the proof to ¯ ¯ illustrate the style. Suppose a − x − y − b is a path in Γ(S). Let c ∈ Γ(S) ¯ ¯ with c ∈ {a, x, y, b}. If a − c is an edge in Γ(S), then diam(Γ(S)) ≤ 3 ¯ ¯ implies there exist w, z ∈ Γ(S) with c − w − z − b a path in Γ(S). This gives a cycle a − c − w − z − b − y − x − a. Note that if w = x, w = y, or z = x, a cycle may still be constructed, just of shorter length. If z = y, then y − x − a − c − w − y is a cycle (or shorter if w = y), and if w = a, then a − z − b − y − x − a is a cycle. In all cases, a cycle has been produced. This completes the proof of 1. The proof of 2. follows that given in [10]. Given a graph G, the girth of G, gr(G), is the length of the shortest cycle of the graph, and gr(G) = ∞ if G has no cycles. The core K of G is the union of all cycles of G, and an end is a vertex of G with only one edge. ¯ Theorem 2.4. Let S be a zero-symmetric semigroup. If Γ(S) contains ¯ ¯ a cycle, then gr(Γ(S)) ≤ 4. Moreover, any vertex in Γ(S) is either a ¯ vertex of the core, K, or is an end of Γ(S). Proof. Theorem 3.3 of [15] proves the ﬁrst statement for the multiplicative semigroup of a general ring. A similar proof applies to zerosymmetric semigroups as well, making the ﬁrst statement true. So we ¯ only need to verify the second statement. If x is a vertex of Γ(S), one of the following is true (by Theorem 2.2 and since K = ∅): 1. x is in the core (i.e., x ∈ K); ¯ 2. x is an end of Γ(S); ¯ 3. a − x − b is a path in Γ(S) where a is an end and b ∈ K;

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¯ 4. a − x − y − b or a − y − x − b is a path in Γ(S) where a is an end and b ∈ K. In the ﬁrst two cases, we are done. In case three, assume a − x − b is a ¯ path in Γ(S) with a an end, b ∈ K, and x ∈ K. Since b ∈ K, there exist ¯ distinct c, d ∈ Γ(S) with c, d ∈ {a, x, b} and b − c − d part of some cycle ¯ ¯ in Γ(S). Thus we have the path a − x − b − c − d. Since diam(Γ(S)) ≤ 3, we must have one of the following: ¯ (i) b − d is an edge in Γ(S), or ¯ (ii) a − x − d is a path in Γ(S), or ¯ (iii) a − x − e − d is a path in Γ(S) for some e ∈ {a, b, c, d, x}. ¯ However, condition (ii) implies that x − d − c − b − x is a cycle in Γ(S) ¯ and condition (iii) implies that x − e − d − c − b − x is a cycle in Γ(S). Both imply that x ∈ K. Hence only condition (i) can hold. Without a loss of generality, we take cd = 0. Note that if xc = x, then x − b − d − xc is a cycle and, thus, x ∈ K. Therefore, xc = x. Also, if dx = x, then x − b − c − dx is a cycle and, thus, x ∈ K. Therefore, dx = x. Finally, the edges a − x and x − b yield four possible cases, each ¯ producing a cycle in Γ(S). A. If ax = 0 and bx = 0, then a − xc − b − x − a is a cycle. B. If ax = 0 and xb = 0, then a − xc − dx − b − x − a is a cycle. C. If xa = 0 and bx = 0, then a − dx − xc − b − x − a is a cycle. D. If xa = 0 and xb = 0, then a − dx − b − x − a is a cycle. Thus, in all cases A through D, x ∈ K, a contradiction. So case three cannot hold. For case four we show that both x and y are in K. Without loss ¯ of generality, assume a − x − y − b is a path in Γ(S). Since b ∈ K, there is some c ∈ K such that c = b and b − c is part of a cycle. Then ¯ a − x − y − b − c is a path in Γ(S). But the distance from a to c is four, a contradiction unless y − c or x − c is an edge. However, if y − c is an edge, then y ∈ K since y − b − c − y is a cycle. By case three, x is also in the core. If instead, x − c is an edge, then x − y − b − c − x is a cycle. Thus x, y ∈ K. ¯ Hence it must be the case that any vertex x of Γ(S) is either an end or in the core. The next theorem and its proof are analogous to those given in [11] for rings. Theorem 2.5. Let N be a left nearring having p ≥ 2 right zero-divisors (including zero). Then |N | ≤ p2 . Proof. Let z be a nonzero right zero-divisor of N and let A = {x ∈ N | zx = 0} be the right ideal in N consisting of all right annihilators

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of z. Since every element of A is a right zero-divisor, A is a ﬁnite right ideal with order α ≤ p. For any two elements r, s ∈ N , the right zerodivisors zr and zs are equal if and only if r = s(mod A). Thus, with each coset r + A we can associate a distinct right zero-divisor zr, being a representative of the residue class. Therefore, the number of distinct residue classes modulo A is less than or equal to p. Hence both the order and index of A are bounded by p and |N | ≤ p2 .

3.

Graphs of Nearrings

Before discussing the results on nearrings, we ﬁrst summarize what is currently known on the zero-divisor graphs of rings. For commutative rings with identity all realizable graphs with ﬁve or fewer vertices are known; see [5] and [16]. Further, all rings, up to isomorphism, that generate these graphs have been determined. In the case of noncommutative rings, not necessarily with identity, all realizable graphs on three or fewer vertices are known; see [15] and [17]. Much of the information concerning ﬁnite nearrings of small order may be found in [7]. The following results will add to this body of knowledge new information concerning the zero-divisor graphs of these nearrings. Theorem 2.5 ensures that to determine all zero-divisor graphs having four or fewer vertices only left nearrings of order less than or equal to twenty-ﬁve need to be considered. Using the SONATA package [2] running in GAP [12], the multiplicative Cayley tables for left nearrings with additive groups of order less than or equal to twenty-ﬁve were created. A computer sort (using PERL [1]) was done to ﬁnd those nearrings having ﬁve or fewer zero-divisors. Then the zero-divisor graphs for these nearrings were constructed to determine all possible zero-divisor graphs with four or fewer vertices that are determined by zero-symmetric nearrings with identity. Detailed information about the computer code used in this paper can be found in [8]. The following theorems list all the zero-symmetric nearrings with identity, up to isomorphism, that produce the zero-divisor graph of indicated type. We label nearrings as found in the Sonata libraries by (Sonata x/y, z) where x/y is the Thomas-Wood group number [19] and z is the Sonata library index number [2]. The searchable Sonata libraries include all nearrings of order less than or equal to ﬁfteen and all nearrings with unity of orders sixteen to thirty-one. Operation tables for nearrings with unity and order less than sixteen are also found in [7]. Theorem 3.1. There are 2 zero-symmetric nearrings N with identity such that Γ(N ) has exactly one vertex. These are the commutative rings Z4 and Z2 [X]/(X 2 ).

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Theorem 3.2. There are 4 zero-symmetric nearrings N with identity such that Γ(N ) is isomorphic to the graph in Figure 2 (left) consisting of exactly two vertices and two directed edges. These are the commutative rings Z9 , Z3 [X]/(X 2 ), and Z2 × Z2 and one nonring with additive group Z3 × Z3 (Sonata 9/2, 215). Theorem 3.3. There is one zero-symmetric nearring N with identity such that Γ(N ) is isomorphic to the graph in Figure 2 (right) consisting of exactly two vertices and one directed edge. This nonring has additive group Z2 × Z2 (Sonata 4/2, 12).

Figure 2. Graph for Theorem 3.2 (left) and graph for Theorem 3.3 (right). The zero-divisor graph of the nearring in the previous theorem is of interest because it is a tournament, a network, and has an odd number of edges. In [17] it was shown that the zero-divisor graph of a ring will never satisfy any of these conditions (if the graph has more than one vertex). Theorem 3.4. There are 16 zero-symmetric nearrings N with identity such that Γ(N ) is isomorphic to the graph in Figure 3 (left). These nearrings are: Four commutative rings: Z4 [X]/(2, X)2 , F4 [X]/(X 2 ), Z2 [X, Y ]/(X, Y )2 , and Z4 [X]/(X 2 + X + 1). One noncommutative ring with additive group Z2 ×Z2 ×Z2 ×Z2 and multiplication deﬁned by (x1 , x2 , x3 , x4 ) · (y1 , y2 , y3 , y4 ) = (x1 y1 + x4 y4 , x2 y1 + x2 y4 + x3 y4 , x3 y1 + x2 y4 + x3 y4 , x1 y4 + x4 y1 + x4 y4 ) (Sonata 16/5, 3). The three non-zero zero-divisors are (0,1,0,0), (0,0,1,0), and (0,1,1,0). Eleven nonrings: one with additive group Z4 × Z2 (Sonata 8/2, 971), one with additive group Z2 × Z2 × Z2 (Sonata 8/3, 661), two with additive group Q × Z2 where Q denotes the quaternion

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group of order 8 (Sonata 16/7, 3 and Sonata 16/7, 4), four with additive group Z2 × Z2 × Z2 × Z2 (Sonata 16/5, 4; Sonata 16/5, 5; Sonata 16/5, 6; and Sonata 16/5, 10), and three with additive group Z4 × Z4 (Sonata 16/3, 48; Sonata 16/3, 49; and Sonata 16/3, 50).

(right). The noncommutative ring listed above is the only noncommutative ring with identity whose zero-divisor graph has four or fewer vertices. This example extends what was known for noncommutative rings with identity. By exhaustion, this graph is the only possible graph on four or fewer vertices for noncommutative rings with identity. Theorem 3.5. There are 6 zero-symmetric nearrings N with identity such that Γ(N ) is isomorphic to the graph in Figure 3 (right). These nearrings include the 4 commutative rings Z6 , Z8 , Z2 [X]/(X 3 ), and Z4 [X]/(2X, X 2 −2) and 2 nonrings with additive groups Z4 ×Z2 (Sonata 8/2, 974) and Z2 × Z2 × Z2 (Sonata 8/3, 677), respectively. If the requirement that the nearring have identity is removed, then other zero-divisor graphs on three vertices can be realized. For example, in [17], two noncommutative rings (without multiplicative identity) were found to have the zero-divisor graphs in Figure 4.

Figure 4. Examples of graphs of rings on three vertices without unity. Theorem 3.6. There is one zero-symmetric nearring N with identity such that Γ(N ) is isomorphic to the graph in Figure 5 (left). This is the commutative ring Z2 × F4 .

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Theorem 3.7. There is one zero-symmetric nearring N with identity such that Γ(N ) is isomorphic to the graph in Figure 5 (center). This is the commutative ring Z3 × Z3 . Theorem 3.8. There are 5 zero-symmetric nearrings N with identity such that Γ(N ) is isomorphic to the graph in Figure 5 (right). These are the commutative rings Z25 and Z5 [X]/(X 2 ), and three nonrings with additive group Z5 × Z5 (Sonata 25/2, 8; Sonata 25/2, 10; and Sonata 25/2, 11).

Figure 5. Graphs for Theorem 3.6 (left), Theorem 3.7 (center), and Theorem 3.8 (right). Theorem 3.9. There is one zero-symmetric nearring N with identity such that Γ(N ) is isomorphic to the graph in Figure 6. The additive group of this nonring is Z3 × Z3 (Sonata 9/2, 211).

Figure 6. Graph for Theorem 3.9. Theorem 3.10. There is one zero-symmetric nearring N with identity such that Γ(N ) is isomorphic to the graph in Figure 7. The additive group of this nonring is Z2 × Z2 × Z2 (Sonata 8/3, 648).

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Figure 7. Graph for Theorem 3.10. This last graph is of interest because it has an odd number of directed edges, a feature that cannot happen in the zero-divisor graph of a ring as seen in [17]. We have found other examples of nonrings whose zerodivisor graphs have an odd number of edges: nine with additive group Z2 × Z2 × Z2 (Sonata 8/3, with indices 634, 638, 644, 647, 669, 693, 696, 707, 711) and two with additive group Z2 × Z2 × Z3 (Sonata 12/2, 2118 and Sonata 12/2, 2180). The zero-divisor graphs of these nonrings are not included since they contain more than four vertices.

4.

Table

For each additive group of order n with 16 ≤ n ≤ 31, the following table gives the total number of nearrings with identity on the given group. Furthermore, the total number of commutative rings, noncommutative rings, zero-symmetric nonrings, and nonzero-symmetric nonrings are given for each of these additive groups. These results follow the example established in [7]. The group names follow the format given in [19]. TW No. 16/1 16/2 16/3 16/4 16/5 16/6 16/7 16/8 16/9 16/10 16/11 16/12 16/13 16/14

Group Z16 Z2 × Z8 Z4 × Z4 Z22 × Z4 Z42 D4 × Z2 Q4 × Z2 Γ2 b Γ2 c1 Γ2 c2 Γ2 d D8 Γ 3 a2 Q8

NR w/1 1 37 51 470 2798 708 4 0 132 40 33 0 0 0

Com R 1 3 6 11 16 0 0 0 0 0 0 0 0 0

Noncom R 0 0 0 4 9 0 0 0 0 0 0 0 0 0

Z-S Nonring 0 33 44 435 2709 692 4 0 131 40 32 0 0 0

Non Z-S 0 1 1 20 64 16 0 0 1 0 1 0 0 0

198 TW No. 17/1 18/1 18/2 18/3 18/4 18/5 19/1 20/1 20/2 20/3 20/4 20/5 21/1 21/2 22/1 22/2 23/1 24/1 24/2 24/3 24/4 24/5 24/6 24/7 24/8 24/9 24/10 24/11 24/12 24/13 24/14 24/15 25/1 25/2 26/1 26/2 27/1 27/2 27/3 27/4 27/5 28/1 28/2 28/3 28/4 † There

G. Alan Cannon, Kent M. Neuerburg, Shane P. Redmond Group Z17 Z18 Z2 × Z23 S3 × Z3 D9 Z23 Z2 Z19 Z20 Z2 × Z10 D10 Q10 Hol(Z5 ) Z21 Z7 Z3 Z22 D11 Z23 Z24 Z6 × Z4 Z32 × Z3 D6 × Z2 A4 × Z2 Q6 × Z2 D4 × Z3 Q4 × Z3 S3 × Z4 D12 Q12 S4 SL2 (F F3 ) Z3 Z8 Z3 D4 Z25 Z5 × Z5 Z26 D13 Z27 Z3 × Z9 Z33 † † Z28 Z2 × Z14 D14 Q14

NR w/1 1 1 17 8 0 0 1 1 9 1 0 0 1 0 1 0 1 1 14 136 10 8 0 7 0 1 0 0 0 0 0 0 1 16 1 0 1 20 202 22 4 1 9 1 0

Com R 1 1 3 0 0 0 1 1 3 0 0 0 1 0 1 0 1 1 3 6 0 0 0 0 0 0 0 0 0 0 0 0 1 3 1 0 1 4 6 0 0 1 3 0 0

Noncom R 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0

Z-S Nonring 0 0 13 8 0 0 0 0 5 1 0 0 0 0 0 0 0 0 10 121 9 8 0 7 0 1 0 0 0 0 0 0 0 12 0 0 0 15 187 21 3 0 5 1 0

Non Z-S 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 8 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 8 1 1 0 1 0 0

is no standard nomenclature for these groups in [19].

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Zero-Divisor Graphs of Nearrings and Semigroups TW No. 29/1 30/1 30/2 30/3 30/4 31/1

5.

Group Z29 Z30 D5 × Z3 D3 × Z5 D15 Z31

NR w/1 1 1 0 0 0 1

Com R 1 1 0 0 0 1

Noncom R 0 0 0 0 0 0

Z-S Nonring 0 0 0 0 0 0

Non Z-S 0 0 0 0 0 0

Open Questions

For a ﬁnite ring R, it is known that Γ(R) is never a network nor a tournament when Γ(R) has more than one vertex [17]. In the case of nonrings, the only example of a network or tournament with four or fewer vertices appears in Theorem 3.3 with Γ(N ) having only two vertices. If N is a ﬁnite nearring with identity and Γ(N ) has ﬁve or more vertices, is Γ(N ) a network or a tournament? In this paper we have investigated zero-symmetric nearrings with identity whose zero-divisor graphs have four or fewer vertices. Can the zerodivisor graphs of zero-symmetric nearrings with identity having more than four vertices be classiﬁed? Can the zero-divisor graphs of nonzerosymmetric nearrings with identity be classiﬁed? What if the requirement that the nearrings have an identity is omitted? These are some of the many avenues of investigation left to explore.

Acknowledgments The authors would like to thank Erhard Aichinger and Ragan Malott for their computer expertise and our algebra colleagues at Southeastern Louisiana University for their help with this paper.

References [1] ActivePerl 5.6.1 for Windows, http://www.activestate.org/Products/ActivePerl (10 September 2004). [2] E. Aichinger, F. Binder, J. Ecker, R. Eggetsberger, P. Mayr, C. N¨obauer, SONATA - System of Near-Rings and Their Applications, Package for the group theory system GAP4. Division of Algebra, Johannes Kepler University, Linz, Austria (1999). [3] D. D. Anderson and M. Naseer, “Beck’s Coloring of a Commutative Ring,” J. Algebra, 159, (1993) 500–514. [4] D. F. Anderson and P. S. Livingston, “The Zero-Divisor Graph of a Commutative Ring,” J. Algebra, 217, (1999) 434–447. [5] D. F. Anderson, A. Frazier, A. Lauve, and P. S. Livingston, “The Zero-Divisor Graph of a Commutative Ring, II,” 61-72, Lecture Notes in Pure and Appl. Math., 202, Marcel Dekker, New York, 2001. [6] I. Beck, “Coloring of Commutative Rings,” J. Algebra, 116, (1988) 208–226.

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[7] F. Binder and C. N¨ obauer, Table of All Nearrings with Identity Up to Order 15. http://verdi.algebra.uni-linz.ac.at/Sonata/encyclo/ (14 June 2003). [8] G. A. Cannon, K. M. Neuerburg, and S. P. Redmond, Computer Search for the Paper “Zero-Divisor Graphs of Nearrings and Semigroups”. http://www.selu.edu/Academics/Faculty/kneuerburg/computercode1.pdf (10 November 2004). [9] G. Chartrand, Graphs as Mathematical Models, Prindle, Weber & Schmidt, Boston, 1977. [10] F. DeMeyer, T. McKenzie, and K. Schneider, “The Zero-Divisor Graph of a Commutative Semigroup,” Semigroup Forum, 65 (2), (2002) 206–214. [11] N. Ganesan, “Properties of Rings with a Finite Number of Zero Divisors, II,” Math. Ann., 161, (1965) 241–246. [12] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.3; 2002. http://www.gap-system.org (14 June 2003) [13] J. D. P. Meldrum, Near-Rings and Their Links with Groups, Pitman Advanced Publishing, Boston, 1985. [14] G. Pilz, Near-Rings, revised ed., North Holland Publishing Co., Amsterdam, 1983. [15] S. P. Redmond, “The Zero-Divisor Graph of a Non-Commutative Ring,” Internat. J. Commutative Rings, 1 (4), (2002) 203–211. [16] S. P. Redmond, “An Ideal-based Zero-divisor Graph of a Commutative Ring,” Comm. Alg., 31 (9), (2003) 4425–4443. [17] S. P. Redmond, “Structure in the Zero-Divisor Graph of a Noncommutative Ring,” Houston J. Math, 30 (2), (2004) 345–355. [18] D. F. Robinson and L. R. Foulds, Digraphs: Theory and Techniques, Gordon and Breach Science Publishers, New York, 1980. [19] A. D. Thomas and G. V. Wood, Group Tables, Shiva Publishing, Orpington, UK, 1980.

ADDITIVE GS-AUTOMATA AND SYNTACTIC NEARRINGS Yuen Fong Department of Mathematics, National Cheng-Kung University Tainan 701, TAIWAN [email protected]

Feng-Kuo Huang Department of Mathematics, National Taitung University Taitung 950, TAIWAN [email protected]

Chiou-Shieng Wang Department of Business Administration, Kao Yuan Institute of Technology Lujhu 821, TAIWAN [email protected]

Abstract

The purpose of this paper is to study those related algebraic structures of additive GS-automata and their associated syntactic nearrings in which the set of states are abelian groups. The properties of reachability and connectivity in GS-automata are studied. The endomorphism semigroups of additive GS-automata are investigated. Every ring can be embedded into the endomorphism ring of an additive GS-automaton. 2000 Mathematics Subject Classiﬁcation: 68Q70; 16Y30.

Keywords: additive GS-automaton, syntactic nearring

1.

Preliminaries

A group semiautomaton (in short, GS-automaton) or group state machine [16] is a triple (Q, X, δ), where (Q, +) as the set of states, is a group, X is the set of inputs and δ : Q × X → Q is the state transition function. The input set X can be extended to the free monoid X ∗ over X and the state transition function δ can be extended to X ∗ by deﬁn-

201 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 201–216. c 2005 Springer. Printed in the Neatherlands.

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ing δ(q, xy) = δ(δ(q, x), y) for all q ∈ Q and x, y ∈ X ∗ . For more on GS-automata, please refer to [5, 6, 7, 15]. Let S = (Q, X, δ) be a GS-automaton. An element x0 ∈ X is said to be a zero-input if δ(0, x0 ) = 0. A GS-automaton S = (Q, X, δ) is called additive if there exists a zero-input x0 ∈ X with the following properties: (i) decomposition property: δ(q, x) = δ(q, x0 ) + δ(0, x) for all q ∈ Q and x ∈ X; (ii) zero-input additivity: δ(q1 − q2 , x0 ) = δ(q1 , x0 ) − δ(q2 , x0 ) for all q1 , q2 ∈ Q. The deﬁnition of additive GS-automaton is motivated by Arbib [19, p.168] and initiated by Fong, Huang and Ke [5]. Here we note that both the decomposition property and the zero-input additivity can imply the existence of the “zero-input” in a GS-automaton. Let x0 ∈ X be chosen such that it satisﬁes the decomposition property: δ(q, x) = δ(q, x0 ) + δ(0, x) for all q ∈ Q and x ∈ X. Then, by letting q = 0 and x = x0 , we have δ(0, x0 ) = δ(0, x0 ) + δ(0, x0 ), which implies δ(0, x0 ) = 0 because Q is a group. Thus x0 is a zero-input in the GS-automaton S. On the other hand, if the element x0 satisﬁes the zero-input additivity: δ(q1 − q2 , x0 ) = δ(q1 , x0 ) − δ(q2 , x0 ) for all q1 , q2 ∈ Q. Then, letting q1 = q2 = 0, we have δ(0, x0 ) = δ(0, x0 ) − δ(0, x0 ), which also implies x0 is indeed a zero-input in the GS-automaton S. However, the following examples show that the decomposition property and the zero-input additivity are independent (i.e., they can not imply each other). Example 1.1. Let (R, +, ·) be any ring. The binary operation · : R × R → R deﬁned by ·(r, s) = rs in R gives the system R = (R, R, ·) a structure as a GS-automaton. Every element in R is a zero-input in R. They all satisfy the zero-input additivity but fail the decomposition property. However, if we choose a speciﬁed element r ∈ R, then the system (R, {r}, ·) is an additive GS-automaton. Example 1.2. Let Mc0 (G) = {φx : G → G | x ∈ G; 0φx = 0 and gφx = x if g = 0 for all g ∈ G} be a left nearring of group mappings on a group (G, +) with order |G| ≥ 3. Choose and ﬁx a nonzero x ∈ G and let φx ∈ Mc0 (G). Deﬁne a function δ : G × {φx } → G by δ(g, φx ) = gφx . Then (G, {φx }, δ) is a GS-automaton and φx is a zero-input which satisﬁes the decomposition property but fails the zero-input additivity. Additive GS-automata can be thought as a generalization for linear sequential machines [17] without output sets. On the other hand, it can

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be viewed as a generalization of several well known algebraic structures. Any given group (G, +) can be naturally associated with an additive GS-automaton where the state transition function δ is deﬁned by the binary operation “ + ” of G. Explicitly, we consider the GS-automaton G = (G, G, δ) where δ : G × G → G is deﬁned as δ(g, x) = g + x for all g, x ∈ G. Then the unity “0” in the group G is a zero-input and δ(g, x) = g + x = δ(g, 0) + δ(0, x), δ(g1 − g2 , 0) = g1 − g2 = δ(g1 , 0) − δ(g2 , 0). Thus G is an additive GS-automaton. This automaton is written as G = (G, G, +) and called the associated GS-automaton [2, 4, 5] for a given group G. The following example shows how to construct an additive GS-automaton from an arbitrary given GS-automaton by adding an extra input. Example 1.3. For any given GS-automaton S = (Q, X, δ), we can always deﬁne an additive GS-automaton by adding a new input into the system. Explicitly, let α be a new input alphabet not in X. Consider the system T = (Q, Y, η) where Y = X ∪{α}. Deﬁne the state transition function η : Q × Y → Q by q + δ(0, x) if x ∈ X, η(q, x) = q if x = α, for all q ∈ Q. The system T is clearly a well-deﬁned GS-automaton. We will show it is additive. Observe that η(0, α) = 0, and so α is a zero-input of the new system T . Now let q, q1 , q2 ∈ Q and x ∈ X, we have η(q, x) = q + δ(0, x) = η(q, α) + η(0, x) and η(q, α) = q = η(q, α) + η(0, α). Hence η satisﬁes the decomposition property. Further η(q1 − q2 , α) = q1 − q2 = η(q1 , α) − η(q2 , α), so η satisﬁes the zero-input additivity. Therefore T = (Q, Y, η) is an additive GS-automaton. We denote this additive GS-automaton by (Q, X ∪ {α}, δα ). Let (P, X, δ) and (Q, Y, µ) be GS-automata. A GS-automaton homomorphism (abbre. GSA homomorphism) from (P, X, δ) into (Q, Y, µ) is a pair of functions (ϕ, f ), where ϕ : P → Q is a group homomorphism and f : X → Y is a mapping such that (δ(p, x))ϕ = µ(pϕ, xf ) for all p ∈ P and x ∈ X. It is called an embedding if both ϕ and f are injective. Here we note that S may not be able to embed into the new system

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T in Example 1.3. If δ(0, x) = 0 for all x ∈ X in the system S, then η(q, x) = q + δ(0, x) = q and η(q, α) = q. Therefore T is a system in which all states are isolated. It is not know whether it is possible to embed any GS-automaton into an additive GS-automaton. The following result addresses this question. Theorem 1.4. A GS-automaton S = (Q, X, δ) can be embedded into an additive GS-automaton T = (Q, X ∪ {α}, δα ) if and only if there exists a group automorphism π of (Q, +) such that (δ(q, x))π = qπ + δ(0, x) for all q ∈ Q, x ∈ X. Proof. Let S = (Q, X, δ) be a GS-automaton and T = (Q, X ∪ {α}, δα ) an additive GS-automaton as deﬁned in Example 1.3. Let π be a group automorphism of Q and 1X : X → X ∪ {α} deﬁned as x1X = x for all x ∈ X. If (π, 1X ) is the embedding from S into T , then (δ(q, x))π = δα (qπ, x1X ) = δα (qπ, x) = qπ + δ(0, x). On the other hand, if π ∈ Aut(Q) exists and (δ(q, x))π = qπ + δ(0, x), then (π, 1X ) is the desired embedding. Proposition 1.5. A homomorphic image of an additive GS-automaton S = (Q, X, δ) is also additive. Proof. Let T = (Qϕ, Xf, µ) be the homomorphic image of S under a GSA homomorphism (ϕ, f ). Let x0 ∈ X be the zero-input in S. Then µ(0, x0 f ) = µ(0ϕ, x0 f ) = (δ(0, x0 ))ϕ = 0ϕ = 0, and thus x0 f is a zeroinput in T . Let q ∈ Q and x ∈ X. Observe that, µ(qϕ, xf ) = (δ(q, x))ϕ = [δ(q, x0 ) + δ(0, x)]ϕ = (δ(q, x0 ))ϕ + (δ(0, x))ϕ = µ(qϕ, x0 f ) + µ(0ϕ, xf ) = µ(qϕ, x0 f ) + µ(0, xf ); and

µ(q1 ϕ − q2 ϕ, x0 f ) = µ((q1 − q2 )ϕ, x0 f ) = (δ((q1 − q2 , x0 ))ϕ = [δ(q1 , x0 ) − δ(q2 , x0 )]ϕ = (δ(q1 , x0 ))ϕ − (δ(q2 , x0 ))ϕ = µ(q1 ϕ, x0 f ) − µ(q2 ϕ, x0 f ).

Therefore T is an additive GS-automaton. This completes the proof. Let S = (Q, X, δ) be a GS-automaton. We call T = (P, X, δ) a group subsemiautomaton of S if P is a subgroup of Q and δ(p, x) ∈ P for all

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p ∈ P , x ∈ X. A normal subgroup K of (Q, +) is called a kernel of a GS-automaton (Q, X, δ) if δ(q + k, x) − δ(q, x) ∈ K for all q ∈ Q, k ∈ K, x ∈ X. Clearly {0} and Q are the trivial kernels of (Q, X, δ). For more details on kernels of a GS-automaton, please refer to [6] or [7]. It is not diﬃcult to see that a group subsemiautomaton of an additive GSautomaton is still additive. The following proposition [5] characterizes the state transition function of an additive GS-automaton. Proposition 1.6. A GS-automaton S = (Q, X, δ) is additive if and only if (i) there exists a group homomorphism Ψ : Q → Q; (ii) there exists a zero-input x0 ∈ X and a map Υ : X → Q with x0 Υ = 0 such that δ(q, x) = qΨ + xΥ. According to Proposition 1.6, throughout, an additive GS-automaton will be written as (Q, X, (Ψ, Υ)) if no confusion arises.

2.

Reachability and connectivity

A state q of a semiautomaton S = (Q, X, δ) is called reachable from a state p if there exists an input word w ∈ X ∗ such that δ(p, w) = q. Two states p, q are called (completely) connected if p can reach q (and) or q can reach p. A subset C of Q is called connected if for any two states p, q ∈ C, there exists a ﬁnite sequence of states x0 = p, x1 , x2 , · · · , xn = q in C such that xi and xi+1 are connected for all i = 0, 1, · · · , n − 1. C is called completely connected if every state in C is reachable by any other. Explicitly, for any given states p, q ∈ C, there exists a word w ∈ X ∗ such that δ(p, w) = q. A maximal connected subset of Q is called a connected component of S. If S is any semiautomaton, then one can always ﬁnd a connected subsemiautomaton for a given state q ∈ Q by ﬁnding the connected component (denoted conn(q)) in S which contains q. Be aware that conn(q) is not necessarily a group subsemiautomaton of a GS-automaton due to the restriction that the set of states shall be a group. Motivated by the method commonly used in the algebraic theory of linear sequential machine, deﬁne the complex of state 0, denoted cp(0), as the group subsemiautomaton generated by the set reach(0) = {q ∈ Q | there exists w ∈ X ∗ such that δ(0, w) = q}. This set reach(0) contains all the states reachable from state 0. The following result is immediate. Proposition 2.1. The set reach(0) is connected and is contained in conn(0).

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In contrast to the linear case [10], reach(0) need not be completely connected even when S is additive as demonstrated in the following example. Example 2.2. Consider the additive GS-automaton S = (Z, {x, y}, (1, Υ)) where Z is the additive group of integers and {x, y} is the set of inputs. The map 1 : Z → Z is the identity map of Z, and the map Υ : {x, y} → Z is deﬁned as xΥ = 0, yΥ = 1. The system S can be represented by the following digraph. ···

y

/ •−2 X x

y

/ •−1 y X x

/ •0 x

y

X

/ •1 x

y

X

/ •2

y

X

/ ···

x

It can easily be seen that reach(0) = N ∪ {0} is connected but not completely connected. Whereas cp(0) = Z = conn(0) is connected. Consider the special situation when the input set X is singleton, say X = {x}. Let S = (Q, {x}, (Ψ, Υ)) be an additive GS-automaton. Then x must be the zero-input, i.e., xΥ = 0. Let Ψ be a group endomorphism of Q such that δ(q, x) = qΨ. It follows immediately that δ(0, x) = 0Ψ = 0. Hence reach(0) = cp(0) = 0. If q ∈ kerΨ, then δ(q, x) = qΨ = 0. That is the state q can reach the state 0 or q ∈ conn(0). Therefore kerΨ ⊆ conn(0). We write this as the following. Proposition 2.3. Let S = (Q, {x}, (Ψ, Υ)) be an additive GS-automaton with single input x. Then the set reach(0) = cp(0) = 0 and kerΨ ⊆ conn(0). Further, if Ψm = Ψ for some positive integer m, then kerΨ = conn(0). Proof. It suﬃces to show that conn(0) ⊆ kerΨ if Ψm = Ψ for some positive integer m. Choose a state q ∈ conn(0). Then, by deﬁnition of conn(0), there exists a ﬁnite sequence of states p0 = q, p1 , p2 , · · · , pn = 0 such that pi and pi+1 are connected for all i = 0, 1, 2, · · · , n − 1. Since pn−1 , pn = 0 are connected and reach(0) = 0, it follows that pn−1 can reach state 0. Thus there exists some word w ∈ X ∗ such that δ(pn−1 , w) = 0. It follows that pn−1 Ψ|w| = 0 where |w| is the length of the word w. If |w| = 0, then pn−1 = 0 ∈ kerΨ. If |w| > 0, then there exists a positive integer r < m such that Ψ|w| = Ψr since Ψm = Ψ by hypothesis. Thus pn−1 ∈ kerΨ|w| = kerΨr ⊆ kerΨm = kerΨ. Further, pn−2 and pn−1 are connected shows that pn−2 can reach pn−1 or pn−1 can reach pn−2 . If pn−2 can reach pn−1 , then there exists a nonnegative integer s such that pn−2 Ψs = pn−1 and thus pn−2 Ψs+1 = 0. If pn−1 can reach pn−2 , then there exists a nonnegative integer t such that pn−1 Ψt =

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pn−2 and thus pn−2 Ψ = pn−2 Ψm = pn−1 Ψm+t = 0. Hence pn−2 ∈ kerΨ. Inductively, q = p0 ∈ kerΨ. This completes the proof. Proposition 2.4. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton. Then every element in reach(0) is a ﬁnite sum of the elements in the set {xΥΨi | x ∈ X for i = 0, 1, 2, · · · }, and conn(0) contains the set kerΨn ∪ reach(0) for all positive integer n. Proof. Let w ∈ X ∗ . Consider ﬁrst the case when |w| = 2, say w = xy where x, y ∈ X. Observe that δ(0, w) = δ(0, xy) = δ(δ(0, x), y) = (0Ψ + xΥ)Ψ + yΥ = xΥΨ + yΥ. Inductively, it can be seen that if the length of the word w is n, say w = x1 x2 · · · xn , then δ(0, w) =

n

xi ΥΨn−i .

i=1

Thus every element in reach(0) is a ﬁnite sum of elements in the set {xΥΨi | x ∈ X for i = 0, 1, 2, · · · }. The second assertion is routine to verify. Theorem 2.5. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton. Then cp(0) is generated additively by the set {xΥΨi | x ∈ X, i = 0, 1, 2, · · · }. Thus cp(0) is Ψ-invariant (i.e., cp(0)Ψ ⊆ cp(0)). Proof. Note that yΥΨj ∈ reach(0) for all y ∈ X and j ∈ N ∪ {0}. Explicitly, consider the word w = y1 y2 · · · yj +1 of length j + 1 where y1 = y and y2 = y3 = · · · = yj +1 = x0 is the zero-input. Then yΥΨj =

j+1

yi ΥΨ(j+1)−i = δ(0, w) ∈ reach(0).

i=1

Thus {xΥΨi | x ∈ X, i = 0, 1, 2, · · · }, + ⊆ reach(0) ⊆ cp(0). On the other hand, cp(0) ⊆ {xΥΨi | x ∈ X, i = 0, 1, 2, · · · }, + by Proposition 2.4. The following corollary considers the special case when the state group is abelian.

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Corollary 2.6. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton. If Q is an abelian group, then n gi Ψi | gi ∈ QX, + for some n ∈ N} cp(0) = { i=0

where QX, + is the group generated additively by the set QX = {δ(q, x) | q ∈ Q, x ∈ X}. Proof. Let A = { ni=0 gi Ψi | gi ∈ QX, + for some n ∈ N}. It is clear that 0 ∈ A and reach(0) ⊆ A by Proposition 2.4. Therefore cp(0) ⊆ A if A is a group subsemiautomaton of S. It suﬃces to show that A is a subgroup = {δ(a, x) | a ∈ A, x ∈ X} ⊆ A. AX nof Q and m i Let i=0 gi Ψ , j=0 hj Ψj ∈ A and x ∈ X. Without loss of generality, n j j we may suppose n ≥ m and rewrite m j=0 hj Ψ as j=0 hj Ψ where hm+1 = hm+2 = · · · = hn = 0. Now n

gi Ψi −

i=0

m

hj Ψj =

j=0

n

gi Ψi −

i=0

=

n

n

hj Ψj

j=0

(gi − hi )Ψi ∈ A,

i=0

and

δ

n

i

gi Ψ , x

=

i=0

= = =

n

i

gi Ψ Ψ + xΥ i=0 (g0 Ψ0 + g1 Ψ1 + · · · + gn Ψn )Ψ + xΥ g0 Ψ1 + g1 Ψ2 + · · · + gn Ψn+1 + (xΥ)Ψ0 n+1

hi Ψi ∈ A,

i=0

where h0 = xΥ = δ(0, x) and hj = gj −1 for all j = 1, 2, · · · , n + 1. Hence A is a GS-automaton. On the other hand, cp(0) is generated by the set {xΥΨi | x ∈ X, i = 0, 1, 2, · · · } by Theorem 2.5. Hence cp(0) contains A and the assertion is proved. Proposition 2.7. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton. If Q is an abelian group, then cp(0) is a kernel of S. Proof. Let k ∈ cp(0), q ∈ Q, x ∈ X. Then δ(q + k, x) − δ(q, x) = ((q + k)Ψ + xΥ) − (qΨ + xΥ) = qΨ + kΨ + xΥ − xΥ − qΨ = kΨ

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By Theorem 2.5, we have δ(q + k, x) − δ(q, x) ∈ cp(0) and thus cp(0) is a kernel of S. The following example shows that cp(0) need not be a kernel of S if Q is not an abelian group. Example 2.8. Let S = (S3 , {x, y}, (1, Υ)) be an additive GS-automaton where S3 = {0, (1, 2), (1, 3), (2, 3), (1, 2, 3), (1, 3, 2)} is the symmetric group of degree 3 and the function 1 is the identity map of S3 . Deﬁne Υ : {x, y} → S3 via xΥ = 0 and yΥ = (1, 2). A routine check will show that cp(0) = {0, (1, 2)} which is not a normal subgroup of S3 and hence is not a kernel of S. We remark that cp(0) is completely connected. Explicitly, cp(0) can be represented by the following digraph. y x

•(1,2)

7 •0 ^

t

x

y

3.

Endomorphisms of additive GS -automata

Given a GS-automaton S = (Q, X, δ), a group endomorphism π of Q is called a GSA endomorphism of S if δ(q, x)π = δ(qπ, x) for all q ∈ Q, x ∈ X. The set of all GSA endomorphisms of S will be denoted by End(S). Observe that, by using function composition as operation, End(S) is a subsemigroup of End(Q), the endomorphism semigroup of the state group Q. We remark here that the deﬁnition of GSA endomorphism is slightly diﬀerent from the previous one deﬁned in Section 1. However, this is commonly used when we investigate mappings of GS-automaton with identical set of inputs [1, 9, 20]. The following proposition characterizes the endomorphisms of an additive GS-automaton. Proposition 3.1. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton and π a group endomorphism of Q. Then the map π is a GSA endomorphism of S if and only if Ψπ = πΨ and Υπ = Υ. Proof. Assume π is a GSA endomorphism of S = (Q, X, (Ψ, Υ)). Then δ(q, x)π = δ(qπ, x) implies (qΨ + xΥ)π = (qπ)Ψ + xΥ for all q ∈ Q, x ∈ X. It follows that qΨπ + xΥπ = qπΨ + xΥ, for all q ∈ Q, x ∈ X. In particular, by letting x = x0 be the zero-input in S, it follows that qΨπ = qπΨ for all q ∈ Q and thus Ψπ = πΨ. It remains to show that

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(xΥ)π = xΥ for all x ∈ X. Since (qΨ)π + (xΥ)π = qπΨ + xΥ and Ψπ = πΨ, it follows immediately that (xΥ)π = xΥ for all x ∈ X. Thus Υπ = Υ. Conversely, δ(q, x)π = (qΨ + xΥ)π = (qΨ)π + xΥπ = (qπ)Ψ + xΥ = δ(qπ, x), for all q ∈ Q and x ∈ X. Therefore π is a GSA endomorphism of S. Corollary 3.2. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton and π ∈ End(S), then cp(0) is π-invariant. Moreover, π acts as an identity map on cp(0). Proof. By Theorem 2.5, it suﬃces to consider the generating elements xΥΨi for cp(0). Since (xΥΨi )π = ((xΥ)π)Ψi = xΥΨi by Proposition 3.1, we see that π ﬁxes every generator of cp(0) and thus ﬁxes every element in cp(0). Corollary 3.3. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton. If S is completely connected, then the identity map is the unique endomorphism of S. Proof. If S is completely connected, then reach(0) = Q = cp(0). Thus the identity map is the only endomorphism of S by Corollary 3.2. The GS-automaton G = (G, G, +) deﬁned in Section 1 is completely connected, thus End(G) = {1} by Corollary 3.3. The remaining results in this section characterize the endomorphisms of additive GS-automata with abelian state groups. Theorem 3.4. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton where Q is an abelian group and 1 is the identity map on Q. Deﬁne E(S) = {α ∈ End(Q) | (qΨ + xΥ)α = qαΨ for all q ∈ Q, x ∈ X}. Then we have: (1) End(S) = {α + 1 | α ∈ E(S)}; (2) αΨ = Ψα for all α ∈ E(S); (3) cp(0)α = 0 for all α ∈ E(S). Proof. (1) Let π ∈ End(S). Since (Q, +) is abelian, the map π − 1 is also a group homomorphism of Q. Further, by Proposition 3.1, (qΨ + xΥ)(π − 1) = (qΨ + xΥ)π − (qΨ + xΥ)1 = qΨπ + xΥ − xΥ − qΨ = q(π − 1)Ψ.

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Hence End(S) ⊆ {α + 1 | α ∈ E(S)}. On the other hand, let α ∈ E(S). Then α + 1 is a group endomorphism of Q, and δ(q, x)(α + 1) = (qΨ + xΥ)(α + 1) = (qΨ + xΥ)α + (qΨ + xΥ)1 = qαΨ + qΨ + xΥ = q(α + 1)Ψ + xΥ = δ(q(α + 1), x). Hence α + 1 ∈ End(S) and so {α + 1 | α ∈ E(S)} ⊆ End(S). For (2) and (3), let α ∈ E(S), then α + 1 ∈ End(S) by (1). From Proposition 3.1, we see that (α + 1)Ψ = Ψ(α + 1) and (xΥ)(α + 1) = xΥ for all x ∈ X. Therefore αΨ = Ψα and (xΥ)α = 0 for all x ∈ X. Thus cp(0)α = 0 by Theorem 2.5. Corollary 3.5. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton where Q is an abelian group, then E(S) is a ring. Proof. Let E(S) = {α ∈ End(Q) | (qΨ + xΥ)α = qαΨ for all q ∈ Q, x ∈ X} ⊆ End(Q). We will show that E(S) is a ring. It is clear that 0 ∈ E(S). If α ∈ E(S) then 0 = (qΨ + xΥ)(α + (−α)) = qαΨ + (qΨ + xΥ)(−α). This implies (qΨ + xΥ)(−α) = −(qαΨ) = q(−α)Ψ. Hence −α ∈ E(S). Moreover, let α, β ∈ E(S), then (qΨ + xΥ)(α − β) = (qΨ + xΥ)α + (qΨ + xΥ)(−β) = qαΨ + q(−β)Ψ = q(α − β)Ψ, and (qΨ + xΥ)(αβ) = (qαΨ)β = q(αβ)Ψ by Theorem 3.4(2). Hence α − β, αβ ∈ E(S) and thus E(S) is a ring. Note that End(S) is not a ring even when S is an additive GSautomaton with abelian state group. The following example demonstrates this assertion. Example 3.6. Let S = (Z, {x, y}, (1, Υ)) be an additive GS-automaton deﬁned in Example 2.2. A routine calculation shows that End(S) = {1} and E(S) = {0}. Thus End(S) is not a ring. We may ask: when is End(S) a ring? The following results address this question. Theorem 3.7. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton where Q is an abelian group. Then End(S) is a centralizer ring of the

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semigroup generated by Ψ in End(Q) if and only if every input in S is a zero-input. Proof. If End(S) is a ring, then the identity map 1 = (1 + 1) − 1 ∈ E(S) by Theorem 3.4(1). Thus cp(0) = 0 by Theorem 3.4(3). Therefore xΥ = 0 for all x ∈ X by Theorem 2.5. Hence every input in S is a zero-input. Conversely, assume every input in S is a zero-input. Then xΥ = 0 by deﬁnition for all x ∈ X. Thus End(S) is a centralizer ring of the semigroup generated by Ψ in End(Q) by Proposition 3.1.

Theorem 3.8. Every ring can be embedded into the endomorphism ring of a suitable additive GS-automaton. Proof. Every ring R is isomorphic to a subring of the endomorphism ring End(A) for some abelian group A. Using this group A, deﬁne an additive GS-automaton S = (A, X, (1, Υ)) where 1 : A → A is the identity map and Υ : X → A is deﬁned as xΥ = 0 for all x ∈ X. Thus every input in S is a zero-input. By Theorem 3.7, End(S) is the centralizer ring of the semigroup {1} in End(A), thus End(S) ∼ = End(A). Hence every ring can be embeded into End(S).

4.

Syntactic nearrings

Given a GS-automaton S = (Q, X, δ), for each x ∈ X, we consider the group mapping fx : Q → Q deﬁned by qffx = δ(q, x) for all q ∈ Q. The syntactic nearring associated to the GS-automaton S = (Q, X, δ), denoted N (S), is the nearring generated by the set {ffx : Q → Q | x ∈ X} ∪ {1}, where 1 denotes the identity mapping of the group Q. For more on syntactic nearrings see [2, 3, 4, 5, 12, 17, 24]. For an additive GS-automaton S = (Q, X, (Ψ, Υ)) with Q an abelian group, N (S) is an abstract aﬃne nearring [5]. Thus some properties of the syntactic nearring N (S) are evident. But here N0 (S) is generated additively by the commutative cyclic semigroup Ψ, · joint with the identity map of Q [5, Theorem 3] and thus N0 (S) is a commutative ring with unity. Hereforth 0 in index and c in index will indicate the 0-symmetric and constant part of a nearring. Thus N0 and Nc will stand for the 0symmetric and the constant part of a nearring N , respectively. Elements of N0 and Nc will be indexed too by 0 and c respectively. For nearring terminology, please refer to [22] but be aware that we are using left nearrings here and [22] is using right nearrings.

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Proposition 4.1. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton where Q is an abelian group. Then a subgroup R of (N (S), +) is a right ideal of N (S) if and only if RΨ ⊆ R. Proof. Let R be a right ideal of N (S). Pick r ∈ R, Ψ ∈ N (S). Then rΨ = (0 + r)Ψ − 0Ψ ∈ R or RΨ ⊆ R. On the other hand, suppose that R is a subgroup of (N (S), +) and RΨ ⊆ R. Obviously RN N0 (S) ⊆ R for N0 (S) is generated by the set {Ψ, 1}. Let r ∈ R and f, g ∈ N (S), then (f + r)g − f g = (f + r)(g0 + gc ) − f (g0 + gc ) = f g0 + rg0 + gc − f g0 − gc = rg0 ∈ R, since g0 ∈ N0 (S) which is generated additively by {Ψ, 1}. Therefore R is a right ideal of N (S) and the assertion holds. Proposition 4.2. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton where Q is an abelian group. If W is a subgroup of (N (S), +) then the following are equivalent. (1) W is an ideal of N (S). W0 ⊆ W. (2) W N0 (S) ⊆ W and N (S)W W0 ⊆ W c . (3) W Ψ ⊆ W , ΨW ⊆ W , and Nc (S)W Proof. (1) ⇒ (2) Let w ∈ W and α ∈ N0 (S). Then wα = (0+w)α−0α ∈ W . Further, N (S)W W0 ⊆ N (S)W ⊆ W is clear. (2) ⇒ (1) Let f, g = g0 + gc ∈ N (S) and w ∈ W . Then (f + w)g − f g = (f + w)(g0 + gc ) − f (g0 + gc ) = f g0 + wg0 + gc − f g0 − gc = wg0 ∈ W, and thus W is a right ideal of N (S). Moreover f w = f (w0 + wc ) = f w0 + wc ∈ W + Wc = W . Therefore W is an ideal of N (S). (2) ⇒ (3) Since Ψ ∈ N0 (S), it follows immediately that W Ψ ⊆ W Wc ⊆ W + Wc = W . Moreover N (S) is an abstract and ΨW ⊆ ΨW W0 + ΨW aﬃne nearring, Nc (S) is an ideal of N (S) and so Nc (S) is right N0 (S) N0 (S) ⊆ Nc (S). Therefore Nc (S)W W0 ⊆ Nc (S). invariant, i.e., Nc (S)N W0 ⊆ N (S)W W0 ⊆ W by hypothesis. Thus On the other hand, Nc (S)W W0 ⊆ W ∩ Nc (S) = Wc . Nc (S)W (3) ⇒ (2) Letw ∈ W and f0 = ni=0 ki Ψi ∈ N0 (S) where k ∈ Z. the other hand, Then wff0 = w( ni=0 ki Ψi ) = ni=0 ki (wΨi ) ∈ W. On n i + f ∈ N (S) and w ∈ W . Then f w = ( let f = f 0 c 0 0 0 0 i=0 ki Ψ )w0 = n i f0 +ffc )w0 = f0 w0 +ffc w0 ∈ W0 +W Wc = i=0 ki (Ψ w0 ) ∈ W0 and f w0 = (f W. Hence N (S)W W0 ⊆ W . This completes the proof.

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From Proposition 4.2, we see that every two sided invariant subgroup T of N (S) is an ideal. The following lemma characterizes the ideal structure of N0 (S). Lemma 4.3. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton with Q an abelian group and M a subgroup of (N N0 (S), +). Then the following are equivalent. (1) M is an ideal of N0 (S). (2) M N0 (S) ⊆ M . (3) N0 (S)M ⊆ M . (4) M Ψ ⊆ M . (5) ΨM ⊆ M . Proof. These equivalent conditions follow immediately from the fact that N0 (S) is a commutative ring generated by the set {1, Ψ}. Proposition 4.4. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton with Q an abelian group and N (S) the syntactic nearring associated with S. Then every right invariant subgroup T of N (S) is two-sided invariant and T = T0 + Nc (S) where T0 is an ideal of N0 (S). Proof. Let T be a subgroup of (N (S), +) satisfying T N (S) ⊆ T . Then T is a nearring. Decompose T as group direct sum T = T0 + Tc where Tc ⊆ Nc (S). Observe that Nc (S) = T Nc (S) ⊆ T N (S) ⊆ T and thus Nc (S) ⊆ Tc . This gives Tc = Nc (S). Since N (S) is abstract aﬃne, Nc (S) is an ideal of N (S) and hence an ideal of T . Now let t ∈ T , f ∈ N (S), then f t = (ff0 + fc )(t0 + tc ) = f0 t0 + fc t0 + tc . Since N0 (S) is commutative, f0 t0 = t0 f0 ∈ T N (S) ⊆ T . On the other hand fc t0 + tc ∈ Nc (S)T + T = Tc T + T = T . It follows that f t ∈ T or N (S)T ⊆ T and thus T is two-sided invariant. Finally, T0 is an ideal of N0 (S) for T0 Ψ ⊆ T0 N0 (S) ⊆ T0 by Lemma 4.3. By using the above results and the fact that N (S) is an abstract aﬃne nearring, the following result describes the main types of substructures of the syntactic nearring N (S). Theorem 4.5. Let S = (Q, X, (Ψ, Υ)) be an additive GS-automaton with Q an abelian group and N (S) the syntactic nearring associated with S. Then: (1) T is a two-sided invariant subgroup of N (S) if and only if T = T0 + Nc (S) is a subgroup of N (S) where T0 is an ideal of N0 (S). (2) L is a left ideal of N (S) if and only if L = L0 + Lc where L0 is an ideal of N0 (S) and Lc is a subgroup of (N Nc (S), +) with Nc (S)L0 ⊆ Lc .

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(3) I is an ideal of N (S) if and only if I = I0 + Ic where I0 is an Nc (S), +) satisfying Ic Ψ ⊆ Ic , ideal of N0 (S) and Ic is a subgroup of (N Nc (S)II0 ⊆ Ic . Proof. (1) The suﬃciency follows from Proposition 4.4, we show only the necessity. Assume T = T0 + Nc (S) where T0 is an ideal of N0 (S). Then (T T0 + Nc (S))N (S) = (T T0 + Nc (S))N N0 (S) + Nc (S) N0 (S) + Nc (S) = T0 N0 (S) + Nc (S)N ⊆ T0 + Nc (S) = T. That is, T is right invariant and thus T is two-sided invariant by Proposition 4.4. (2) Let L be a left ideal of N (S). Write L as the group direct sum L = L0 + Lc . Since ΨL0 ⊆ N0 (S)L0 ⊆ L0 by hypothesis, then L0 is an ideal of N0 (S) by Lemma 4.3. Further, Nc (S)L0 ⊆ N (S)L0 ⊆ L and Nc (S)L0 ⊆ Nc (S), thus Nc (S)L0 ⊆ L ∩ Nc (S) = Lc . On the other hand, let L = L0 + Lc be a subgroup of (N (S), +) where L0 is an ideal of N0 (S) and Nc (S)L0 ⊆ Lc , then N (S)L = (N N0 (S) + Nc (S))L0 + Lc = N0 (S)L0 + Nc (S)L0 + Lc ⊆ L0 + Lc = L. Therefore L is a left ideal of N (S). (3) Let I be an ideal of N (S). Then I0 is an ideal of N0 (S) by (2) above. The remaining assertions follow from (2) and Proposition 4.1. On the other hand, let I = I0 + Ic be a subgroup of (N (S), +) such Nc (S), +) satisfying that I0 is an ideal of N0 (S) and Ic is a subgroup of (N Ic Ψ ⊆ Ic and Nc (S)II0 ⊆ Ic . Clearly, I is a left ideal of N (S) by (2). Moreover, IΨ = (II0 + Ic )Ψ = I0 Ψ + Ic Ψ ⊆ I0 + Ic = I. It follows that I is a right ideal of N (S) by Proposition 4.1. This completes the proof. Corollary 4.6. Every right invariant subgroup of N (S) is an ideal of N (S). Proof. This follows immediately from Proposition 4.4 and Theorem 4.5.

References [1] M. A. Arbib, Automata automorphisms, Inform. Contr. 11 (1967) 147–154.

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[2] J. R. Clay, Y. Fong, On syntactic nearrings of even dihedral groups, Results Math. 23 (1993) 23–44. [3] Y. Fong, On the structure of abelian syntactic near-rings, in: Algebraic structure and number theory (Hong Kong, 1988), 114–123, World Sci. Publishing, 1990. [4] Y. Fong, J. R. Clay, Computer program for investigating syntactic nearrings of ﬁnite group semiautomata, Bull. Inst. Math. Acad. Sinica 16 (1988) 295–304. [5] Y. Fong, F.-K. Huang, W.-F. Ke, Syntactic near-rings associated with group semiautomata, Pure Math. Appl. Ser. A 2 (1992) 187–204. [6] Y. Fong, F.-K. Huang, C.-S. Wang, Group semiautomata and their related topics, in: Words, Languages and Combinatorics II (Kyoto, 1992), 155–169, World Sci. Publishing, 1994. [7] Y. Fong, F.-K. Huang, R. Wiegandt, Radical theory for group semiautomata, Acta Cybernet. 11 (1994) 169–188. [8] F. G´ ´ecseg, I. Pe´ ´ ak, Algebraic theory of automata, Akad´ ´emiai Kiad´ ´ o, 1972. [9] S. C. Geller, P. Natarajan, K. C. Smith, Endomorphisms of group automata, Math. Panon. 4 (1993) 79–93. [10] M. A. Harrison, Lectures on linear sequential machines, Academic Press, 1969. [11] G. Hofer, Ideals and reachability in machines, in: Near-rings and near-ﬁelds (T¨ u ¨bingen, 1985), 123–131, North-Holland, 1987. [12] —, Reachability in machines, in: Rings, modules and radicals (Hobart, 1987), 88–96, Pitman Res. Notes Math. Ser. 204, Longman Sci. Tech., 1989. [13] —, Syntactic rings, Results Math. 15 (1989) 245–254. [14] —, Left ideals and reachability in machines, Theoret. Comput. Sci. 68 (1989) 49–56. [15] G. Hofer, G. Pilz, Group automata and near-rings, Contri. General Algebra 2 (Klagenfurt, 1982), 153–162, H¨ ¨ older-Pichler-Tempsky, 1983. [16] M. Holcombe, Algebraic automata theory, Cambridge Studies in Advanced Mathematics 1, Cambridge University Press, 1982. [17] —, The syntactic near-ring of a linear sequential machine, Proc. Edinburgh Math. Soc. 26 (1983) 15–24. [18] —, A radical for linear sequential machines, Proc. Roy. Irish Acad. sec. A 84 (1984) 27–35. [19] R. E. Kalman, P. L. Falb, M. A. Arbib, Topics in mathematical system theory, McGraw-Hill Book Co., 1969. [20] C. J. Maxson, K. C. Smith, Endomorphisms of linear automata, J. Comput. System Sci. 17 (1978) 98–107. [21] —, Automorphisms of linear automata, J. Comput. System Sci. 19 (1979) 18–26. [22] G. Pilz, Near-rings, North-Holland, 1983. [23] —, Near-rings and nonlinear dynamical system, in: Near-rings and near-ﬁelds (T¨ u ¨bingen, 1985), 211–232, North-Holland, 1987. [24] —, Strictly connected group automata, Proc. Roy. Irish Acad. sec. A 86 (1986) 115–118.

ON THE NILPOTENCE OF THE S -RADICAL IN MATRIX NEAR-RINGS John F.T. Hartney Univ. of Witwatersrand South Africa [email protected]

Anthony M. Matlala Univ. of Witwatersrand South Africa [email protected]

Abstract

1.

Let R be a zero-symmetric near-ring with identity and Mn (R) the matrix near-ring associated with R. We prove that the s-radicals Js (R) Js (R))+ ⊆ Js (M Mn (R)), where Mn (R) satisﬁes and Js (Mn (R)) satisfy (J the DCCL. We also show that A+ ⊆ A , where A and A are the ssocles of R and Mn (R), respectively. The s-socle of a near-ring R is the unique minimal ideal modulo which Js (R) is non-zero and nilpotent. We further conjecture about the relationship between A and A∗ .

Introduction

The relationship between special ideals of a near-ring, for example the Jacobson-type radicals, and the ideals of its associated Matrix near-ring has been of a great interest since the Matrix near-rings were ﬁrst deﬁned by J.D.P. Meldrum and A.P.J. van der Walt in 1986, [6]. We focus our attention on the relationship between the s-radical and the s-socle of these near-rings. Throughout this paper the near-ring R is assumed to be a right zerosymmetric near-ring with identity 1. If the near-ring R satisﬁes the descending chain condition for R-subgroups (left ideals), we say R satisﬁes the DCCS (DCCL). Our R-groups G will be assumed to be unitary. That is 1 · g = g for all g ∈ G. Kernels of R-homomorphisms will be called R-kernels. For a natural number n we deﬁne Rn to be the direct sum of n copies of the group (R, +) and its elements will be written as 217 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 217–224. c 2005 Springer. Printed in the Neatherlands.

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ρ¯ = r1 , r2 , . . . , rn , ri ∈ R. M0 (G) will denote the set of zero ﬁxing maps of the group G into itself. For r ∈ R and 1 ≤ i, j ≤ n we deﬁne the function fijr : Rn → Rn by r πj (¯)) for each ¯ ∈ Rn , where ιi : R → Rn and πi : Rn → R fij (¯) = ιi (rπ are the i-th injection and projection functions, respectively. The subnear-ring of M0 (Rn ) generated by the set {ffijr |r ∈ R, 1 ≤ i, j ≤ n} is called the n × n matrix near-ring over R and it is denoted by Mn (R). For an ideal I R, there are two ways to construct an ideal in Mn (R) which relate naturally to I (see [8] ), namely I + := Id ffija |a ∈ I, 1 ≤ i, j ≤ n and

ρ ∈ Rn }. I ∗ := {U ∈ Mn (R)|U ρ¯ ∈ I n , ∀¯

We refer the reader to [6] and [8] for further results in matrix near-rings. In section 2 we collect necessary deﬁnitions and known results needed in the sequel and in section 3 our two main results are proved.

2.

Preliminaries

We will from now on assume that R satisﬁes the DCCS, unless otherwise stated. Deﬁnition. [1] An R-group Ω of type-0 is said to be of type-s if for all k ω ∈ Ω, ω = 0, Rω := Ωi , where each Ωi is of type-0 and an R-kernel i=1

of Rω. An ideal of R is s-primitive if it is the annihilating ideal of an R-group of type-s. The s-radical, Js (R), is the intersection of all s-primitive ideals of R. R is said to be s-primitive if the zero ideal is an s-primitive ideal. For the basic results on the s-socle and its relationship with nil-rigid series we refer the reader to [2]. We remind the reader that in the nil-rigid series L1 < C1 < L2 < C2 < · · · < Lα < Cα = R Ci−1 ) = Li /C Ci−1 , where i = 2, 3, . . . , α , and for R, we have J0 (R/C Soi(R/Lj ) = Cj /Lj , for j = 1, 2, . . . , α. Other than in the statement of the next theorem nil-rigid series will play no further role in this article. The following two theorems are of fundamental importance in the work that follows.

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Theorem 2.1. [2] Let R be a near-ring which satisﬁes the DCCS with a nil-rigid length α > 1, that is Lα < Cα = R. Then there exists an ideal A of R such that A is uniquely minimal amongst all ideals B of R for which Js (R/B) is non-zero and nilpotent. Moreover, A ≤ Js (R) ∩ Cα−1 and Js (R/A) = Js (R)/A. Theorem 2.2. [9] If Ω is a monogenic R-group, then Ω has no nontrivial R-subgroups (resp. R-kernels) iﬀ Ωn has no non-trivial Mn (R)subgroups (resp. Mn (R)-kernels ). Deﬁnition. The unique minimal ideal A of Theorem 2.1 is called the s-socle of R.

3.

Proofs of the main theorems

J.D.P. Meldrum and J.H. Meyer in [5] conjectured that : If R is a zero-symmetric near-ring with identity, then (J J0 (R))+ ⊆ J0 (Mn (R)). Certainly, if J0 (R) is nilpotent, the result follows from Corollary 8 of [8]. For the s-radical of a near-ring R, we can at least show that if Mn (R) satisﬁes the DCCL, then (J Js (R))+ ⊆ Js (Mn (R)). We need the following theorem to prove this result. Theorem 3.1. [3] Let R be a near-ring which satisﬁes DCCL and let Ω be a faithful R-group. Any R-group of type-0 is the homomorphic image of some monogenic R-subgroup Rω of Ω. Thus, if Mn (R) satisﬁes the DCCL, then any Mn (R)-group Γ of type-s is a homomorphic image of some monogenic Mn (R)-subgroup of Rn , say K n . Let Ψ : K n → Γ be the Mn (R)-homomorphism and Ln := ker(Ψ), so that K n /Ln ∼ =Mn (R) Γ. It is well known that [7] (K/L)n ∼ = K n /Ln .

(3.1)

Lemma 3.2 Let R be a zero-symmetric near-ring with identity such that Mn (R) satisﬁes the DCCL, and K n /Ln be an Mn (R)-group of types. Then (0 : K/L) is an intersection of s-primitive ideals of R. Proof. For any k ∈ K, consider M := (Mn (R))(k, 0, ..., 0 + Ln ), a monogenic Mn (R)-subgroup of K n /Ln . l Mi , where Now K n /Ln is an Mn (R)-group of type-s, hence M = i=1

each Mi is of type-0 and an Mn (R)-kernel of M . Suppose

k, 0, ..., 0 + Ln = (k11 , ..., kn1 + Ln ) + ... + (k1l , ..., knl + Ln ),

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where each k1i , ..., kni + Ln is an Mn (R)-generator of Mi . This decomposition is unique modulo Ln , so that we can choose kj ∈ Mj such that k, 0, ..., 0 + Ln = (k1 , 0, ..., 0 + Ln ) + ... + (kl , 0, ..., 0 + Ln ). It follows readily that (Rk + L)/L = (Rk1 + L)/L ⊕ · · · ⊕ (Rkl + L)/L, where each (Rkj + L)/L is of type-0 and an R-kernel of (Rk + L)/L. l Thus (0 : k + L) = (0 : kj + L) is the intersection of maximal left j=1 (0 : k + L) an intersection of ideals. Consequently, (0 : K/L) = k∈K

s-primitive ideals, by Theorem 2.11 of [1]. We can now prove our ﬁrst main theorem. Theorem 3.3. Let R be a zero-symmetric near-ring with identity such that Mn (R) satisﬁes DCCL. Then (J Js (R))+ ⊆ Js (Mn (R)). Proof. Let Γ be an Mn (R)-group of type-s, then Γ ∼ =Mn (R) K n /Ln , for some Mn (R)-subgroups of K n , and Ln of Rn . By Lemma 3.2, (0 : Js (R))+ , σ ∈ Js (R), we have K/L) ⊇ Js (R). Thus, for any fijσ ∈ (J σ n fij (k1 , .., kn +L ) = 0, .., 0, σkj , 0, .., 0+Ln = 0, 0, . . . , 0+Ln . Hence fijσ ∈ (0 : K n /Ln ), and therefore fijσ ∈ (0 : Γ). Since Γ is arbitrary, we have fijσ ∈ Js (Mn (R)). We use the following example of a ﬁnite Matrix near-ring from J.D.P. Meldrum and J.H. Meyer [5] to show that the inclusion in Theorem 3.3 can be strict. Example 1. Let G := Z2 ⊕ Z2 ⊕ Z2 ⊕ Z2 , N := Z2 ⊕ Z2 ⊕ Z2 ⊕ {0}, and M := Z2 ⊕ Z2 ⊕ {0} ⊕ {0}. Let Mi , i ∈ {1, 2, 3}, be the 2-element subgroups of M , and Nj , j ∈ {1, 2, 3, 4}, be the 2-element subgroups of N but not of M , and g1 , g2 , . . . , g8 be elements of G \ N. Deﬁne Mi ) ⊆ Mi , ∀i ∈ {1, 2, 3}; f (N Nj ) ⊆ a near-ring R := {f ∈ M0 (G) | f (M ∀ ∈ {1, 2, 3, 4}; Nj , ∀j ∀g, g ∈ G, g − g ∈ M ⇒ f (g) − f (g ) ∈ M ; ∀g, g ∈ G, g − g ∈ N ⇒ f (g) − f (g ) ∈ N }. Also, deﬁne F := {f ∈ R | f (gi ) ∈ M, i = 1, 2, . . . , 8; otherwise ¯0} and H := {f ∈ R | f (gi ) ∈ N, i = 1, 2, . . . , 8; otherwise ¯0} It was observed in [5] that (a) M and N are R-kernels of G, and F and H are R-kernels of (R, +), (b) J0 (R) = AnnR N ∩ AnnR (G/N ), (c) the Mi ’s, i = 1, 2, 3; the Nj ’s, j = 1, 2, 3, 4; and G/N are R-groups of type-2. Thus J0 (R) = J2 (R),

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(d) [(J J0 (R))+ ]2 = (0), and (e) J0 (Mn (R)) has an element V such that V 2 = 0. Therefore (J J0 (R))+ = J0 (M2 (R)). We note that J0 (R) = Js (R) = Js (R))+ = (J J0 (R))+ J0 (M2 (R)) ⊆ Js (M2 (R)). J2 (R), and thus (J We note that since the s-socle A ⊆ Js (R) by deﬁnition, then A+ ⊆ (J Js (R))+ by Proposition 5 of [8]. Theorem 3.4. Let R be a zero-symmetric near-ring with identity such Js (R))+ is nilpotent modulo A+ . that Mn (R) satisﬁes the DCCS. Then (J Proof. We need to show that if U ∈ (J Js (R))+ then U m ∈ A+ , for some σ positive integer m. If U = fij , σ ∈ Js (R), with σ m ∈ A, then from 0 if i = j fij σ m m (ffij ) = if i = j, fijσ Js (R))k ⊆ A for some positive we see that (ffijσ )m ∈ A+ . In fact, if (J σk σ1 σ2 ···σk σ1 σ2 integer k, then f11 f11 · · · f11 = f11 ∈ A+ , where each σi ∈ Js (R), i = 1, 2, . . . , k. This provides the starting point for a recursive proof based on the way in which (J Js (R))+ is generated by the set {ffijσ | σ ∈ Js (R), 1 ≤ i, j ≤ n}. We recall the following from [8]. Deﬁnition. Let D be an ideal of Mn (R), then D∗ := {r ∈ R|r = πj (U ρ¯), for some U ∈ D, ρ¯ ∈ Rn , 1 ≤ j ≤ n} is an ideal of R. We have the following chain of ideals in Mn (R). Proposition 3.5. [8] If D is an ideal of Mn (R), then (D∗ )+ ⊆ D ⊆ (D∗ )∗ . This enables us to prove the following lemma. Lemma 3.6. Let R be a zero-symmetric near-ring with identity such that Mn (R) satisﬁes the DCCS. If D is an ideal of Mn (R) contained in Js (R))+ ]/D is nilpotent, then A+ ⊆ D. (J Js (R))+ such that [(J Proof. For any σ ∈ Js (R), we have fijσ ∈ (J Js (R))+ . Since [(J Js (R))+ ]m ⊆ D for some positive integer m, we have (ffijσ )m ∈ D. For any r1 , ..., rn ∈ Rn , πi (ffiiσ )m r1 , ..., rn = πi 0, ..., σ m ri , ..., 0 = σ m ri ∈ D∗ . In particular, πi (ffiiσ )m 1, ..., 1 = πi 0, ..., σ m , ..., 0 = σ m ∈ D∗ . Thus Js (R)/D∗ is a nil ideal of R/D∗ and by the DCCS, Js (R)/D∗ is nilpotent.

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By the minimality of A we have A ⊆ D∗ . Therefore, by Proposition 3.5, A+ ⊆ (D∗ )+ ⊆ D. Hence A+ is the unique minimal ideal modulo which (J Js (R))+ is nonzero nilpotent. Js (R))+ ⊆ Js (Mn (R)) Let us denote the s-socle of Mn (R) by A. Since (J + Js (R)) + A is nilpotent modulo and Js (Mn (R))/A is nilpotent, then (J + A. Which implies that (J Js (R)) is nilpotent modulo A+ ∩ A. By the + minimality of A we have Theorem 3.7. Let R be a zero-symmetric near-ring with identity such that Mn (R) satisﬁes the DCCS. Then A+ ⊆ A. Clearly, A+ ⊂ A∗ , but it is not clear what the relationship between and A is. However, we do have a relationship if A = (0). To show this we need the following lemma.

A∗

Lemma 3.8 Let R be a zero-symmetric near-ring with identity such that Mn (R) satisﬁes the DCCS, and let ∆ be a monogenic R-group. If ∆ is an R-group of type-0 but not of type-s, then the Mn (R)-group ∆n is also of type-0 but not of type-s. Proof. If ∆n is of type-s then for any monogenic R-subgroup Rδ of ∆, k Hin , where each we have (Rδ)n is a direct-sum of the form (Rδ)n = i=1

Mn (R)-kernel Hin is an Mn (R)-group of type-0, for i = 1, 2, ....k. As in k the proof of Lemma 3.2 this will imply that Rδ = Hi , where each Hi i=1

is an R-group of type-0. That is, ∆ is of type-s. Theorem 3.9. Let R be a zero-symmetric near-ring with identity such that Mn (R) satisﬁes the DCCS. If A = (0), then A∗ = (0). Proof. Let us ﬁrst note that A = (0) if and only if Js (Mn (R)) = J0 (Mn (R)). Assume that (0) = J0 (R) = Js (R). Then there exists an Rgroup ∆ of type-0 which is not of type-s. It follows from Lemma 3.8 that ∆n is an Mn (R)-group of type-0 which is not of type-s. This contradicts the fact that Js (Mn (R)) = J0 (Mn (R)). Therefore J0 (R) = Js (R), hence A = (0). It follows that A∗ = (0). Thus, if A∗ = (0), then A = (0). We conjecture that A∗ ⊆ A. In fact, examples of near-rings exist where A properly contains A∗ . The following example illustrates this. Example 2. Let G := Z2 ⊕ Z4 ⊕ Z2 ⊕ Z2 , S := Z2 ⊕ Z4 ⊕ Z2 ⊕ {0}, and

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S := {0} ⊕ {0} ⊕ Z2 ⊕ Z2 . Let S1 := {(0, 0, 0, 0), (0, 2, 1, 1)}, S2 := {(0, 0, 0, 0), (0, 2, 0, 1)}, S3 := {(0, 0, 0, 0), (0, 2, 1, 0)}, S4 := {(0, 0, 0, 0), (0, 0, 1, 1)}, S5 := {(0, 0, 0, 0), (0, 0, 0, 1)}, S6 := {(0, 0, 0, 0), (0, 0, 1, 0)}, S7 := {(0, 0, 0, 0), (0, 2, 0, 0)}. Deﬁne a near-ring R by Si ) ⊆ Si , ∀i ∈ {1, 2, . . . , 7}; R := { f ∈ M0 (G) | f (S ) ⊆ S , f (S ∀s, s ∈ S , s − s ∈ S5 ⇒ f (s) − f (s ) ∈ S5 ; ∀g, g ∈ G, g − g ∈ S ⇒ f (g) − f (g ) ∈ S }. We note that (a) Each Si is an R-subgroup of G, for each i ∈ {1, 2, . . . , 7}. (b) G is a monogenic R-group generated by (1, 3, 1, 1), and S is an R-kernel of G. Thus G is not of type-0, so that J0 (R) = (0). (c) S is an R-group of type-0 but not of type-2, because it has Rsubgroups S3 , S6 and S7 none of which is an R-kernel. Thus J0 (R) = J2 (R) (d) S is an R-subgroup of G because all its proper subgroups are Rgroups. (e) S5 is an R-kernel of S . (f) All the type-0 R-subgroups of G are also of type-s, hence Js (R) = J0 (R). Therefore, the s-socle A of R is zero. Hence, A∗ = (0). We now look at the R-subgroups of (R, +) but we ﬁrst deﬁne the following maps in R. Let x ¯ := (1, 2, 1, 1), and ¯0 := (0, 0, 0, 0). Deﬁne k1 (¯ x) := (0, 2, 1, 1); k2 (¯ x) := (0, 2, 0, 1); k0 to be the zero-map; x) := (0, 2, 1, 0); k4 (¯ x) := (0, 0, 1, 0); k5 (¯ x) := (0, 0, 0, 1); k6 (¯ x) := k3 (¯ x) := (0, 2, 0, 0); such that kj (¯) = ¯0, if y¯ = x, ¯ for (0, 0, 1, 1); k7 (¯ each j ∈ {1, 2, . . . , 7}. Deﬁne K := { k0 , k1 , k2 , . . . , k7 }, H := {k0 , k4 , k5 , k6 }, and T := {k0 , k5 }, and observe that (1) K and H are non-monogenic R-groups, and can be written as grouptheoretic direct-sums as follows; K = Rk4 ⊕ Rk5 ⊕ Rk7 , and H = Rk4 ⊕ Rk5 . (2) The maps kj , for j = 1, 2, . . . , 7, are deﬁned in such a way that we have the following R-isomorphisms between R-subgroups of (R, +) and the R-subgroups of G: Rk4 ∼ =R S6 ; T ∼ =R S7 , =R S5 ; Rk7 ∼ ∼ and S =R H. (3) T = Rk5 is an R-kernel of H. Consider K 2 as an M2 (R)-group under the action 2 as deﬁned in [4]. Let k, k ∈ K 2 , where k = a1 k4 +a2 k5 +a3 k7 and k = b1 k4 +b2 k5 +b3 k7 . There is an isomorphism Ψ : K 2 → K 2 deﬁned by Ψ(k, k ) = a1 k4 , b1 k4 , a2 k5 , b2 k5 , a3 k7 , b3 k7 .

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U ∈ M2 (R) acts on K 2 by action 2 as follows: U k, k = Ψ−1 U Ψk, k = Ψ−1 (U a1 , b1 )k4 , U (a2 , b2 )k5 , U (a3 , b3 )k7 a + f b , a, b ∈ R , we obtain Taking U := f11 21

U k4 + k5 + k7 , 0 = Ψ−1 U 1, 0k4 , U 1, 0k5 , U 1, 0k7 = Ψ−1 ak4 , bk4 , ak5 , bk5 , ak7 , bk7 = ak4 + ak5 + ak7 , bk4 + bk5 + bk7 . Since a and b can be arbitrarily chosen in R, all the 64 elements of K 2 can be obtained by using k4 + k5 + k7 , 0 as an M2 (R)-generator. So then K 2 = M2 (R)k4 + k5 + k7 , 0. In a similar manner, it can be shown that the subgroup H 2 of K 2 is a monogenic M2 (R)-group under action 2, such that H 2 = M2 (R)k4 + k5 , 0. We note that K 2 has no M2 (R)-kernels, hence it is an M2 (R)-group of type-0. Since H 2 is a monogenic M2 (R)-subgroup of K 2 which is not a direct sum of M2 (R)-kernels, it follows that K 2 is not of type-s. Hence Js (M2 (R)) = J0 (M2 (R)), therefore the s-socle A of M2 (R) is non-zero. From this it follows that A∗ = (0) ⊂ A .

References [1] Hartney, J.F.T., A radical for near-rings, Proc. Roy. Soc. Edinburgh 93A (1982), 105 - 110. [2] Hartney, J.F.T., On the decomposition of the s-radical of a near-ring, Proc. Edinburgh Math. Soc. 33 (1990), 11-22. [3] Hartney, J.F.T., s-Primitivity in Matrix near-rings, Quaestiones Math. 18 (1995), 487 - 500. [4] Meldrum, J.D.P. and Meyer, J.H., Modules over matrix near-rings and the J0 radical, Monatsh. Math. 112 , 125 - 139 (1991). [5] Meldrum, J.D.P. and Meyer, J.H., The J0 -radical of a Matrix near-ring can be intermediate, Canad. Math. Bull. 40 (2), 1997, 198 - 203. [6] Meldrum, J.D.P. and van der Walt, A.P.J., Matrix near-rings, Arch. Math. 47 (1986), 312 - 319. [7] Meyer, J.H., Left ideals and 0-primitivity in matrix near-rings, Proc. Edin. Math. Soc. 35 (1992), 173 - 187. [8] van der Walt, A.P.J., On two-sided ideals in Matrix near-rings, In: Near-rings and Near-ﬁelds, (ed, G. Betsch), North Holland 1987, 267 - 271. [9] van der Walt, A.P.J., Primitivity in matrix near-rings, Quaestiones Math. 9 (1986), 459 - 469.

A RIGHT RADICAL FOR RIGHT D.G. NEAR-RINGS John F.T. Hartney University of South Africa, South Africa [email protected]

Danielle S. Rusznyak University of the Witwatersrand South Africa [email protected]

Abstract

1.

Rahbari embarked on developing a right representation theory for right d.g. near-rings and proved interesting right structure theorems. Our main focus is connections between left and right representation. We discuss a Jacobson-type radical, r J0 (R), for right d.g. near-rings. The radical r J0 (R) is deﬁned using annihilators of certain d.g. right Rgroups which are the equivalent of type-0 R-groups from left representation. We then explore connections in near-rings with suitable chain conditions between r J0 (R), the (left) radicals and the intersection of all maximal right ideals, denoted r J 1 (R). In particular we prove that 2 J2 (R) = r J0 (R) for near-rings R satisfying the descending chain condition for left R-subgroups of R+ .

Introduction

Our near-ring R is distributively generated (d.g.) by the semi-group S, right distributive and possesses a multiplicative identity 1. For the standard deﬁnitions and results we refer the reader to [5] and [6]. As there is a lack of left-right symmetry in near-rings, the natural representation for right d.g. near-rings is the left one. A right representation theory for right d.g. near-rings was developed in [7]. We use the deﬁnitions in [7] with a naming convention that parallels that for left representation. If R satisﬁes the descending chain condition for left ideals (right ideals), we will say R satisﬁes the DCCL (respectively DCCR). If R satisﬁes the 225 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 225–234. c 2005 Springer. Printed in the Neatherlands.

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descending chain condition for left R-subgroups, we will say R satisﬁes the DCCLS. Deﬁnition 1.1 [7] An additive group Ω is a right R-group if there is a mapping Ω × R → Ω such that for all ω, ω1 , ω2 ∈ Ω, x, y ∈ R, (i) ω(xy) = (ωx)y (ii) (ω1 + ω2 )x = ω1 x + ω2 x (iii) ω1 = ω If, in addition, Ω possesses a set of generators ∆ satisfying (iv) δ(x + y) = δx + δy for all δ ∈ ∆, and (v) ∆S ⊆ ∆ then we say Ω is a distributively generated (d.g.) right R-group. The existence of a set ∆ is crucial to right representation theory. We call ∆ a distributive basis of Ω. It is clear that R+ is a d.g. right R-group with distributive basis S and that a right ideal is a right R-group but is not necessarily distributively generated. If I is a right ideal of R, then R/I is a d.g. right R-group. Right R-homomorphisms and right R-subgroups of a right R-group are deﬁned in the obvious way. Deﬁnition 1.2 1. An element ω of a right R-group Ω is said to be distributive if for all x, y ∈ R, ω(x + y) = ωx + ωy. 2. A d.g. right R-group Ω is monogenic if there is a distributive element ω such that Ω = ωR. 3. A non-zero monogenic d.g. right R-group is (right) type-0 if it does not possess any proper non-zero normal right R-subgroups. One shows routinely that R/M is a d.g. right R-group of type-0 if and only if M is a maximal right ideal of R. We remark that if ω is a distributive element of a right R-group Ω, then (ω : 0) = {r ∈ R | ωr = 0} is a right ideal of R. Moreover, if Ω is a d.g. right R-group then (Ω : 0) = {r ∈ R | ωr = 0 for all ω ∈ Ω} is a two-sided ideal of R. We call (ω : 0) the annihilating right ideal of ω and (Ω : 0) the right annihilating ideal of Ω. We note that (Ω : 0) = (∆ : 0) for any distributive basis ∆ of Ω. Remark 1.3 If Ω = ωR is a d.g. right R-group of type-0 then, as for left representation, Ω ∼ = R/(ω : 0) and (ω : 0) is a maximal right ideal of R.

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Deﬁnition 1.4 1. We deﬁne r J0 (R) = ∩{(Ω : 0) | Ω is a d.g. right R-group of type-0}. 2. The right annihilating ideal of a d.g. right R-group of type-0 is said to be right 0-primitive. Thus r J0 (R) is the intersection of all right 0-primitive ideals of R. Deﬁnition 1.5

r

J 1 (R) = ∩{Y | Y is a maximal right ideal of R}. 2

As for the corresponding left radicals, we have that largest ideal of R contained in r J 1 (R).

r

J0 (R) is the

2

Proposition 1.6 Let I be an ideal of R and Γ a group. 1. If Γ is a d.g. right R-group of type-0 with I ⊆ (Γ : 0)R , then γ(r+I) := γr for all γ ∈ Γ and r ∈ R makes Γ into a d.g. right R/I-group of type-0. 2. If Γ is a d.g. right R/I-group of type-0, then γr := γ(r + I) for all γ ∈ Γ and r ∈ R makes Γ into a d.g. right R-group of type-0. Proof. We only prove 1. Suppose that ∆ = {δi } is a distributive basis of Γ. To check that the action is well-deﬁned, suppose that r1 + I = r2 + I. Then r1 −r2 ∈ I ⊆ (Γ : 0)R = (∆ : 0)R and so for all δi ∈∆, δi (r1 −r2 ) = for all δi ∈ ∆. 0 i.e., δi r1 = δi r2 i with i = Consequently, if γ = i i δ ±1, then γr1 = ( i i δi )r1 = i i (δi r1 ) = i i (δi r2 ) = ( i δi )r2 = γr2 . Veriﬁcation of the axioms is routine and will be omitted. To prove that Γ is type-0 as a d.g. right R/I-group, suppose that Λ is a normal right R/I-subgroup of Γ. The action of R on Λ is deﬁned via its action on Γ, i.e., λr := λ(r + I). Hence Λ is a normal right R-subgroup of Γ. Theorem 1.7 R → r J0 (R) is a radical map. Proof. We need to show that r J0 (R/ r J0 (R)) = 0 and if θ is a near-ring homomorphism, then θ( r J0 (R)) ⊆ r J0 (θ(R)). Firstly, let r + r J0 (R) ∈ r J0 (R/ r J0 (R)) and let Γ be a d.g. right R-group of type-0. Since r J0 (R) ⊆ (Γ : 0)R , Γ is a d.g. right R/ r J0 (R)-group of type-0 by Proposition 1.6. Hence Γ(r + r J0 (R)) = {0} ⇒ Γr = {0} ⇒ r ∈ (Γ : 0)R . Since Γ is arbitrary, r ∈ r J0 (R). Thus r + r J0 (R) = r J0 (R), i.e., r J0 (R/ r J0 (R)) = 0. Next, let θ : R → R be a near-ring homomorphism and let r ∈ r J0 (R). Let Γ be a d.g. right θ(R)-group of type-0. Since θ(R) ∼ = R/kerθ, Γ can be considered as a d.g. right R-group of type-0 with kerθ ⊆ (Γ : 0)R . Therefore Γr = {0}. Now Γθ(r) = Γ(r + kerθ) = Γr = {0}, so θ(r) ∈ r J0 (θ(R)).

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In the case of near-rings R that satisfy the DCCL, any left R-group of type-0 can be obtained as a subfactor of a faithful left R-group [2]. One can prove the same result for d.g. right R-groups of type-0. Although the proof for the right case is fairly similar, we give it here as an indication of the extent to which adjustments are necessary to accommodate the lack of left distribution. Indeed, brute force is required to get the complete result. We ﬁrst need: Deﬁnition 1.8 Let Ω be a right R-group and Λ a right R-subgroup of Ω. An element ω of Ω is said to be distributive modulo Λ if for all x, y ∈ R, ω(x + y) − ωy − ωx ∈ Λ. Lemma 1.9 Let Λ be a normal right R-subgroup of the right R-group Ω and I be a right ideal of R. If ω ∈ Ω is distributive modulo Λ, then ωI + Λ is a right R-subgroup of Ω. Theorem 1.10 [8] Let R be a d.g. near-ring satisfying the DCCR and Ω a faithful d.g. right R-group. Any d.g. right R-group of type-0 is the (right) R-homomorphic image of some right R-subgroup of Ω. Proof. Let Γ = γR be a d.g. right R-group of type-0 with Y = (γ : 0). Then Γ ∼ = R/Y , and since Γ is of type-0, Y is a maximal right ideal of R. If Ω has distributive basis ∆, then since Ω is faithful, (0) = (Ω : 0) = ∩δ∈∆ (δ : 0), and as R satisﬁes the DCCR, we may assume that (0) = ni=1 (δi : 0) ⊆ Y . Now (δi : 0) is a right ideal of R, so R/(δi : 0) is a d.g. right R-group. If (δi : 0) ⊆ Y for some i, we have the right R-isomorphisms of factor groups Γ∼ = R/Y ∼ = (R/(δi : 0)) / (Y /(δi : 0)), so that Γ is the homomorphic image of the monogenic d.g. right Rsubgroup R/(δi : 0) ∼ = δi R of Ω. If (δi : 0) ⊆ Y for all i, then using the fact that R has DCCR, and reindexing if necessary, we may assume that there exists some k, 2 ≤ k ≤ n, such that B=

k−1

(δi : 0) ⊆ Y

and (δk : 0) ∩ B ⊆ Y.

i=1

We observe that B is a right ideal of R and as Y is a maximal right ideal, we have R = B + Y and γR = γB. Since R is unitary, there exists e ∈ B such that γ = γe.

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Consider the element δk e. If δk e = 0, then e ∈ (δk : 0) ∩ B ⊆ Y , and we would have γ = γe = 0. Hence δk e = 0 and we consider two cases. Case 1: δk e is a distributive element of Ω. Then (δk e : 0) is a right ideal of R. If x ∈ (δk e : 0), then 0 = (δk e)x = δk (ex), so that ex ∈ (δk : 0) ∩ B ⊆ Y , and we have γx = (γe)x = γ(ex) = 0. So x ∈ Y . Consequently, (δk e : 0) ⊆ Y and the result follows as before. Case 2: δk e is not a distributive element of Ω. We observe that if x, y ∈ R, then γe(x + y) = γ(x + y) = γx + γy = γex + γey = γ(ex + ey), so that e(x + y) − (ex + ey) ∈ Y . We also see that e ∈ B ⇒ e(x + y) − (ex + ey) ∈ B, so that e is distributive modulo Y ∩ B. Using Lemma 1.9, e(δk : 0) + (Y ∩ B) is a right R-subgroup of R+ . Deﬁne ψ : δk (e(δk : 0) + (Y ∩ B)) → γ(e(δk : 0) + (Y ∩ B)) = γ(δk : 0) by ψ(δk (ex + l)) = γ(ex + l) = γx, for x ∈ (δk : 0) and l ∈ Y ∩ B. Now δk (ex+l) = 0 ⇒ ex+l ∈ (δk : 0) so that ex+l ∈ (δk : 0)∩B ⊆ Y . It follows that 0 = γ(ex + l) = γex = γx. Hence ψ is well-deﬁned. Using the fact that e is distributive modulo Y ∩ B, we can show that ψ is a right R-homomorphism. Finally, as γ(δk : 0) is a non-zero normal right R-subgroup of Γ and as Γ is of type-0, we have that Γ = γ(δk : 0) is the homomorphic image of the right R-subgroup δk (e(δk : 0) + (Y ∩ B)) of Ω.

2.

An example

We use the example of the d.g. near-ring T from Laxton [4] to illustrate that the notions introduced in section 1 occur quite naturally. Consider the d.g. near-ring R generated by all inner automorphisms of a ﬁnite non-abelian simple group G. The following results hold for this well known d.g. near-ring [1]: 1(i) J2 (R) = Js (R) = J 1 (R) = J0 (R) = (0). 2

(ii) R = L1 ⊕ L2 ⊕ · · · ⊕ Ln , where each left ideal Li is an R-group of type-2 and R-isomorphic to G. The number of summands n is equal to |G| − 1. (iii) There exists ei ∈ Li such that Rei = Li , i = 1, . . . , n, 1 = ni=1 ei and {ei }ni is an orthogonal idempotent set.

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Now let T be the right d.g. near-ring generated by the endomorphisms {φx }x∈R , where φx · y = yx for all y ∈ R. We have that T is a subnear-ring of M0 (R+ ) with R+ a faithful left T -group [4]. The right Rsubgroups of R are precisely the left T -subgroups of R. Moreover, there is a one-to-one correspondence between the right R-subgroups of R+ and the subgroups of G. Under this correspondence normal subgroups of G correspond to right ideals of R and so R is a faithful T -group of type-0. From [4] we may choose G such that (0) = J0 (T ) = J 1 (T ) = J2 (T ). We 2 have the following results concerning the structure of T : 2(i) The identity 1T of T is of the form 1T = φe1 + φe2 + · · · + φen = φe1 +e2 +···+en and for i, j = 1, 2, . . . , n, φei φej = φej ei =

φ0 φei

if i = j . if i = j

(ii) For each i = 1, . . . , n, φei T is a d.g. right T -group with distributive basis {φei φx | x ∈ R} = {φxei | x ∈ R}. We have that for all t, t ∈ T, r ∈ R (−t + φei t + t ) · r

= = = = = = = = =

−t · r + (φei t) · r + t · r −t · r + φei · (t · r) + t · r −t · r + (t · r)ei + t · r xei for some x ∈ R, since Rei is normal in R (xei )ei since e2i = ei (−t · r + (t · r)ei + t · r)ei (−t · r + t · r + t · r)ei φei · (−t · r + t · r + t · r) (φei (−t + t + t )) · r

Hence −t + φei t + t = φei (−t + t + t ) ∈ φei T and thus the d.g. right T -group φei T is a right ideal of T for each i = 1, . . . , n. (iii) Now in the decomposition of R = ⊕ni=1 Rei , we have the (left) R-isomorphism Rek ∼ = Rej . Suppose ek → rej under this isomorphism. Deﬁne a map Ψ : φek T → φej T by Ψ(φek t) = φrej t = φej (φr t). If φek t = 0, then for all s ∈ R, 0 = φek t · s = (t · s)ek . Hence (t · s)rej = 0 for all s ∈ R by the R-isomorphism and thus (φrej t) · s = 0 for all s ∈ R. It follows that Ψ is well deﬁned. Using the fact that the φx are distributive elements of T we verify routinely that Ψ is a right T -isomorphism. (iv) Clearlyfrom t = ( ni=1 φei )t = ni=1 (φei t) for all t ∈ T we have n φ T . If x ∈ φ T ∩ T = e e i i i=1 j= i φej T , then x = φei t =

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231

so that x = φei φei t = j= i φei φej t = 0 by the orthogonal idempotence of the the set {φei }. We have established that n φei T. T = j= i φej tj

i=1

That is, T is a direct sum of right ideals, each of which is a d.g. right T -subgroup. (v) If ∆ is a normal subgroup of R, then X∆ = {t ∈ T |t · r ∈ ∆ for all r ∈ R} is a right ideal of T . On the other hand, if X is a right ideal of T and ω is a distributive element of R, then X ·ω = {x·ω|x ∈ X} is a normal subgroup of ωR. If ω = 1, then X · 1 = X · R. (vi) Put Ωi = L1 ⊕ · · · ⊕ Li−1 ⊕ Li+1 ⊕ · · · ⊕ Ln . Then R = Li ⊕ Ωi . We have that Xi := XΩi = {t ∈ T |t · R ⊆ Ωi } is a right ideal of T . It is not hard to see from the deﬁnition that φej ∈ Xi for all j = i so that φej T ⊆ Xi for j = i. Using the modular law Xi ∩ φei T ). on the direct sum in (iv), we have Xi = ⊕j= i φej T ⊕ (X Now φei t ∈ Xi ∩ φei T implies (φei t) · r = (t · r)ei ∈ Rei ∩ Ωi by the deﬁnition of Xi . Thus for all r ∈ R (φei t) · r = 0 and hence φei t = 0. It follows that Xi = ⊕j= i φej T . (vii) Let X be a maximal right ideal of T = ⊕ni=1 φei T = φei T ⊕ Xi . If φei T ⊆ X for each i = 1, . . . , n, then X = T . Hence there exists i such that φei T ⊆ X and we have T = φei T + X. Let Y = {t ∈ T | φei t ∈ X}. Then Y is a right ideal of T . For j = i we have φej ∈ Y and so Xi ⊆ Y . It is readily shown that φei T ∩ Y = φei T ∩ X and that Y = (φei T ∩ Y ) ⊕ Xi = (φei T ∩ X) ⊕ Xi . From the right T -isomorphism T /X = (φei T +X)/X ∼ = φei T /(φei T ∩X) and the fact that T /X is a d.g. right T -group of type-0, we have either φei T = φei T ∩ X or φei X = φei T . In either case it can be shown that Y is a maximal right ideal of T so that T /Y is a d.g. right T -group of type-0 with Y = (φei T ∩ X) ⊕ Xi .

3.

Relationships between the radicals Using a proof that mirrors that for left representation [4], we have

Proposition 3.1 Every right 0-primitive ideal is a prime ideal. If R satisﬁes the DCCLS, then any prime ideal is left 0-primitive [4], so in this case, every right 0-primitive ideal is left 0-primitive. Thus it follows that J0 (R) ⊆ r J0 (R).

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Theorem 3.2 If R satisﬁes the DCCLS, then r

J0 (R) ⊆ J2 (R).

Proof. We ﬁrst show that every maximal ideal M of R is right 0-primitive. If M is also maximal as a right ideal, then R/M is a d.g. right R-group of type-0. Since M is an ideal of R and R is unitary, we have that M = (R/M : 0), so M is a right 0-primitive ideal. If M is not maximal as a right ideal, then by Zorn’s Lemma there exists a maximal right ideal, say X, containing M , and R/X is therefore a d.g. right R-group of type-0. Now we see that M ⊆ (R/X : 0), since RM ⊆ M ⊆ X. But M is a maximal ideal of R, so (R/X : 0) = M , i.e., M is right 0-primitive. Finally, if R satisﬁes the DCCLS, then J2 (R) = ∩{I | I is a maximal ideal of R} [3], and so r J0 (R) ⊆ J2 (R). In [3] it is stated without proof that the intersection of all maximal right ideals contains all the nilpotent left R-subgroups of R+ . We prove this result. Theorem 3.3 Every maximal right ideal of R contains all the nilpotent left R-subgroups of R+ . Proof. Let Ω be a nilpotent left R-subgroup of R+ , say Ωp = (0). Let Y = {Σi (xi + ωi ri − xi ) | ri , xi ∈ R, ωi ∈ Ω}. Then Y is a right ideal of R containing Ω. Suppose that there exists a maximal right ideal B of R such that Ω ⊆ B. Then the right ideal Y is not contained in B so R = Y + B and the identity can be written as 1 = y + b, with y ∈ Y and b ∈ B. Let ω1 , . . . , ωp−1 ∈ Ω. Then ω1 . . . ωp−1 = = = = =

(y + b)ω1 . . . ωp−1 yω1 . . . ωp−1 + bω1 . . . ωp−1 [Σi (xi + ωi ri − xi )]ω1 . . . ωp−1 + b (for some b ∈ B) Σi (xi ω1 . . . ωp−1 + ωi ri ω1 . . . ωp−1 − xi ω1 . . . ωp−1 ) + b b ,

since ri ω1 ∈ Ω and so ωi ri ω1 . . . ωp−1 ∈ Ωp = (0). Hence Ωp−1 ⊆ B. Repeating this procedure we arrive at the contradiction Ω ⊆ B. Hence: Corollary 3.4 of R+ .

r

J 1 (R) contains all the nilpotent left R-subgroups 2

We remark that the right ideal Y in the proof of Theorem 3.3 is actually a two-sided ideal. As this result is important in the sequel, we state it more formally as follows.

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Theorem 3.5 Let X be a right ideal of R and let Ω be a left R-subgroup of R+ contained in X. Then Y = {Σi (xi +ωi ri −xi ) | ri , xi ∈ R, ωi ∈ Ω} is an ideal of R contained in X and containing Ω. Moreover, Y ⊆ (R/X : 0). Proof. Clearly Y is a right ideal of R containing Ω. Moreover, it is the smallest right ideal containing Ω, so Y ⊆ X. To show that Y is also a left ideal of R, write r ∈ R as r = j j sj where j = ±1 and sj ∈ S. If y = Σi (xi + ωi ri − xi ) ∈ Y it follows that sj y = Σi (sj xi + sj ωi ri − sj xi ) ∈ Y since Ω is a left R-group, and so ry ∈ Y . Theorem 3.6 Let P be a right 0-primitive ideal of R. Then P contains all nilpotent left R-subgroups of R+ . Proof. As P is a right 0-primitive ideal of R, by Remark 1.3 there is a maximal right ideal X of R such that P = (R/X : 0). If Ω is a nilpotent left R-subgroup of R+ , then by Theorem 3.3, Ω ⊆ X. If Y is the ideal of R as in Theorem 3.5, we have that Ω ⊆ Y ⊆ P . If R satisﬁes the DCCLS, then J2 (R) is the smallest ideal of R containing all nilpotent left R-subgroups of R+ [3]. Hence J2 (R) ⊆ r J0 (R). This, together with Theorem 3.2, proves: Theorem 3.7 If R satisﬁes the DCCLS, then J2 (R) = r J0 (R). To summarize, for a near-ring R with DCCLS we have J0 (R) ⊆ J 1 (R) ⊆ Js (R) ⊆ J2 (R) = r J0 (R) ⊆ r J 1 (R). 2

2

It is not known whether J2 (R) = r J 1 (R). Laxton states without proof 2 in [3] that the intersection of all the maximal right ideals of R contains all the nilpotent right R-subgroups of R+ . He does however prove that J2 (R) contains all the nilpotent right R-subgroups. For the sake of completeness, we prove that r J 1 (R) contains all nilpotent right R2 subgroups, even though this proof uses an argument similar to that given in Theorem 3.3. Theorem 3.8 R+ .

r

J 1 (R) contains all the nilpotent right R-subgroups of 2

Proof. Let Ω be a nilpotent right R-subgroup of R+ , say Ωp = (0). Deﬁne Y = {Σi (xi + ωi − xi ) | xi ∈ R, ωi ∈ Ω}. Then Y is a right ideal of R containing Ω. Suppose that there exists a maximal right ideal B of

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R such that Ω ⊆ B. Then Y ⊆ B so R = Y + B and the identity can be written as 1 = y + b, with y ∈ Y and b ∈ B. Let ω1 , . . . , ωp−1 ∈ Ω. Then ω1 . . . ωp−1 = = = = =

(y + b)ω1 . . . ωp−1 yω1 . . . ωp−1 + bω1 . . . ωp−1 [Σi (xi + ωi − xi )]ω1 . . . ωp−1 + b (for some b ∈ B) [Σi (xi ω1 . . . ωp−1 + ωi ω1 . . . ωp−1 − xi ω1 . . . ωp−1 )] + b b ,

since ωi ω1 . . . ωp−1 ∈ Ωp = (0). Hence Ωp−1 ⊆ B. Repeating this procedure we ﬁnally arrive at the contradiction Ω ⊆ B. We conjecture that if R satisﬁes the DCCLS then J2 (R) = r J 1 (R). 2

References [1] A. Fr¨ ¨ ohlich. The near-ring generated by the inner automorphisms of a ﬁnite simple group. The Journal of the London Mathematical Society, 33:95–107, 1958. [2] J. F. T. Hartney. s-Primitivity in matrix near-rings. Quaestiones Mathematicae, 18:487–500, 1995. [3] R. R. Laxton. A radical and its theory for distributively generated near-rings. The Journal of the London Mathematical Society, 38:40–49, 1963. [4] R. R. Laxton. Prime ideals and the ideal-radical of a distributively generated near-ring. Mathematische Zeitschrift, 83:8–17, 1964. [5] J. D. P. Meldrum. Near-rings and their links with groups. Number 134 in Research Notes in Mathematics. Pitman Advanced Publishing Program, London, 1985. [6] G. Pilz. Near-Rings: The Theory and its Applications. Number 23 in NorthHolland Mathematics Studies. North-Holland Publishing Company, Amsterdam, 1977. [7] M. H. Rahbari. Some aspects of near-ring theory. M.Phil dissertation, University of Nottingham, 1979. [8] D. S. Rusznyak. D.g. near-rings and matrix d.g. near-rings. M.Sc research report, University of the Witwatersrand, 2001.

FROM INVOLUTION SETS, GRAPHS AND LOOPS TO LOOP-NEARRINGS Helmut Karzel Zentrum Mathematik, T.U. M¨ Munchen Germany D - 80290 Munchen, M M¨ [email protected] muenchen.de

Silvia Pianta Dipartimento di Matematica e Fisica Universit` ` a Cattolica Via Trieste, 17 I - 25121 Brescia, Italy [email protected]

Elena Zizioli∗ Dipartimento di Matematica, Facolt` a ` di Ingegneria Universita ` degli Studi di Brescia Via Valotti, 9 I - 25133 Brescia, Italy [email protected]

Abstract

This is a general frame for a theory which connects the areas of loops, involution sets and graphs with parallelism. Our main results are stated in §5, §6 and §7. In §5 we derive a partial binary operation from an involution set and we discuss if such operation is a Bol operation or a K-operation, in §6, we relate involution sets with loops. In §7 we look for the possibility to construct loop-nearrings by considering the automorphism groups of loops. 2000 Mathematics Subject Classiﬁcation: 20N02, 20N05, 05C15.

Keywords: graphs, loops, involution sets, loop-nearrings

∗ Research

supported by M.I.U.R.

235 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 235–252. c 2005 Springer. Printed in the Neatherlands.

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Introduction Let (N, +, ·) be a nearﬁeld, for a ∈ N , let a+ : N → N ; x → a + x and : N → N ; x → a · x. If G := {a+ ◦ b· | a, b ∈ N, b = 0} then (N, G) is a sharply 2-transitive permutation group and in the ﬁnite case each sharply 2-transitive group can be represented in this way by a suitable ﬁnite nearﬁeld. For inﬁnite sharply 2-transitive groups there is such a characterization only if we replace the notion of nearﬁeld by the more general notion of near-domain (N, +, ·). For a nearﬁeld one can prove that (N, +) is a commutative group whereas in a near-domain, (N, +) is not even a group but only a loop which is characterized by some additional properties, for instance by: (B) ∀x, y ∈ N : x+ y + x+ = (x + (y + x))+ (Bol condition) (ν) ∀x, y ∈ N : ν(x + y) = ν(x) + ν(y) where ν(x) = (x+ )−1 (0). Condition (B) is the (left) Bol identity which characterizes the class of Bol loops; it is well known that in any Bol loop (N, +) all elements have an inverse −a such that a+(−a) = (−a)+a = 0 hence ν(a) = −a and condition (ν) is nothing else than the so called automorphic inverse property (see e.g. [3], [10]). The loops which are characterized by ( B) and (ν) are called Bruck-loops or K-loops (cf. [10]). In this case (N, I) with I := {a+ ◦ ν | a ∈ N } is an invariant regular set of involutions. Such observations inspire several generalizations. In fact a nearﬁeld can be deﬁned as a group (N, +) together with a ﬁxed element 1 ∈ N ∗ := N \{0} and a map · : N → Aut(N, +)∪{0}; x → x· such that ∀a, b ∈ N : a· (1) = a and (a· (b))· = a· ◦ b· holds; setting a · b := a· (b), then (N, +, ·) satisﬁes the usual axioms for a nearﬁeld. If, in this deﬁnition, one replaces Aut(N, +) by End(N, +) (i.e. the semigroup of all endomorphisms of (N, +)) and cancels the condition “a· (1) = a” but keeps “(a· (b))· = a· ◦ b· ” then one obtains the concept of a nearring. If one replaces the group (N, +) by a K-loop then one obtains, with a further condition, the notion of a near-domain. Finally if (N, +) is just a loop and · : N → End(N, +); x → x· a map with (a· (b))· = a· ◦ b· , for a, b ∈ N , then (N, +, ·) is a loop-nearring (see [11]). In order to be not too general we are interested in loop-nearrings where for the loop (N, +) the set I is still a set of involutions: such loops are characterized by the condition: () ∀a, b ∈ N a − (a − b) = b. In this paper we lay a foundation for this further research. Recent papers established connections between K-loops, invariant reﬂection structures and certain graphs with parallelism (cf. [7], [8], [13]). Here we are going to set up a general frame for these subjects. In fact the a·

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notion of an involution set gives rise from one side to a (partial) binary operation and from the other side to the geometric structure of a graph with parallelism. In this note we study the connections between these structures and translate properties of involution sets into conﬁgurational properties of the associated graphs. In the case that the involution set (P, I) contains a semiregular point o ∈ P , we introduce a partial binary operation “ + ” on P◦ × P (where P◦ is the orbit of the point o). For an element a ∈ P◦ we can deﬁne the permutation a+ : P → P ; x → a + x of SymP and we can ask for instance when “ + ” is a Bol operation and when “ + ” is a K-operation (cf. (5.2)). In the case P = P◦ we obtain Bol loops, resp. K-loops. In particular there is a correspondence between left loops ∗ (P, +) (a slight generalization of loops) satisfying condition (), reﬂection structures (P, I) and graphs with parallelism. We show that the automorphism group of the left loop (P, +) coincides with the group of automorphisms of (P, I), and of the graph, ﬁxing the point o (in other words the automorphisms of the 1-factorization of the graph ﬁxing the vertex o) and we relate such automorphism groups to the construction of loop-nearrings.

1.

Notation

In this note let P and E be two non empty sets and let I ⊆ P ×E be an incidence relation. For E ∈ E, p ∈ P let [E] := {x ∈ P | (x, E) ∈ I} and [[p] := {X ∈ E | (p, X) ∈ I}, moreover let J := {σ ∈ SymP | σ 2 = id} and for any I ⊆ J let I ∗ := I \ {id}. For a subset I ⊆ J, the pair (P, I) is called an involution set. The triple Γ := (P, E, I) is called a graph if: ∀E ∈ E, 1 ≤ |[E]| ≤ 2 and the elements of P , resp. of E, are called vertices, resp. edges (for general background on graphs see e.g. [2]). A graph Γ = (P, E, I) is called simple if ∀A, B ∈ E: [A] = [B] ⇒ A = B (in this case each edge P E ∈ E can be identiﬁed with the set [E], i.e. E ⊆ ∪ P where 2 P := {{x, y} | x, y ∈ P, x = y}), 2 complete if ∀a, b ∈ P ∃E ∈ E: {a, b} = [E], complete simple if ∀a, b ∈ P ∃1 E ∈ E: {a, b} = [E], i.e. E can be P identiﬁed with ∪ P. 2

∗ The

use of the terms “left” or “right” loop is not unique (cf. [10] p.24). We follow here the suggestion of the referee and call left loop what we previously (e.g. in [8]) called right loop.

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Given a set K = ∅, a surjective map c : E → K is called a (edge-) coloring of the graph Γ if: (C) ∀D, E ∈ E with D = E and [D] ∩ [E] = ∅: c(D) = c(E) and a parallelism if the parallel axiom is satisﬁed [ ] : c(E) = k; (P) ∀p ∀ ∈ P , ∀k ∈ K, ∃1 E ∈ [p let such E be denoted by Ep,k , then we can deﬁne a map ε : P × K → E; (p, k) → Ep,k = ε(p, k). Thus, for each k ∈ K, the set Fk := ε(P × {k}) = {E Ep,k | p ∈ P } is a partition of P , that is a 1-factor of the graph Γ = (P, E, I), and the set F := {F Fk | k ∈ K} is a 1- factorization of Γ (cf. [1]). Remark. It is straightforward to verify that condition (C) is equivalent to requiring that ∀p ∀ ∈ P the map c|[p] is injective, while (P) is equivalent to its bijectivity. Hence (P ) ⇒ (C). The quadruple Γc := (P, E, I, c) is called a colored graph, resp. a graph with parallelism, if c is a coloring, resp. a parallelism, of Γ. If c is a parallelism we shall write also Γ = (P, E, I, ) for Γc and for A, B ∈ E we set A B ⇔ c(A) = c(B). Then the parallel axiom (P ) assumes the usual form: ∀ ∈ P , ∀D ∈ E ∃1 E ∈ [p [ ] : E D; (P ) ∀p we set (p D) := E. From these deﬁnitions it follows: (1.1) Let Γ = (P, E, I, ) be a graph with parallelism, then: (1) ∀a, b ∈ P : |[a]| = |[b]| (= |K|), (2) ∀p ∀ ∈ P , ∀k ∈ K, ∀q ∈ [E Ep,k ]: Eq,k = Ep,k . Ep,k ] ∩ [Eq,k ] = ∅ and c(E Ep,k ) = c(Eq,k ) = k; Ep,k ] ⇒ [E Proof. (2) q ∈ [E hence, by (C), Ep,k = Eq,k . In a graph with parallelism we can introduce further concepts: Let A, B, C ∈ E, then (A, B, C) is called a trapezium if A C, A ∦ B, [A] ∩ [B] = ∅ = [C] ∩ [B] and [B] ⊆ [A] ∪ [C]. Let T r be the set of all trapeziums. If (A, B, C) ∈ T r, let [A] = {a, a1 }, [C] = {c, c1 } and a, c ∈ [B] (in particular a1 = a if |[A]| = 1 , resp. c1 = c if |[C]| = 1 ). A trapezium (A, B, C) is called proper if A = C. In this case [B] = {a, c} with a = c and we denote by T rp the set of all the proper trapeziums. For A, B ∈ E with A ∦ B let T r(A, B) := {(A , B , C ) ∈ T r | A A, B B}. By the trapezium axiom we mean the following statement:

From Involution Sets, Graphs and Loops to Loop-nearrings

(Tr) ∀(A, B, C) ∈ T rp ∃k ∈ K : ∀(X, Y, Z) ∈ T r(A, B) : {x1 , z1 }† and by the local trapezium axiom (T rl ) where l ∈ K: (Trl ) ∀(A, B, C) ∈ T rp with c(A) = l ∃k ∈ K: ∀(X, Y, Z) ∈ T r(A, B) :

239 [E Ex1 ,k ] =

[E Ex1 ,k ] = {x1 , z1 }.

A trapezium (A, B, C) ∈ T r is called closed if [a1 ] ∩ [c1 ] = ∅ and any element of [a1 ] ∩ [c1 ] is a closure of (A, B, C). If D ∈ [a1 ]∩[c1 ] is a closure of (A, B, C) then (A, D, C) is also a closed trapezium and B is a closure of (A, D, C). If there is a D ∈ [a1 ] ∩ [c1 ] with D B then (A, B, C, D) is called a parallelogram. A subset Q ⊆ P is componentwise closed in the graph Γ = (P, E, I) if [E] ⊆ Q for any E ∈ E with [E] ∩ Q = ∅. We may consider the subgraph of Γ spanned by Q, i.e. (Q, EQ , I) with EQ := {E ∈ E | [E] ⊆ Q}. Since P \ Q is also componentwise closed, the graph Γ can be decomposed into the two subgraphs (Q, EQ , I) and (P \ Q, EP \Q , I) and E = EQ ∪˙ EP \Q . For any vertex p ∈ P of the graph Γ, the connected component of p in Γ is trivially componentwise closed. Conversely, any componentwise closed subset of vertices of Γ is a union of connected components of the graph. Now let (P, I) be an involution set. For a, b ∈ P let [a ↔ b] := {γ ∈ I | γ(a) = b}. A point p ∈ P is called semiregular, resp. transitive, resp. regular if ∀x ∈ P : |[p [ ↔ x]| ≤ 1, resp. |[p [ ↔ x]| ≥ 1, resp. |[p [ ↔ x]| = 1. By Ps , resp. Pt , resp. Pr we denote the set of all semiregular, resp. transitive, resp. regular, points of (P, I). Clearly Pr = Ps ∩ Pt . If Pr = ∅, o ∈ Pr is ﬁxed and ∀x ∈ P , x◦ := [o ↔ x], then I = {x◦ | x ∈ P } and (P, ◦ ; o) is called (according to [7]) a reﬂection structure. Given an involution set (P, I), for a ∈ P we shall consider the following conditions: (Ba ) For α ∈ [a ↔ a] : αIα = I, (Ba ) For α ∈ [a ↔ a]: ∀ξ ∈ I : ξαIαξ = I, (B• ) ∀ξ ∈ I : ξIξ = I. An involution set (P, I) is called invariant if (B • ) is satisﬁed. (1.2) For an involution set (P, I) such that there exists a point a ∈ P with [a ↔ a] = ∅ the following statements are equivalent: † Recall

that by our convention {x} := X ∩ Y , {z} := Z ∩ Y and [X] = {x, x1 }, [Z] = {z, z1 }.

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(1) (Ba ) and (B a ) are satisﬁed, (2) (B • ) is satisﬁed. (1.3)

If (P, I) is an invariant involution set and Pr = ∅ then P = Pr .

Proof. Let o ∈ Pr be ﬁxed and for x ∈ P let x◦ := [o ↔ x]. Then for a, b ∈ P , a◦ (a◦ (b))◦ a◦ ∈ I and a◦ (a◦ (b))◦ a◦ (a) = a◦ (a◦ (b))◦ (o) = a◦ (a◦ (b)) = b and so a◦ (a◦ (b))◦ a◦ ∈ [a ↔ b]. If γ ∈ [a ↔ b] then a◦ γa◦ ∈ I and a◦ γa◦ (o) = a◦ γ(a) = a◦ (b) hence a◦ γa◦ = (a◦ (b))◦ i.e. γ = a◦ (a◦ (b))◦ a◦ . An invariant involution set (P, I) with Pr = ∅ is called invariant reﬂection structure since for each o ∈ Pr and ◦ : P → I; x → x◦ := [o ↔ x] the triple (P, ◦ ; o) is an invariant reﬂection structure in the sense of [7]. A subset Q ⊆ P is called an involution subset of (P, I) if I(Q) = Q and a faithful involution subset of (P, I) if moreover : ∀α, β ∈ I, α|Q = β|Q ⇔ α = β. (1.4) Let Q be an involution subset of an involution set (P, I), then: (1) (Q, I|Q ) is an involution set, (2) P \ Q is an involution subset. In an involution set (P, I), for p ∈ P let p := I(p) be the orbit of the element p. Then p is an involution subset of (P, I). Conversely, any involution subset of (P, I) is a union of orbits. The involution set is connected if P = p. Let (P, I) and (P , I ) be involution sets. A bijection α : P → P is called an isomorphism from (P, I) onto (P , I ) if αIα−1 = I . (P, I) and (P , I ) are called isomorphic if there is an isomorphism between (P, I) and (P , I ). An isomorphism from (P, I) onto (P, I) is called an automorphism; let Aut(P, I) denote the group of all automorphisms of (P, I). From the deﬁnitions we get immediately: (1.5) Let (P, I) be an involution set. Then (1) Aut(P, I) = N (I) (where N (I) := {σ ∈ SymP | σIσ −1 = I} is the normalizer of the set I of involutions), (2) (P, I) is invariant if and only if I ⊆ N (I).

2.

Connections between involution sets and graphs with parallelism

Let (P, I) be an involution set, let E := {({x, α(x)}, α) | x ∈ P, α ∈ I} and let I := {(y, ({x, α(x)}, α)) ∈ P × E | y ∈ {x, α(x)}}. Then for E = ({x, α(x)}, α) ∈ E, [E] = {x, α(x)} and for p ∈ P , [[p] = {({p, α(p)}, α) | α ∈ I}. Hence Γ(P, I) := (P, E, I) is a graph.

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241

If we consider the function c : E → I; ({x, α(x)}, α) → α it turns out immediately that ∀x ∈ P the restriction c|[x] is a bijection. Thus c is a parallelism; hence we may set Γ (P, I) = (P, E, I, ) and call Γ the graph derivation. Now we can prove: (2.1) Let (P, I) be an involution set and let (P, E, I, ) = Γ (P, I) be the derived graph with parallelism, then: (1) ∀A, B ∈ E with A non parallel to B, ∃X, Y ∈ E with X A, Y B, [X] ∩ [Y ] = ∅ and [X] = [Y ], or equivalently (2) ∀α, β ∈ I(=: K), α = β, ∃x ∈ P : [E Ex,α ] = [Ex,β ]. Proof. Let α := c(A), β := c(B) hence α, β ∈ I with α = β, therefore there is a vertex x ∈ P such that α(x) = β(x). The edges X := ({x, α(x)}, α), Y := ({x, β(x)}, β) ∈ E have the properties c(X) = α = c(A) ( i.e. X A) ; c(Y ) = β = c(B) (i.e. Y B) and [X]∩[Y ] = {x, α(x)}∩{x, β(x)} = {x}. Since α(x) = β(x), [X] = [Y ].

(2.2) Let (P, I) be an involution set and let Γ be the derived graph. Then Ps = ∅ if and only if there exists x ∈ P such that ∀α, β ∈ I, α = β, [E Ex,α ] = [Ex,β ]. Such a vertex x will be called semiregular for the graph Γ . Now let (P, E, I, ) be a graph with parallelism. By (1.1.2) each color α ∈ K deﬁnes a permutation α ˜ ∈ J ⊆ SymP as follows: ∀x ∈ ˜ = [E Ex,α ] \ {x} =: x if P, α ˜ (x) = x if |[E Ex,α ]| = 1 and α(x) := {˜ ˜ (x)} = [E Ex,α ] and α ˜ 2 = id. Set K α | α ∈ K} |[E Ex,α ]| = 2, so {x, α and Inv(P, E, I, ) := (P, K) and call “Inv” the involution derivation. to the graph (P, E, I, ). So “Inv” associates the involution set (P, K) (2.3) The map : K → K; α → α ˜ is injective if and only if (P, E, I, ) satisﬁes one of the equivalent conditions (2.1.1) or (2.1.2). Proof. Let α, β ∈ K, α = β. Then α ˜ = β˜ ⇔ ∃p ∈ P : ˜ ˜ (p)} = [Ep,β ] = {p, β(p)}. ∃p ∈ P : [E Ep,α ] = {p, α

˜ α(p) ˜ = β(p) ⇔

can be described by the surjection The set E of edges of Γ (P, K) ε :

→ E P ×K (x, α ˜ ) → ({x, α ˜ (x)}, α ˜)

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and there is exactly one map δ : E → E such that the following diagram is commutative: P ×K ε↓ E

(id,)

−→ δ

−→

; (x, α) → P ×K ↓ ↓ε E

;

Ex,α

(x, α ˜) ↓

(2.1)

→ ({x, α ˜ (x)}, α ˜)

˜ (x)} = [({x, α ˜ (x)}, α ˜ )] = [δ(E Ex,α )]. ThereWe have [E Ex,α ] = {x, α fore the map: (id, δ) : P × E → P × E is an isomorphism from the graph (P, E, I, ) onto the graph Γ (Inv(P, E, I, ) if and only if the map is injective, i.e. if and only if condition (2.1.2) is satisﬁed. On the other hand if (P, I) is an involution set, then Inv(Γ (P, I)) = (P, I). Thus: (2.4) The composition Inv ◦ Γ is the identity on the class of involution sets while Γ ◦ Inv is the identity only on the class of graphs with parallelism satisfying condition (2.1.2). For the sequel it is useful to introduce in the graph with parallelism derived from an involution set (P, E, I, ) = Γ (P, I) the notion of trapezium with origin x and directions α , β, for a given point x ∈ P and any two distinct involutions α, β ∈ I, in the following way: τ (x; α, β) := (({x, α(x)}, α), ({x, β(x)}, β), ({β(x), αβ(x)}, α)). (2.5) If α, β ∈ I with α = β and if A := ({a, α(a)}, α), B := ({b, β(b)}, β)) then τ (x; α, β) ∈ T r(A, B) for any point x ∈ P .

3.

Translation of properties

In this section let (P, I) be an involution set and (P, E, I, ) = Γ (P, I) be the derived graph with parallelism. We are going to translate properties of (P, I) into the language of the corresponding graph (P, E, I, ) and viceversa. We recall that for a, b ∈ P , [a ↔ b] := {σ ∈ I | σ(a) = b}. (3.1) Let o ∈ P and i ∈ {1, 2, 3}; then the following statements (i) and (i ) are equivalent: (1) o ∈ Ps , (1 ) ∀A, B ∈ [o] : [A] = [B] ⇒ A = B (i.e. o is semiregular for the graph Γ ), (2) o ∈ Pt , (2 ) ∀x ∈ P ∃E ∈ [o] : x ∈ [E] (i.e. the neighbourhood of the vertex o, given by E∈[o] [E], coincides with P ),

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(3) o ∈ Pr , (3 ) The conditions (1 ) and (2 ) are satisﬁed for o. The proof of (3.1) is a straightforword consequence of the deﬁnitions of the previous sections. (3.2) For i ∈ {1, 2, . . . , 6} the following statements (i) and (i ) are equivalent: (1) (P, I) is semiregular, i.e. P = Ps , (1 ) Γ (P, I) is a simple graph , (2) (P, I) is ﬁxed point free, i.e. ∀α ∈ I : Fix α := {x ∈ P | α(x) = x} = ∅ (⇒ I ⊆ J ∗ ), (2 ) ∀E ∈ E: |[E]| = 2 (i.e. the graph does not contain loops), (3) (P, I) is transitive, i.e. P = Pt , (3 ) Γ (P, I) is a complete graph, (4) (P, I) is regular, i.e. P = Pr , (4 ) Γ (P, I) is a complete simple graph, (5) (P, I) is regular and (P, I ∗ ) is ﬁxed point free, (5 ) (P, E ∗ , I, ) := Γ (P, I ∗ ) is a graph where E ∗ can be identiﬁed P with and E = E ∗ ∪ Pˆ with Pˆ := {({p}, id)| p ∈ P }, 2 (6) (P, I) is invariant, (6 ) (P, E, I, ) satisﬁes the trapezium axiom. Proof. (4) ⇔ (4 ) is a consequence of “(1) ⇔ (1 )” and “(3) ⇔ (3 )”. Let p, q ∈ P . Since (P, I) is regular there is exactly one (5) ⇒ (5 ). / I ∗ and σ ∈ I with σ(p) = q. If p = q, since I ∗ is ﬁxed point free, σ ∈ so σ = id. Therefore id ∈ I and {({p}, id) | p ∈ P } ⊆ E. Now let p = q, since (P, I) is regular and since id ∈ I, σ is ﬁxed point free, i.e. σ ∈ I ∗ and E := ({p, q}, σ) ∈ E ∗ is the only edge with [E] = {p, q}. So we obtain E = E ∗ ∪ {({p}, id) | p ∈ P }. Let p, q ∈ P . By (5 ) id ∈ I and if p = q there is (5 ) ⇒ (5). exactly one edge A = ({a, b}, α) ∈ E ∗ with {p, q} = [A] = {a, α(a)}, hence exactly one α ∈ I with α(p) = q. For each x ∈ P, X := ({x, α(x)}, α) ∈ E ∗ and so α(x) = x. This shows that I = {id} ∪ I ∗ acts regularly on P and I ∗ is ﬁxed point free. (6) ⇔ (6 ) see proposition (3.3) of [8]. (3.3) Let Q ⊆ P , then for i ∈ {1, 2} the following statements (i) and (i ) are equivalent: (1) Q = p with p ∈ P , i.e. Q consists of a single orbit under I, (1 ) Q is a connected component of Γ (P, I), (2) Q is an involution subset of (P, I), (2 ) Q is componentwise closed in Γ (P, I).

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Helmut Karzel, Silvia Pianta and Elena Zizioli

Generalized, partial and proper binary systems

Let A, B be two non empty sets and + : A × B → B; (a, b) → a + b be a map, then (A × B, +) is called a generalized binary system and “+” is called a generalized binary operation. If (A × B , + ) is another binary operation, and σ : A → A and τ : B → B are two bijections such that ∀(a, b) ∈ A × B : σ(a) + τ (b) = τ (a + b) then (σ, τ ) is called an isomorphism from (A × B, +) onto (A × B , + ) and an automorphism if moreover A = A , B = B and + = + . In a generalized binary system (A × B, +) deﬁne for all a ∈ A the map a+ : B → B; x → a + x, and let A+ := {a+ | a ∈ A}. (A × B, +) is called of exponent 2 if ∀a ∈ A, (a+ )2 = id. If (σ, τ ) is an isomorphism between (A × B, +) and (A × B , + ) then for all a ∈ A : (σ(a))+ = τ a+ τ −1 , hence τ A+ τ −1 = (A )+ , and we say that the mapping sets (B, A+ ) and (B , (A )+ ) are isomorphic. If moreover for (A × B, +) and (A × B , + ) the maps +

: A → A+ ⊆ B B ; a → a+ and

+

: A → (A )+ ⊆ B B ; a → a+

are injective then (A × B, +) and (A × B , + ) are isomorphic if and only if there is a bijection τ : B → B such that τ A+ τ −1 = (A )+ , so a permutation ω ∈ SymB induces an automorphism of (A × B, +) if and only if ωA+ ω −1 = A+ . From now on, we suppose that (A × B, +) is a generalized binary system with A ⊆ B. In this case we call (A × B, +) a partial binary system, and “+” a partial binary operation. If A = B, we set (A, +) = (A × A, +) and call (A, +) a (proper) binary system. We can consider the following notions. (1) An element o ∈ A is called neutral if ∀(x, y) ∈ A×B : x+o = x and o + y = y. By this deﬁnition (A × B, +) has at most one neutral element. If (A × B, +) has a neutral element then, for a ∈ A an element b ∈ B is called a right inverse if a + b = o. (2) We say that (A × B, +) satisﬁes the automorphic inverse property if (A × B, +) has a neutral element and there is a map ν : B → B such that ν(A) = A, and ∀(a, b) ∈ A × B : a + ν(a) = o and ν(a + b) = ν(a) + ν(b). (3) If A + A ⊆ A and A+ ⊆ SymB, we set for all x, y ∈ A, δx,y := ((x + y)+ )−1 x+ y + and ∆ := {δx,y | x, y ∈ A}. Then for δ ∈ ∆, δ(A) = A and (A × B, +) may satisfy one of the conditions: (A)

∀δ ∈ ∆:

δA+ δ −1 = A+

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(K)

∀x, y ∈ A:

245

δx,y = δx,y+x .

(4) We say that “+” is a Bol-operation if the two Bol conditions : (B1)

∀x, y ∈ A :

x + (y + x) ∈ A

(B2)

∀x, y ∈ A :

x+ y + x+ = (x + (y + x))+

are satisﬁed and a K-operation if “+” is a Bol operation satisfying the automorphic inverse property. (5) (A × B, +) is called a left loop if A = B (i.e. we set (A, +) := (A × B, +)), A+ ⊆ SymA and there is a neutral element 0 ∈ A. (A, +) is a loop if moreover + A := {+ a | a ∈ A} ⊆ SymA, where + a : A → A; x → x + a. A loop (P, +) is called Bol-, resp. Bruck-loop = K-loop , if “+” is a Bol-, resp. K- operation. We remark: (4.1) A K-loop (P, +) can be also characterized as a loop satisfying the automorphic inverse property and conditions (A) and (K) (cf. [9]). (4.2) If (A × B, +) is a binary system of exponent 2 then (B, A+ ) is an involution set, if moreover A ⊆ B and “+” is a Bol operation then (B, A+ ) is an invariant involution set.

5.

Semiregular, transitive and regular points and derived binary operations

Let (P, I) be an involution set. In this section we investigate the properties of the sets Ps , Pt and Pr and later we show how to derive a partial binary system from any involution set, or equivalently from any graph with parallelism, admitting at least one regular point. We note that in the corresponding graph (P, E, I, ) the points p ∈ Ps , resp. Pt , resp. Pr are characterized by: for each x ∈ P , x = p there is at most, resp. at least, resp. exactly one, edge E ∈ E with p, x ∈ [E]. (5.1)

Let o ∈ P and o := I(o) then:

(1) If o ∈ Ps then o is a faithful involution subset, (2) If (o, I|o ) is invariant and Ps ∩ o = ∅ then o ⊆ Ps . Proof. (1) If α, β ∈ I with α = β and o ∈ Ps then α(o) = β(o) , i.e. α|o = β|o . (2) By assumption there is a point p ∈ Ps ∩ o, hence, by (1), o = p is faithful. Now if I|o is invariant I is invariant i.e. it is contained in the normalizer N (I) in Sym o then I ⊆ N (I) and by (1.5) I ≤ Aut(o, I|o ). Consequently if p ∈ Ps ∩ o then o = I(p) ⊆ Ps .

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Now assume Ps = ∅. Fix an element o ∈ Ps and set P◦ := I(o) ∪ {o} (if o ∈ Pr then P◦ = P ). Then for each x ∈ I(o), |[o ↔ x]| = 1; set / I (o). Finally for x ∈ P◦ , y ∈ P , x◦ := [o ↔ x] and o◦ := id if o ∈ deﬁne x+o := x◦ o◦ and let νo : P → P ; x → o◦ (x) be the negative map with respect to o. Moreover let (P P◦ )◦ := {x◦ | x ∈ P◦ } and +o + ◦ P◦ ) if o ∈ I(o) and I ∪ {id} = (P P◦ ) := {x o | x ∈ P◦ }. Then I = (P ◦ / I (o). (P P◦ ) if o ∈ We call loop derivation in o the partial binary operation +o : P◦ × P → P ; (x, y) → x +o y := x+o (y). Note that ∀x ∈ P◦ : x +o νo (x) = x +o o◦ (x) = o hence we set −o x := νo (x). (5.2) The partial binary system (P P◦ × P, +o ) has the following properties: (1) ∀(a, b) ∈ P◦ × P : o +o b = b, () : a −o (a −o b) = b and

a +o o = a, ∃1 x ∈ P : a +o x = b,

P◦ × P, +o ) (i.e.(P P◦ × P, +o ) sat(2) (o◦|P , o◦ ) is an automorphism of (P ◦ isﬁes the automorphic inverse property) ⇔ (Bo ) is satisﬁed (i.e. in P◦ , (P P◦ )◦ ), this case o◦ Io◦ = I in (P (3) (i) If (P P◦ × P, +o ) is of exponent 2 then o◦ |P◦ = idP◦ . P◦ × P, +o ) is of exponent 2. (ii) If o◦ = id then (P (iii) If P = o then:

(P P◦ × P, +o ) is of exponent 2 ⇔ o◦ = id,

(4) (P P◦ × P, +o ) satisﬁes (B1) ⇔ ∀α, β ∈ I :

αo◦ βo◦ α(o) ∈ P◦ ,

P◦ )◦ (5) (P P◦ ×P, +o ) is a Bol operation, i.e. satisﬁes (B1) and (B2) ⇔ (P o satisﬁes (B ), (6) (P P◦ × P, +o ) is a K-operation, i.e. satisﬁes (B1), (B2) and the automorphic inverse property ⇔ (P P◦ )◦ satisﬁes (B o ) and (Bo ). If [o ↔ o] = ∅ then I is invariant ⇔ I satisﬁes (B o ) and (Bo ), (7) (P P◦ × P, +o ) is associative, i.e. ∀a, b ∈ P◦ : a +o b ∈ P◦ and P◦ )◦ (⇒ “ +o ” a+o b+o = (a +o b)+o ⇔ ∀α, β ∈ I : αo◦ β ∈ (P is commutative, i.e. ∀a, b ∈ P◦ : a +o b = b +o a). Proof. (1) o +o b = o◦ o◦ (b) = b and a +o o = a◦ o◦ (o) = a◦ (o) = a , a−o (a−o b) = a◦ o◦ o◦ (a◦ o◦ o◦ (b)) = a◦ (a◦ (b)) = b, a+o x = a◦ o◦ (x) = b hence x := o◦ a◦ (b) is the solution.

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(2) “⇒” ∀(a, b) ∈ P◦ × P : o◦ (a +o b) = o◦ a◦ o◦ (b) = o◦ (a) +o o◦ (b) = (o◦ (a))◦ o◦ o◦ (b) = (o◦ (a))◦ (b) ⇔ o◦ a◦ o◦ = (o◦ (a))◦ ∈ I hence (Bo ) is satisﬁed. “ ⇐ ” Let a = α(o) ∈ I(o) = P◦ hence o◦ (a) = o◦ αo◦ (o) ⊆ I(o) = P◦ ) = P◦ , o◦|P is a permutation of P◦ and P◦ by (Bo ). Therefore o◦ (P ◦ (o◦ (a))◦ = o◦ a◦ o◦ since o◦ a◦ o◦ (o) = o◦ a◦ (o) = o◦ (a). Moreover if b ∈ P then o◦ (a +o b) = o◦ a◦ o◦ (b) = (o◦ (a))◦ o◦ o◦ (b) = o◦ (a) +o o◦ (b). (3) (i) Let a ∈ P◦ then id = a+o a+o = a◦ o◦ a◦ o◦ hence a◦ o◦ = o◦ a◦ and so a = a◦ o◦ (o) = o◦ a◦ (o) = o◦ (a), i.e. o◦ |P◦ = idP◦ . (ii) Let a ∈ P◦ then a+o a+o = a◦ o◦ a◦ o◦ = a◦ a◦ = id. (iii) By (ii), we have only to show “ ⇒ ” . By (i) P◦ ⊆ Fix o◦ . Now let x ∈ P◦ and y ∈ Fix o◦ , then o◦ x◦ (y) = x◦ o◦ (y) = x◦ (y) and so I(P P◦ ) ⊆ I(Fix o◦ ) ⊆ Fix o◦ ; by induction on the length of words formed on I, o ⊆ Fix o◦ , i.e. o◦ = id. P◦ )◦ → P◦ ; α → α(o) is a bijection and (4) Since o ∈ Ps the map (P ◦ ◦ α(o) +o (β(o) +o α(o)) = αo βo α(o). This shows the equivalence. (5) Let α, β ∈ I and a := α(o), b := β(o) (i.e. a, b ∈ P◦ ). Then a+o b+o a+o (o) = a +o (b +o a) and so a+o b+o a+o = αo◦ βo◦ αo◦ becomes equivalent to (a +o (b +o a))+o = (a +o (b +o a))◦ o◦ if, ﬁrstly a +o (b +o a) ∈ P◦ , and secondly αo◦ βo◦ α ∈ I. If αo◦ βo◦ α ∈ I then a +o (b +o a) = αo◦ βo◦ α(o) ∈ I(o) ⊆ P◦ and so (a +o (b +o a))◦ ∈ I implying a+o b+o a+o = (a +o (b +o a))+o . (6) The ﬁrst equivalence is a consequence of (2) and (5) and the second of (1.2) . (7) Let a = α(o), b = β(o) then a +o b ∈ P◦ ⇔ αo◦ β(o) ∈ P◦ , thus P◦ )◦ ⇔ αo◦ β = (αo◦ β(o))◦ ⇔ αo◦ β(o) ∈ P◦ and a+o b+o = αo◦ β ∈ (P αo◦ βo◦ = (αo◦ β(o))◦ o◦ = (a +o b)+o . Now, if αo◦ β is an involution then αo◦ β = βo◦ α, i.e. a+o b+o = b+o a+o that implies, in turn, a +o b = b +o a. Remark. If (P P◦ ×P, +o ) is a Bol operation of exponent 2 then (P P◦ ×P, +o ) is also a K- operation.

6.

Left loops, loops and their corresponding involution sets and graphs

Let (P, +) be a binary system with a neutral element o ∈ P and with P + ⊆ SymP . Then (P, +) is a left loop and one can deﬁne the maps: ν : P → P ; x → −x := (x+ )−1 (o) (i.e. ∀x ∈ P : x + (−x) = o), and ◦ : P → P P ; x → x◦ := x+ ν, P ◦ := {x◦ | x ∈ P }. For the proofs of (6.1), (6.2), (6.3), (6.4) see [7] and [6]. (6.1) The following statements are equivalent:

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(1) ∀a ∈ P

∃1 x ∈ P : x + a = o,

(2) ν ∈ SymP , (3) P ◦ ⊆ SymP . (6.2) P ◦ ⊆ J ⇔ (P, +) satisﬁes the condition (). (6.3) If (P, +) satisﬁes the condition () then (P, P ◦ ) is an involution set with o ∈ Pr or equivalently (P, ◦ ; o) is a reﬂection structure. (6.4) If (P, I) is an involution set with Pr = ∅ then in each point o ∈ Pr , the loop derivation turns P into a left loop (P, +o ) satisfying (). (6.5) Let (P, I) be an involution set with P = Ps and Pr = ∅, let o ∈ Pr , (P, +o ) the loop derivation in o and (P, E, I, ) the graph derivation. Then the following statements are equivalent: (1) νo ∈ Aut(P, +o ), (2) (P, I) satisﬁes (Bo ), (3) (P, E, I, ) satisﬁes the local trapezium axiom (T ro◦ ). Proof. “(1) ⇔ (2)” is proved in (5.2.2).“(2) ⇒ (3)”: Let (A, B, C) ∈ T rp with c(A) = o◦ , a := [A] ∩ [B], β := c(B), let (X, Y, Z) ∈ T r(A, B, C) and x := [X] ∩ [Y ]. Then [A] = {a, o◦ (a)} and since [A] ∩ [C] = ∅, β(a) = a, o◦ (a), and by (2.5) τ (a; o◦ , β) = (A, B, C) and τ (x; o◦ , β) = (X, Y, Z), moreover o◦ (x) =: x1 ∈ [X] and o◦ β(x) =: z1 ∈ [Z]. By (2), there exists γ ∈ I with o◦ βo◦ = γ, i.e. with γo◦ = o◦ β. Then γ(x1 ) = γo◦ (x) = o◦ β(x) = z1 , and so [ε(x1 , γ)] = {x1 , z1 }. “(3) ⇒ (2)”. Let β ∈ I \ {o◦ } and p ∈ P ; by P = Ps , o◦ (p) = β(p) hence τ (p; o◦ , β) ∈ T rp . In particular τ (o; o◦ , β) ∈ T rp . By the local trapezium axiom T ro◦ there is a γ ∈ I such that ∀x ∈ P, τ (x; o◦ , β) ∈ T r(τ (o; o◦ , β)) is such that [ε(x1 , γ)] = {x1 , γ(x1 )} = {x1 , z1 } (where x1 := o◦ (x) and z1 := o◦ β(x)) , i.e. γ(x1 ) = z1 . Now o◦ (p) = β(p) implies p = βo◦ (p) hence τ (o◦ (p); o◦ , β) ∈ T rp and τ (o◦ (p); o◦ , β) ∈ T r(τ (o; o◦ , β)). Thus [ε(o◦ (o◦ (p)), γ)] = [ε(p, γ)] = {p, γ(p)} = {p, o◦ βo◦ (p)}, i.e. o◦ βo◦ (p) = γ(p) and so o◦ βo◦ = γ. (6.6) Let (P, I) be an involution set with Pr = ∅, o ∈ Pr and (P, +o ) the loop derivation in o. Then: (1) (P, +o ) is a loop with the property () ⇔ P = Pr ,

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(2) (P, +o ) is a K-loop ⇔ (P, I) is invariant. Proof. See [8], (3.4.(2),(3)). (6.7) Let (P, I) be an involution set with P = Pr ; for p ∈ P let p˜ := [[p ↔ p], (P, +p ) the loop derivation in p, and νp the negative map of (P, +p ) (cf. §5) and let If = {α ∈ I | Fix α = ∅} and P˜ := {˜ p | p ∈ P }. Moreover let (P, E, I, ) be the graph derivation of (P, I). Then: (1) ∀α, β ∈ I, α = β ⇒ Fix α ∩ Fix β = ∅, (2) If = P˜ and ∀α ∈ If , ∀a ∈ Fix α : (3) ˙ α∈IIf Fix α = P , (4) id ∈ I ⇔ P˜ = {id} ⇔ ∀ ∀p ∈ P,

a ˜ = α,

νp = id,

(5) ∀ ∀p ∈ P : νp ∈ Aut(P, +p ) ⇔ ∀α ∈ If : αIα = I ⇔ (P, E, I, ) satisﬁes the local trapezium axiom T rα for all α ∈ If , (6) Let o ∈ P be ﬁxed, + := +o , ν := νo and 2P := {x ∈ P | Fix(x+ ν) = ∅} then the equivalent conditions of (5) are valid if and only if () ∀(a, b) ∈ 2P × P :

a+ νb+ νa+ = (a − (b − a))+ .

Proof. (1) Assume a ∈ Fix α ∩ Fix β; then α, β ∈ [a ↔ a] hence α = β by |[a ↔ a]| = 1. ˜, α ∈ [a ↔ a] (2) Clearly P˜ ⊆ If . Let α ∈ If and a ∈ Fix α; then a hence α = a ˜. (3) and (4) are consequences of (1) and (2), and (5) is a consequence of (6.5) and (2). (6) If (P, +) is the loop derivation of (P, I) in o then I = P + ν and so If = (2P )+ ν. Therefore the second condition of (5) has the translation: ∀(a, b) ∈ 2P × P : a+ νb+ νa+ ν = c+ ν for a suitable c ∈ P . Since c = c+ ν(o) = a+ νb+ νa+ ν(o) = a − (b − a) we obtain a+ νb+ νa+ = (a − (b − a))+ . (6.8) If (P, +) is a loop with the properties () and () and if 2P = P then (P, +) is a 2-divisible K-loop. Proof. From () and 2P = P , it follows that I = P + ν is invariant, hence by (6.6.2), (P, +) is a K- loop and so, by (4.1), (P, +) satisﬁes the condition (K). Therefore for all a ∈ P , δa,a = δa,0+a = δa,0 = id and so a = a + (a − a) = (a + a) + δa,a (−a) = (a + a) − a = (a + a)+ ν(a), i.e. if a ∈ Fix(x+ ν) then x = a + a.

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7.

Helmut Karzel, Silvia Pianta and Elena Zizioli

Automorphisms and loop-nearring construction

Assume now that the graph (P, E, I, ), hence the involution set (P, E), o) is a has a regular point o ∈ Pr : this is equivalent to saying that (P, E; reﬂection structure and, by (6.3), (6.4), it gives rise to a left loop (P, +) with the property (). Our aim is now to connect the automorphisms of these three structures. A permutation σ ∈ SymP is called automorphism of Γ = (P, E, I, ) if: ∀E, F ∈ E with E F : σ(E) ∈ E and σ(E) σ(F ). Let us denote with AutΓ the automorphism group of Γ . It follows immediately (cf. [8]): (7.1)

AutΓ = Aut(P, E = I).

then σ(o) is again a regular point and σ is an (7.2) If σ ∈ Aut(P, E) o) and the isomorphism between the left loop (P, +) associated to (P, E; left loop (P, + ) associated to (P, E; σ(o)). Proof. See [8]. On the other hand, we have: (7.3) Let (P, +) and (P, + ) be left loops with negative maps ν and ν , respectively, satisfying () and let σ : P → P be an isomorphism. If (P, I; o) and (P, I ; o := σ(o)) are the associated reﬂection structures, then I = σIσ −1 .

Proof. Since σ ◦a+ (x) = σ(a+x) = σ(a)+ ◦σ(x) hence σ ◦P + = P + ◦σ and since o = σ(o) = σ(x+ν(x)) = σ(x)+ σν(x) thus ν (σ(x)) = σν(x) i.e. ν ◦ σ = σ ◦ ν. Therefore σI = σP + ν = P + σν = P + ν σ = I σ so I = σIσ −1 . and From (7.2) and (7.3) we obtain: σ ∈ Aut(P, +) ⇔ σ ∈ Aut(P, E) σ(o) = o. Thus (7.4) The automorphisms of the left loop are exactly the automorphisms of the graph with parallelism ﬁxing the regular point o. The study of automorphisms and, more generally, of endomorphisms of left loops is connected to the construction of certain structures with two binary operations which generalize the notion of a nearﬁeld (for general references on this subject see e.g. [4], [5], [12]). We recall that a nearﬁeld (N, +, ·) is a non empty set N provided with two binary operations “+” and “·” such that: 1. (N, +) is a group (with neutral element 0),

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2. (N ∗ := N \ {0}, ·) is a group, 3. ∀a, b, c ∈ N a(b + c) = ab + ac. For any a ∈ N let a· : N → N ; x → ax and let N · := {a· | a ∈ N }. Conditions 1. and 2. were weakened by: 1’. (N, +) is a left loop, 2’. (N, ·) is a semigroup, and in that way the notion of a nearﬁeld was extended to: - (N, +, ·) is a nearring if 1., 2’. and 3. are valid, - (N, +, ·) is a near-domain if (N, +) is a K-loop, 2. and 3. are valid and moreover ∀a, b ∈ N : δa,b ∈ N · , - (N, +, ·) is a left loop-nearring if 1’., 2’. and 3. are satisﬁed. Thus in a left loop-nearring (N, +, ·) one can consider besides the permutations a+ also the maps a· which are, by 3., endomorphisms of (N, +) fulﬁlling the property: 4. ∀a, b ∈ N (a· (b))· = a· ◦ b· . It is straightforward to verify that, for any a ∈ N , the set ker a· := {x ∈ N | a · x = 0} is a left subloop of (N, +). So we can distinguish between the cases: ker a· = N , ker a· = {0}, N , ker a· = {0} ∧ a· (N ) = N and ker a· = {0} ∧ a· (N ) = N (i.e. a· ∈ SymN ). We consider the following subsets: A := {a ∈ N | ker a· = N } the set of zero-multipliers and U := {u ∈ N | a· ∈ SymN }. Then (A, ·) is a sub-semigroup of (N, ·) and if U = ∅ then each u ∈ U deﬁnes an automorphism u· of (N, +) hence (U, ·) is a sub-semigroup of (N, ·) and U · := {u· | u ∈ U } is a sub-semigroup of Aut(N, +). Planar nearrings N satisfy the condition N = A ∪ U . Here we have: (7.5) Let (N, +, ·) be a left loop-nearring such that N = A ∪ U then U · ≤ Aut(N, +) and “0· = 0 (0 is the zero-map)⇒ 0· = id”. Proof. By 4., (U · , ◦) is a semigroup. Let a ∈ U , then: a· ◦ ((a· )−1 (a))· = (a· ((a· )−1 (a))· = a· ⇒ ((a· )−1 (a))· = 0 4 4.

hence by N = A ∪ U : ((a· )−1 (a)) ∈ U and so ((a· )−1 (a))· = id. Since a· ◦ ((a· )−2 (a))· = a· ((a· )−2 (a))· = ((a· )−1 (a))· = id 4.

⇒ (a· )−1 = ((a· )−2 (a))· . If 0· is not the zero-map then, by N = A ∪ U , 0 ∈ U hence 0· = (0· )−1 (0))· = id. Conversely we can state the following:

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Helmut Karzel, Silvia Pianta and Elena Zizioli

(7.6) Let (P, +) be a left loop and let · : P → End(P, +); p → p· be a map such that condition 4. is fulﬁlled. Then (P, +, ·) with a · b := a· (b) is a left loop-nearring. Remark. In [11] some solutions to the problem of constructing proper loop-nearrings are given for the particular situation where P · := {p· | p ∈ P } is contained in the set Aut(P, +) ∪ {0} or, in other words, where the loop-nearring satisﬁes the assumptions of (7.5).

References [1] Andersen, L. D.: Factorization of graphs. CRC Handbook of Combinatorial Designs, Colburn,C.J. and Dinitz, J.H. eds, CRC Press, Boca Raton, FL, (1996), 653-667 `s, B.: Modern Graph Theory. GTM 184 Springer Verlag, New York [2] Bolloba (1998) [3] Bruck, R.H.: A Survey of Binary Systems. Springer Verlag, Berlin - Heidelberg - New York (1966) [4] Clay, J. R.: Nearrings. Geneses and Applications. Oxford University Press, New York (1992) [5] Cotti Ferrero, C., Ferrero, G.: Nearrings. Some developements linked to semigroups and groups. Advances in Mathematics, Kluwer Academic Press, Dordrecht (2002) [6] Gabrieli,E.,Im.B., Karzel, H.: Webs related to K-loops and Reﬂection Structures. Abh. Math. Sem. Univ. Hamburg 69 (1999), 89-102 [7] Karzel, H.: Recent Developments on Absolute Geometries and Algebraization by K-Loops. Discr. Math. 208/209 (1999), 387-409 [8] Karzel, H., Pianta, S. and Zizioli, E.: Loops, Reﬂection Structures and Graphs with parallelism. Results Math. 42 (2002), 74-80 [9] Kreuzer, A. : Inner mappings of Bol loops. Math. Proc. Cambridge Philos. Soc. 123 (1998), 53-57 [10] Kiechle, H.: Theory of K-loops. Lecture Notes in Mathematics 1778 Springer Verlag, Berlin (2002) [11] Pianta, S.: Loop-nearrings. Proceedings of the Conference: Nearrings and Nearﬁelds. Hamburg (2003) [12] Pilz, G.: Near-rings. North-Holland- American Elsevier, Amsterdam (1983) [13] Zizioli, E.: Connections between Loops of exponent 2, Reﬂection Structures and Complete Graphs with Parallelism. Results Math. 38 (2000), 187-194

SEMI-NEARRINGS OF BIVARIATE POLYNOMIALS OVER A FIELD Kent M. Neuerburg Mathematics Department Southeastern Louisiana University Hammond, LA 70402 USA [email protected]

Abstract

The idea of the composed product of univariate polynomials may be generalized to bivariate polynomials. Using these root-based compositions, we deﬁne operations on bivariate polynomials analogous to addition and composition of univariate polynomials. We investigate the seminearring of bivariate polynomials determined by these operations looking at its properties and internal algebraic structures. 2000 Mathematics Subject Classiﬁcation: 16Y30.

Keywords: seminearring, composed products

1.

Introduction

If R is a commutative ring with identity, the nearring of polynomials (R[x], +, ◦) is well known. In [4], the concept is extended to multivariate polynomials in R[x1 , . . . , xn ] = R[x] by deﬁning the composition of f (x) and g(x) by f ◦ g = f (g(x), . . . , g(x)). We would like to introduce an alternative for bivariate polynomials over a ﬁeld in which the operations closely mimic that of univariate polynomials. The operations used are determined by root-based composition of the component polynomials. Such root-based compositions are generally referred to as composed products. We show that such operations lead to a seminearring and then consider the ideal structure.

2.

The composed product

The idea of the composed product of polynomials has its origin in the study of groups over ﬁnite ﬁelds. Let F be the ﬁeld of q elements, with q a power of a prime, and let Γ denote the algebraic closure of F. Let G ⊆ Γ be nonempty and be a binary operation on Γ such that (G, ) 253 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 253–262. c 2005 Springer. Printed in the Neatherlands.

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forms a group. Let f, g ∈ F[x] be nonconstant monic polynomials all of . . , am we may write whose roots lie in G. If the roots of f are a1 , a2 , . f (x) = (x − ai ). Similarly, we may write g(x) = (x − bj ). We deﬁne the composed product of f, g ∈ F[x] as (f

(x − ai bj ).

g)(x) = i

j

We note that the degree of f g is the product of the degrees of f and g. Further, we observe that this represents an entire class of root-based compositions of polynomials distinguished by the choice of the operation . It should be noted that while the historic roots of the composed product may lie in the theory of ﬁnite ﬁelds, there is, in general, no reason to restrict oneself to ﬁnite ﬁelds. In fact, one may consider the composed product deﬁned over any ﬁeld K. Further, since one immediately passes to the algebraic closure of K, one may assume that K is an algebraically closed ﬁeld. For the remainder of this paper we also assume that char(K) = 0. Finally, one may move from (G, ) being a group to the case where G is merely closed under the -operation.

3.

The univariate case

Let (K, +, ·) be any ﬁeld. We begin by specifying two versions of the composed product determined by the ﬁeld addition and ﬁeld multiplication in K. Deﬁnition 3.1. Let f (x) = (x − ai ) and g(x) = (x − bj ) be in K[x] and let G = K, then the composed addition of f and g is (x − (ai + bj )).

(f g) = i

j

Further, the composed multiplication of f and g is (x − (ai · bj )).

(f • g) = i

j

Since K is algebraically closed, we know every f (x) ∈ K[x] splits over K; i.e, we can write f (x) = (x − ai ). Taking G = K we have the following theorem. Theorem 3.2. With the - and •-operations deﬁned above, (K[x], ) forms a commutative semigroup with identity, z(x) = x and (K[x], •) forms a commutative semigroup with identity i(x) = x − 1.

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Proof. It’s easy to see from the deﬁnition of the -operation that (K[x], ) is a commutative semigroup with identity z(x) = x. We also note that since degx (f g) = (degx f )(degx g), the only elements of K[x] having a -inverse are the linear polynomials f (x) = x − a, a ∈ K. Similarly, the deﬁnition of the •-operation leads one to conclude that (K[x], •) forms a commutative semigroup with identity element i(x) = x−1 where 1 is the multiplicative identity of K. Analagous to the case of the -operation, the only •-invertible elements are the linear polynomials f (x) = x − a, a ∈ K∗ . ), g(x) = Finally, we observe that if f (x) = (x − a i (x − bj ) and )) = (x − ai (bj + h(x) = (x − ck ) then f • (g h) = f• ( (x − (bj + ck ck )) = (x − (ai · bj + ai · ck )) = (x − (ai · bj )) (x − (ai · ck )) = (f • g) (f • h). Hence, the •-operation is left distributive over the operation. A similar argument shows that the •-operation is also right distributive over the -operation. Thus, both distributive laws hold in (K[x], , •). We note that (K[x], , •) is not a semiring since degree arguments will show that f • z = z = z • f if and only if the degx f = 1.

4.

The bivariate case

Certainly, the notion of the composed product may be generalized to the case of bivariate polynomials. This generalization is speciﬁcally deﬁned and explored in [6]. In particular, suppose K is an algebraically closed ﬁeld and that f (x, y), g(x, y) ∈ K[x, y] with the property that as polynomials in y, f and g both split over K[x]: m

(y − ai (x))

f (x, y) = i=1

and n

(y − bj (x)).

g(x, y) = j=1

With the roots being polynomials in x, there are some obvious choices for the diamond operations – addition and multiplication. However, we now have an additional operation that was not available to us in the univariate case: function composition. We next deﬁne these three speciﬁc root-based composition operations.

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Deﬁnition 4.1. With K, f (x, y), and g(x, y) as above, we take G = K[x]. Deﬁne the composed addition of f and g to be (y − (ai (x) + bj (x)),

(f g)(x, y) = i

j

the composed multiplication of f and g to be (f • g)(x, y) =

(y − (ai (x) · bj (x)), i

j

and the composed product of f and g to be (f

(y − (ai (x) ◦ bj (x)),

g)(x, y) = i

j

where ai (x) + bj (x) is the usual addition in the ring K[x], ai (x) · bj (x) is the usual multiplication in K[x], and ai (x) ◦ bj (x) represents the usual function composition ai (bj (x)). Theorem 4.2. Let m

(y − ai (x)), ai (x) ∈ K[x]}.

M = {f (x, y) ∈ K[x, y] | f (x, y) = i=1

Then (M, , ) forms a seminearring with unity. Proof. As in the univariate case, it is clear from the deﬁnition of the -operation that (M, ) forms a commutative semigroup with identity z(x, y) = y. The only elements of M with additive inverse are those polynomials linear in y, f (x, y) = y − a(x) with a(x) ∈ K[x]. Similarly, (M, ) forms a non-commutative semigroup with identity i(x, y) = y − x. For univariate polynomials a, b, and c we know that (a + b) ◦ c = (a ◦ c) + (b ◦ c). Therefore we conclude that we also have (f g) h = (f h) (g h). Likewise, since we know that in general c ◦ (a + b) = (c ◦ a) + (c ◦ b) we also have that, in general, h (f g) = (h f ) (h g). Thus, it immediately follows that is right distributive over , but is not left distributive over . Let us look at some of the basic properties of this seminearring. (1) Under the -operation, the zero element is neither left-cancellative nor right-cancellative. (2) Because of degree arguments, the only elements of M that could have annihilators under the -operation are the linear forms in y and their annihilators must also be linear in y.

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(3) If a(x) ∈ K[x] and {a1 , . . . , an } are the zeros (working in K[x]) of a(x) then the right annihilators of y − a(x) are {f (x, y) = (y − ai )|1 ≤ i ≤ n}. (4) If c ∈ K then the left annihilators of f (x, y) = y − c are {f (x, y) = y − (x − c) · q(x) | q(x) ∈ K[x]}. The proofs of the above properties all follow easily from the deﬁnitions. We provide the following as an example. Theorem 4.3. The -idempotents of (M, , ) are {(y − a0 ) | a0 ∈ K} {i(x, y) = y − x}. Proof. If f (x, y) f (x, y) = f (x, y) we must have degy f = 1 since degy (f f ) = (degy f )(degy f ) = degy f . Let f (x, y) = y − a(x) where a(x) = an xn +an−1 xn−1 +· · ·+a0 . If f f = f we must have a(x)◦a(x) = a(x); i.e., an (an xn + · · · + a0 )n + · · · + a0 = an xn + an−1 xn−1 + · · · + a0 . Comparing degrees, we see the above equality can only happen if n ≤ 1. Assuming this restriction upon the degrees, we have a1 (a1 x + a0 ) + a0 = a1 x + a0 which is a21 x + a1 a0 + a0 = a1 x + a0 . Comparing coeﬃcients we have the system a21 = a1 a1 a0 + a0 = a0 If a0 = 0 then we must have a1 = 0 so f (x, y) = y − a0 . On the other hand, if a0 = 0 then either a1 = 0 making f (x, y) = y − 0 or a1 = 1 giving f (x, y) = y − x.

5.

Structure results

5.1

Sub-seminearrings n

Let M0 = {f (x, y) =

(y − ai (x)) | ai (0) = 0, ∀ 1 ≤ i ≤ n}. We obi=1

serve that M0 will be closed under both the - and -operations. Note that the additive identity z(x, y) = y ∈ M0 as is the multiplicative identity, i(x, y) = y − x. With the rest of the properties being inherited from (M, , ), we have shown the following.

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Theorem 5.1. With M0 as deﬁned above, (M M0 , , ) forms a sub-seminearring of (M, , ). n

(y − ai (x)) be an element of (M, , ) that commutes

Let f (x, y) = i=1

with z(x, y) = y − 0. Then we must have n

(y − 0)

n

y = i=1

n

= (y − 0)

(y − ai (x))], which, by assumption,

[ i=1

n

(y − ai (x))]

=[

(y − 0)

i=1 n

(y − ai (0))

= i=1

Which implies that ai (0) = 0 for all 1 ≤ i ≤ n. We have just proved the following corollary. Corollary 5.2. The centralizer of zero, C(z), in (M, , ) is M0 . Next, consider the set M1 = {f (x, y) ∈ M | degy f = 1}. By considering the degrees of the composed sum and composed product, we see that M1 is closed under both operations. Further, both the additive and multiplicative identities are contained in M1 . Thus we have the following. M1 , , ) Theorem 5.3. Given M1 = {f (x, y) ∈ M | degy f = 1}, (M forms a sub-seminearring of (M, , ). Corollary 5.4. Deﬁned as above, (M M1 , , ) is a nearring. Further, under the mapping φ : (M M1 , , ) → (K[x], +, ◦) deﬁned by φ : y−a(x) → a(x), we see that (M M1 , , ) ∼ = (K[x], +, ◦). Proof. We know that (M M1 , , ) is a seminearring. But, if f (x, y) = y − a(x) ∈ M1 , we have that g(x, y) = y − (−a(x)) ∈ M1 with f (x, y) g(x, y) = y − 0 = z(x, y) Hence, M1 is a commutative semigroup with identity under the -operation, making (M M1 , , ) a nearring. To see that (M M1 , , ) ∼ = (K[x], +, ◦), let f = (y − a(x)), g = (y − b(x)) ∈ M1 with φ(f ) = φ(g). Then a(x) = b(x) which implies that f = g. For every a(x) ∈ K[x] we have a(x) = φ(y − a(x)) Thus φ is both injective and surjective. Finally, φ(f g) = φ(y − (a(x) + b(x))) = a(x) + b(x) = φ(f ) + φ(y). Similarly, φ(f g) = φ(y − (a(x) ◦ b(x))) = a(x) ◦ b(x) = φ(f ) ◦ φ(y).

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Semi-nearrings of Bivariate Polynomials over a Field

5.2

Internal Structures and Ideals

We now turn to some of the internal structures assosicated with the seminearring (M, , ). In particular, we will investigate the center of M as well as some of the ideal structure of M . We begin by looking at the center of M . Theorem 5.5. Let K be a ﬁeld of characteristic zero. Let M be deﬁned as in Theorem 4.2 above and let N = (K[x], +, ◦) be the standard unim

(y − ai (x))

variate nearring of polynomials. The polynomial f (x, y) = i=1

is an element of the center of M if and only if ai (x) is an element of the center of N for all 1 ≤ i ≤ m. m

(y − ai (x)) and that ai (x) is an element of

Proof. Assume f (x, y) = i=1

n

(y − bj (x)) be

the center of N for all 1 ≤ i ≤ m. Then let g(x, y) = j=1

any element of M . We have m

f (x, y)

n

(y − ai (x))]

g(x, y) = [

i=1 m n

(y − bj (x))]

[

j=1

(y − ai (x) ◦ bj (x))

= i=1 j=1 n m

(y − bj (x) ◦ ai (x))

= j=1 i=1 n

m

(y − bj (x))]

=[ j=1

(y − ai (x))]

[ i=1

= g(x, y)

f (x, y)

Hence f (x, y) is in the center of M . m

(y − ai (x)) is in the center of M . We

Next, assume that f (x, y) = i=1

will show that ai (x) is in the center of N for all 1 ≤ i ≤ m. If f is in the center of M , then f g = g f for all g ∈ M . In particular, if we

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set g(x, y) = y − c for some constant c ∈ K we have m

f (x, y)

(y − ai (x))]

g(x, y) = [

[y − c]

i=1 m

(y − ai (x) ◦ (c))

= i=1 m

(y − ai (c))

= i=1

Which should equal m

g(x, y)

f (x, y) = [y − c]

(y − ai (x))]

[ i=1

m

(y − c ◦ ai (x))

= i=1 m

(y − c)

= i=1 m

m

(y − ai (c)) =

Hence, i=1

(y − c). This implies ai (c) = c for all 1 ≤ i ≤ i=1

m. Further, since c ∈ K is arbitrary and char(K) = 0, we must have ai (x) = x for all 1 ≤ i ≤ m. It is easy to see that ai (x) = x is an element of the center of N . We next turn to the ideal structure of M . In [5], van Hoorn and van Rootselaar deﬁne the ideal of a seminearring to be the kernal of an admissible morphism. Ahsan [2] generalizes this idea to the notion of an S-ideal. Deﬁnition 5.6. Let (R, +, ·) be a seminearring. A right (resp. left) S-ideal of R is a subset I ⊆ R with the properties that (1) for all x, y ∈ I, x + y ∈ I, (2) for all x ∈ I and for all y ∈ R, y · x ∈ I (resp. x · y ∈ I). If I is both a right S-ideal and a left S-ideal, we say it is an S-ideal of R. For seminearrings we have the following types of ideals, see [3].

Semi-nearrings of Bivariate Polynomials over a Field

261

Deﬁnition 5.7. Let R be a seminearring and P be an S-ideal of R. We say P is prime if IJ ⊆ P =⇒ = I ⊆ P or J ⊆ P for all S-ideals I, J of R. The S-ideal P is completely prime if for a, b ∈ R, ab ∈ P =⇒ = a∈P or b ∈ P . If P is minimal in the set of prime S-ideals we say P is minimal prime. Using these deﬁnitions we may identify two classes of S-ideals in M . Each of these classes contains inﬁnitely many S-ideals. Theorem 5.8. For each n ∈ Z+ , let In = {f (x, y) ∈ (M, , )| degy f = kn, k ∈ Z+ }, then In is an S-ideal of M . Proof. Let f, g ∈ In with degy f = k1 n and degy g = k2 n. Also, let h ∈ M with degy h = l. Now degy (f g) = (degy f )(degy g) = (k1 n)(k2 n) = (k1 k2 n)n showing f g ∈ In . Similarly, deg(f h) = (degy f )(degy h) = (k1 n)(l) = (k1 l)n; whence, f h ∈ In . Corollary 5.9. Let p ∈ Z+ be a prime and let Ip = {f (x, y) ∈ M | degy f = kp, k ∈ Z+ }. Then Ip is completely prime. Proof. By the previous theorem, we know Ip is an S-ideal. Now let g) = f, g ∈ (M, , ) with f g ∈ Ip . Then p divides degy (f =⇒ p divides degy f or p divides degy g. Thus, f ∈ Ip (degy f )(degy g) = or g ∈ Ip making Ip completely prime.

6.

Extensions

Recall that the characteristic of K = 0. Let H be the set of holomorphic power series over K. Then if n

(y − ai (x)) with ai (x) ∈ H

f (x, y) = i=1

and

m

(y − bj (x)) with bj (x) ∈ H

g(x, y) = j=1

we can extend our earlier deﬁnitions of the composed addition, composed multiplication, and composed product in the obvious way. If we set U = {f (x, y) = (y − ai (x)) | ai (x) ∈ H for 1 ≤ i ≤ n} then we see that, just as in the polynomial case, (U, , ) forms a seminearring with -identity z(x, y) = y and -identity i(x, y) = y − x. Of course, (M, , ) is a sub-seminearring of (U, , ). The theorems proved above for the polynomial case have the obvious and immediate analogues when one moves to holomorphic power series.

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Kent M. Neuerburg

We note that the original concept of the bivariate composed product was developed working in the Puiseux ﬁeld over K, see [6]. Indeed, if we take K to be an algebraically closed ﬁeld and f (x, y) = y n + a1 (x)y n−1 + · · · + an (x) ∈ K((x))[y] with n > 0 and assume that char(K) = 0 then there exists a positive integer m such that n

(y − pi (t))

m

f (t , y) = i=1

1

Eliminating the parameter yields pi (t) = pi (x m ) which is called the Puiseux series associated to f (x, y) (see, for example, [1]).

7.

Open questions

There are, of course, many directions in which to proceed. For example, (1) Can one describe the centers of M0 , M1 , or U ? (2) What if any distributive elements are there in M0 , M1 , M , or U ? (3) Are there other ideals in M0 , M1 , M , or U and what are they? (4) What changes occur when we allow K to have positive characteristic? Is there an analogue to U in this case?

References [1]

Abhyankar, Shreeram, Algebraic Geometry for Scientists and Engineers, American Mathematical Society, Providence, 1990, 89-98.

[2]

Ahsan, Javed, Seminearrings characterized by their S-ideals. I. Proc. Japan. Acad., 71, Ser. A. 1995 101-103.

[3]

Ahsan, Javed and Zhongkui, Liu, Strongly idempotent seminearrings and their prime ideal spaces. Nearrings, Nearﬁelds, and K-Loops, G Saad and M.J. Thompson eds., Kluwer Academic Publishers, Netherlands, 1997, 151-166.

[4]

Gutierrez, Jaime and Ruiz de Velasco, Carlos, Polynomial near-rings in several variables, Near-rings and near-ﬁelds (Stellenbosch, 1997), Yuen Fong, Carl Maxson, John Meldrum, Gunter ¨ Pilz, Andries van der Walt and Leon van Wyk eds., Kluwer Academic Publishers, Dordrecht, 2001, 94-102.

[5]

van Hoorn, Willy G. and van Rootselaar, B., Fundamental notions in the theory of seminearrings. Compositio Math., 18, 1967, 65-78.

[6]

Mills, Donald and Neuerburg, Kent M., A bivariate analogue to the composed product of polynomials. Algebra Colloquium, 10, 2003, no. 4, 451-460.

AUTOMORPHISM GROUPS EMITTING LOCAL ENDOMORPHISM NEARRINGS II Gary L. Peterson James Madison University Harrisonburg, Virginia 22807 USA [email protected]

Introduction The basic theory of local right nearrings with identity was developed by C. J. Maxson in [5]. We shall be dealing with left nearrings R with identity in this note. One of the various equivalent ways of deﬁning localness for nearrings from [5] modiﬁed to the left nearring setting is to say that R is local if L = {r ∈ R|r does not have a right inverse} is a right R-subgroup of R. Local endomorphism nearrings have been studied in [4], [8], [9], and [10] and an application of them to the study of ﬁnite p-groups G in which I(G) = A(G) appears in [11]. The main purpose of this paper is to continue the work begun in [9] dealing with the study of ﬁnite groups G for which the endomorphism nearring of G generated by a group of automorphisms of G is local. This note consists of two sections. The ﬁrst reviews and extends general results from [4], [8], [9], and [10]. The second deals with giving necessary and suﬃcient conditions for a group from a class of groups arising in an example in [9] to give rise to a local endomorphism nearring. Some of the results of this paper were obtained while the author was a guest of the Institut f¨ ur Algebra, Stochastik und Wissenbasierte Mathematische Systeme at Johannes Kepler Universit¨¨at in Linz, Austria in May and June, 2002. The author thanks the university for its hospitality during his stay.

1.

General Results

Throughout, R will denote the endomorphism nearring of an additive group G generated by a semigroup of endomorphisms S of G containing 263 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 263–276. c 2005 Springer. Printed in the Neatherlands.

264

Gary L. Peterson

the identity map on G. We will write our maps on the right making R into a left nearring. Moreover, we will assume that R satisﬁes the descending chain condition on right ideals (dccr). In its early stages in [4], [8], and [9], the study of local endomorphism nearrings focused on the case when R is a compatible endomorphism nearring of G; that is, when the inner automorphism group of G is contained in R. (In [4], [8], and [9], the terminology of [7] was followed where such endomorphism nearrings are called tame. Here, as was done in [10], we are following the terminology of Stuart Scott introduced in [12] calling such nearrings compatible. Tameness is a weaker condition in Scott’s terminology.) In [10], it was shown that the compatibility assumption can be relaxed obtaining many of the same results about local endomorphism nearrings as in the compatible case. To see how this is done, we begin by recalling from [10] that R is weakly compatible if there exists a subideal series S 0 = G0 < G1 < . . . < Gn = G

(1.1)

of G such that each factor Gi+1 /Gi is a compatible R-module. (Some of the results we are about to state hold in the less restrictive weakly tame setting, but we will not get into this here to keep the presentation simpler.) Any endomorphism nearring of a solvable group G is seen to be weakly compatible by using the derived series of G as the subideal series S. Since R satisﬁes dccr, we may use the the subideal series S of (1) to form the S-socle series of G 0 = S0 < S1 < S2 < . . . < St = G

(1.2)

by forming the socle series of each compatible factor Gi+1 /Gi . Each Si is a direct sum S-socle factor Si+1 /S Si = ⊕ Mij Si+1 /S j

where each Mij is a type 2 R-module. In fact, each type 2 R-module is isomorphic to some Mij [10, Theorem 1.6]. We know from [10, Theorem 3.1] that localness imposes a strong condition on type 2 R-modules: J2 (R) for all i and j. FurTheorem 1.1. If R is local, then Mij ! R/J Si is either an elementary abelian p-group or a direct ther, each Si+1 /S sum of copies of the additive group of the rational numbers. We get a converse to Theorem 1.1 if we impose nilpotency on G [10, Theorem 3.3]:

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Theorem 1.2. Suppose that G is a nilpotent group. If Mij ! R/J J2 (R) for all i and j, then R is local. Note that if G is a ﬁnite group in Theorem 1.1, then G is a p-group. Likewise G is a p-group in Theorem 1.2 if G is ﬁnite; otherwise, choosing minimal R-ideals from distinct Sylow subgroups of G would give us nonisomorphic minimal R-ideals which cannot both be isomorphic to R/J J2 (R). Thus Theorem 1.2 is the converse of Theorem 1.1 for ﬁnite groups. (It also is the converse if R is compatible since G is nilpotent when R is local [8, Theorem 2.1].) The next result gives us an alternative way to using Theorems 1.1 and 1.2 in checking for localness. Theorem 1.3. Suppose that G is a nilpotent group with Mij ! Mkl for all i, j, k, and l. The following are equivalent: (i) R is local. Mij )is a division ring for some i and j. (ii) R/AnnR (M Mij ) = AnnR (m) for any 0 = m ∈ Mij . (iii) For some i and j, AnnR (M Proof. (i) ⇒ (ii) By [5, Theorem 3.2], R/J J2 (R) is a nearﬁeld. Since Mij ) is a maximal ideal of R, this forces AnnR (M Mij ) = J2 (R) for AnnR (M Mij ) is distributively generated, it must then any i and j. As R/AnnR (M be a division ring. Mij ) is a division ring and let 0 = m ∈ (ii) ⇒ (iii) Suppose R/AnnR (M Mij . We have that Mij ) ⊆ AnnR (m) ⊂ R. AnnR (M Mij ) is a division ring, AnnR (M Mij ) is a maximal right Since R/AnnR (M Mij ) = AnnR (m). ideal of R and hence we must have AnnR (M (iii) ⇒ (i) Choose 0 = m ∈ Mij . We then have Mij ). Mij = mR ! R/AnnR (m) = R/AnnR (M Mkl ) = AnnR (M Mij ) for all k and l, Since AnnR (M R/J J2 (R) = R/(∩AnnR (M Mkl )) = R/AnnR (M Mij ) = R/AnnR (m) ! Mij and this implication now follows from Theorem 1.2. In the case of ﬁnite groups, we have the following result the ﬁrst three parts of which appear in [10, Theorem 3.2]: Theorem 1.4. If G is ﬁnite and R is local, then (i) G is a p-group. (ii) R/J J2 (R) is a ﬁnite ﬁeld of characteristic p.

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(iii) If A = R ∩ Aut(G) (Aut( ( G) the automorphism group of G), ∗ J2 (R) − {0}, and δ : A → (R/J J2 (R))∗ is the mul(R/J J2 (R)) = R/J tiplicative homomorphism obtained by restricting the natural projection from R onto R/J J2 (R) to A, then ker(δ) is the Sylow p-subgroup P of A and P has a cyclic complement K in A with |K| dividing pn − 1 where |R/J J2 (R)| = pn . (iv) K acts as a group of ﬁxed point free automorphisms of G. Proof. Since (i) − (iii) have been proved in [10], we need only prove (iv). As the action of R/J J2 (R) on itself is multiplication by the elements of this ﬁeld, each nonidentity element of K acts ﬁxed point free J2 (R) for all i and j by Theorem 1.1, each on R/J J2 (R). Since Mij ! R/J nonidentity element of K must then act ﬁxed point free on each Mij . This in turn forces each nonidentity element of K to act ﬁxed point free on G. Part (i) of Theorem 1.4 tells us that we may restrict our attention to p-groups G when studying weakly compatible endomorphism nearrings of ﬁnite groups that are local. When doing so, we may then omit mention of the endomorphism nearrings being weakly compatible since p-groups are solvable groups. Further, the fact that ﬁnite p-groups satisfy the stronger condition of being nilpotent gives us that G has a socle series terminating in G [10, Corollary 1.8] which is less cumbersome to work with than the S-socle series mentioned earlier. Because of this, we will henceforth use the socle series for the subideal series S in (1), which is its own S-socle series, in our future work. Finally, while it is a simple consequence, the observation in part (iv) that K acts as a group of ﬁxed point free automorphisms went unnoticed in the past. With it we then have yet another connection between nearrings and ﬁxed point free automorphism groups and Frobenius groups. This raises the question of whether there are results about endomorphism nearrings containing ﬁxed point free automorphisms beyond those of the local situation under consideration here. Returning to the development in the local case, suppose that A is a group of automorphisms of a ﬁnite p-group G. Do we know of any cases where the endomorphism nearring R of G generated by A is local? If A is itself a p-group, then [10, Theorem 3.5], which we restate as our next result, tells us that R is local. Theorem 1.5. Let A be a p-group of automorphisms of a ﬁnite p-group G. If R is the endomorphism nearring of G generated by A, then R is local.

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The determination of those automorphism groups A of ﬁnite p-groups G for which the endomorphism nearring generated by A is local then reduces to studying the case when A is not a p-group. This is the focus of [9]. There it is noted that if R is local, then the splitting of R ∩ Aut(G) into the semidirect product of its Sylow p-subgroup and a cyclic complement in Theorem 1.4(iii) also occurs for A. While the work of [9] is done under the assumption that R is compatible, the compatibility assumption plays no role in this decomposition. Indeed, one way of seeing how to obtain this splitting of A is to observe that any Sylow p-subgroup Q of A is contained in P of Theorem 1.4 since P is a normal subgroup of R ∩ Aut(G). It then follows that Q = A ∩ P is a normal subgroup of A. Now, by the Schur-Zassenhaus Theorem, Q has a complement L in A. Since L is isomorphically embedded in Kδ under the mapping δ of Theorem 1.4, L is cyclic. Using the notation of [9], we will henceforth restrict our attention to endomorphism nearrings R of ﬁnite p-groups G generated by automorphism groups A that are the semidirect product of a normal Sylow p-subgroup P and a nontrivial cyclic complement K = α of order k in our search for local endomorphism nearrings. The next lemma will allow us to ignore P in our search. Lemma 1.6. Suppose that A is a group of automorphisms of a ﬁnite p-group G and A has a normal Sylow p-subgroup P . Write each socle series factor as a direct sum Mij Soci+1 (G)/Soci (G) = ⊕ j

where each Mij is a type 2 R-module, R being the endomorphism nearring of G generated by A. Then P acts trivially on each Mij . If S is the endomorphism nearring of G generated by a complement K of P in A, then the socle series of G relative to S is the same as the socle series of G relative to R and J2 (S) = J2 (R) ∩ S. Further, R is local if and only if S is local. Proof. Since the semidirect product of G and P is a p-group and hence is a nilpotent group, we have [M Mij , P ] < Mij for all i and j. But P being normal in A gives us that [M Mij , P ] is an R-subgroup of Mij and hence [M Mij , P ] = 0 since Mij is of type 2. Thus P acts trivially on each Mij . The trivial action of P on each Mij now gives us that each Mij is a type 2 S-module and hence socle series of G relative to R and S are the same. Because of this, J2 (S) = AnnS (M Mij ) = ( AnnR (M Mij )) ∩ S = J2 (R) ∩ S. i,j

i,j

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To prove the ﬁnal part, ﬁrst notice the trivial action of P on each Mij gives us that 1 − ρ ∈ i,j AnnR (M Mij ) = J2 (R) for each ρ ∈ P . Thus J2 (R). If R is local, we have that Mij is RR/J J2 (R) = (S + J2 (R))/J isomorphic to Mkl for all i, j, k, and l by Theorem 1.1 and consequently Mij is S-isomorphic to Mkl for all i, j, k, and l. By Theorem 1.4, J2 (R))/J J2 (R) is a ﬁeld and hence S/J J2 (S) = S/(J J2 (R)∩ R/J J2 (R) = (S +J Mij ), we now S) is a ﬁeld (isomorphic to R/J J2 (R)). As J2 (S) = AnnS (M have S is local by Theorem 1.3. For the converse, observe that S local gives us Mij is S-isomorphic to Mkl for all i, j, k, and l which in turn gives us that these are R-isomorphic as well since the action of P on each J2 (S) is a ﬁeld, we then get (S + J2 (R))/J J2 (R) = Mij is trivial. As S/J Mij ) is a ﬁeld and another application of Theorem R/J J2 (R) = R/AnnR (M 1.3 gives us that R is local. Next, so that we are dealing with endomorphism nearrings that have the possibility of not being rings, we will further restrict our attention to the case where G is nonabelian. Often the study of questions about endomorphism nearrings become simple in the case when G is abelian, but this is not the case in the study of localness. In fact, the determination of the local endomorphism rings of ﬁnite abelian p-groups is an open problem to this author’s knowledge. When G is a nonabelian, we have: Theorem 1.7. If R is local, then K has odd order. Proof. Suppose 2 divides |K|. By part (iv) of Theorem 1.4, K contains a ﬁxed point free automorphism of order 2 of G. But then G is abelian by [1, V, 8.18a]. Another result about |K| when G is nonabelian is the following theorem from [9] giving us a number theoretic condition insuring when R is not local. As with the splitting of A mentioned earlier, the compatibility assumption assumed throughout in [9] is not necessary here either. We are going to sketch the proof of this theorem for two reasons. One reason is to illustrate that compatibility never comes into play in this proof. The other reason is that we will note a corollary to this proof that will be useful in the next section. Readers desiring more details ﬂeshing out this proof should consult [9]. In the statement of this theorem, n denotes the positive integer so that |M Mij | = pn where Mij is an arbitrarily chosen type 2 summand of one of the factors Soci+1 (G)/Soci (G). Also, recall that k denotes the order of K = α.

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Theorem 1.8. If k does not divide pi +pj −1 for some 0 ≤ i ≤ j ≤ n−1, then R is not local. Proof. Suppose that R is local. Set L1 = G/(pG + γ2 (G)) and L2 = γ2 (G)/(pγ2 (G) + γ3 (G)) where γ2 (G) and γ3 (G) are the terms G and [G , G] of the lower central series of G. By Maschke’s Theorem, we can decompose L1 into a direct sum of type 2 R-modules L1 = H1 ⊕ H2 ⊕ · · · ⊕ Hm .

(1.3)

From Theorem 1.4 we obtain that R/J J2 (R) is the ﬁeld F = Zp [λ] where λ is the image of α under δ. Each Hr is a vector space over Zp and α acts as a linear transformation on Hr . The eigenvalues of α on Hr are i the conjugates of λ which are given by λp , i = 0, 1, . . . , n − 1. We next form the tensor products Li = Li ⊗Zp F (i = 1, 2) and choose a basis of i eigenvectors u0r , u1r , . . . , un−1r for Hr ⊗Zp F with uir (α ⊗ 1) = λp uir . Commutators give us a bilinear mapping from L1 × L1 onto L2 which extends to a bilinear map from L1 × L1 onto L2 . Since the vectors [uir , ujr ] span L2 and since i

j

[uir , ujs ](α ⊗ 1) = λp +p [uir , ujs ],

(1.4) i

j

it follows that the eigenvalues of α⊗1 on L2 are found among the λp +p . Since L2 contains an R-module isomorphic to H1 , λ is an eigenvalue of α on L2 . But then i j λ = λp +p for some 0 ≤ i ≤ j ≤ n − 1 and hence k divides some pi + pj − 1 for some 0 ≤ i ≤ j ≤ n − 1, contrary to our assumption. Notice that if R is local and k > 1, Theorem 1.8 tells us that k divides pi + pj − 1 for some 1 ≤ i ≤ j ≤ n − 1. Also, since pi + pj − 1 is always odd for such values of i and j, Theorem 1.7 is obtained as a consequence of Theorem 1.8. Finally, with λ as in the proof of Theorem 1.8, we have Mij | = pn |Zp [λ]| = |M from which it follows that k does not divide ph −1 for any positive integer h < n. In the proof of Theorem 1.8, L1 is the same as G/Φ(G) where Φ(G) is the Frattini subgroup of G. If L1 is a type 2 R-module (that is, m = 1 in equation (3)), equation (4) gives us that the eigenvalues of α on L2 i j are found among the λp +p with i < j since [ui1 , uj1 ] = 0 when i = j. Thus we have:

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Corollary 1.9. If G/Φ(G) is a type 2 R-module, k > 1, and R is local, then k divides pi + pj − 1 for some 1 ≤ i < j ≤ n − 1. In [9], a group is given where R is local when k does divide pi + pj − 1 for some 1 ≤ i ≤ j ≤ n − 1. The group is a member of a class of matrix groups arising in the study of Suzuki 2-groups (see [2]) and Zassenhaus groups (see [3]). (This author must, however, correct a misstatement made in [9] that is not related to the subject at hand. The group given there is not itself a Suzuki 2-group as stated because it satisﬁes [2, Theorem VIII.6.9d].) In the next section we will give necessary and suﬃcient conditions for a member of this class of matrix groups to give rise to a local endomorphism nearring.

2.

The groups A(m, θ)

The class of matrix groups referred to at the end of the last section are constructed as follows: Let F denote the ﬁnite ﬁeld F = GF(pm ) with m > 1 and let θ be a ﬁeld automorphism of F . Our class of groups is the set of matrices under the form ⎧⎡ ⎫ ⎤% ⎨ 1 0 0 %% ⎬ A(m, θ) = ⎣ a 1 0 ⎦%% a, b ∈ F ⎩ ⎭ b aθ 1 % which forms a group of order p2m under matrix multiplication. Further, A(m, θ) is nonabelian if and only if θ = 1. The work we are about to do does not require A(m, θ) to be nonabelian. Consequently, it also may be of value to readers interested in endomorphism rings of abelian groups. Using G to denote A(m, θ) as an additive group and ⎡ ⎤ 1 0 0 u(a, b) = ⎣ a 1 0 ⎦ , b aθ 1 our group operation on G becomes u(a, b) + u(a , b ) = u(a + a , b + b + aθ · a ). Note that N = {u(0, b)|b ∈ F } is a normal subgroup of G with N and G/N both isomorphic to (F, +) as groups. Also, it is easily seen that N = Φ(G).

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For each λ ∈ F ∗ we obtain an automorphism α of G by setting u(a, b)α = u(λa, λ · λθ · b). Let K = α and A be any automorphism group of G that is the semidirect product of its Sylow p-subgroup P and K. (By Lemma 1.6, the choice of Sylow p-subgroup P is immaterial.) Let |Zp [λ]| = pn . Note that Zp [λ] is a submodule of G/N of type 2. Consequently we will want n > 1 (or equivalently, λ not in Zp ) in our search for local endomorphism nearrings of nonabelian groups with K nontrivial by Theorem 1.8. A local endomorphism nearring with k = |K| = |λ| dividing some pi + pj − 1 as in Theorem 1.8 is obtained in [9] by using F = GF (24 ), θ to be the Frobenius automorphism xθ = x2 of F , λ to be a primitive ﬁfth root of unity in F , and (unnecessarily) P = Inn(G). In fact, we can give a complete characterization of when endomorphism nearrings (including endomorphism rings) constructed in the manner here are local: u

Theorem 2.1. Write xθ = xp with 0 ≤ u < m. The endomorphism nearring R of G generated by A is local if and only if there exists 1 ≤ v < n with k = |K| dividing pu − pv + 1. Proof. By Lemma 1.6 we may assume A = K. First suppose we have for F over L such a v. Let L = Zp [λ] and d1 = 1, d2 , . . . , dl be a basis i of R (where l = m/n). Notice that the action of an element i εi α (εi = ±1) on G/N is the same as multiplication by the element i εi λi of L. Thus G/N is the direct sum of the type 2 R-modules G/N = L ⊕ d2 L ⊕ · · · dl L Further L ! di L for each i as R-modules under the map a → di a. Note that k and pu + 1 are relatively prime since a nontrivial common divisor of them would divide pu +1−(pu −pv +1) = pv giving us p dividing u k. Since λ · λθ = λp +1 , λ · λθ = λ and hence Zp [λ] = Zp [λ · λθ].

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i Because the action of i εi α on N is multiplication by the element i i i εi λ (λθ) , as with G/N we get that N = L ⊕ d2 L ⊕ · · · dl L and L ! di L as R-modules for each i. Observe that N ≤ Soc(G). We are going to apply Theorem 1.3 with the summand L of N for Mij to obtain that R is local. (In fact, we could use any of our summands of N or G/N for Mij .) Since the action of R on L is multiplication by elements of L, we have AnnR (m) = AnnR (L) for any 0 = m ∈ L. To complete our proof in this direction, it suﬃces to show the summand L of G/N , let us call this M1 , is R-isomorphic to the summand L of N , which we will call M2 . Deﬁne ϕ : M1 → M2 by v

(u(a, b) + N )ϕ = u(0, ap ), a ∈ L. It is easily seen that ϕ is a well-deﬁned group isomorphism. To see that it is the desired R-isomorphism, we need only show α commutes with ϕ. As v

v

(u(a, b) + N )αϕ = (u(λa, λ · λθ · b) + N )ϕ = u(0, λp ap ), v

v

u

v

(u(a, b) + N )ϕα = u(0, ap )α = u(0, λ · λθ · ap ) = u(0, λ1+p ap ), and pu + 1 ≡ pv mod k, we have the required commuting of α and ϕ. Now suppose that R is local. We still have the additive group of L = Zp [λ] is a type 2 R-module of G/N . As the additive group of Zp [λ · λθ] is a type 2 R-module of N , we must have Zp [λ] ! Zp [λ · λθ] as R-modules by Theorem 1.1. Hence Zp [λ] = Zp [λ · λθ] since that latter is a subﬁeld of the former. This allows us to retain our notation of M1 for the copy of L in G/N and M2 for the copy of L in N . Let ϕ denote our isomorphism from M1 to M2 (which must exist by localness) and write the action of ϕ on an element u(a, b) + N of M1 as (u(a, b) + N )ϕ = u(0, aβ). Since ϕ is an additive isomorphism, β is an additive isomorphism of L. Moreover, the fact that ϕ is an R-isomorphism, or equivalently, (u(a, b) + N )αϕ = (u(a, b) + N )ϕα, gives us u(0, (λa)β) = u(0, λ · λθ · aβ). Hence (λa)β = λ · λθ · aβ.

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In particular, λβ = λ · λθ · 1β. By induction we obtain (λi )β = λi · (λi )θ · 1β for any positive integer i from which it follows that (p(λ))β = p(λ · λθ) · 1β

(2.1)

for any polynomial p(λ) in λ over Zp . Deﬁne γ : M1 → M2 by (u(a, b) + N )γ = u(0, aβ · (1β)−1 ). The map γ is easily seen to be an additive isomorphism which also is an R-isomorphism since (u(a, b) + N )αγ = (u(λa, λ · λθ · b) + N )γ = u(0, (λa)β · (1β)−1 ) = u(0, λ · λθ · aβ · (1β)−1 ) = (u(a, b) + N )γα. Setting

aβ · (1β)−1 = aδ,

we obtain a notation for γ similar to that with β in ϕ: (u(a, b) + N )γ = u(0, aδ). We are going to show that δ is a ﬁeld automorphism of L. It is certainly an additive isomorphism of L. To see it preserves multiplication, ﬁrst write a ∈ L as a = p(λ) where p(λ) is a polynomial in λ over Zp . Using equation (5), aδ = (p(λ))δ = (p(λ))β · (1β)−1 = p(λ · λθ). Hence if c ∈ L is expressed as a polynomial q(λ) over Zp , we have (ac)δ = (p(λ)q(λ))δ = p(λ · λθ)q(λ · λθ) = aδ · cδ completing the proof that δ is a ﬁeld automorphism. Since δ is a ﬁeld automorphism of L, there is an integer 0 ≤ v < n so that v aδ = ap

274 for all a ∈ L. As we have

Gary L. Peterson

λδ = λβ · (1β)−1 = λ · λθ, v

u

λp = λλp . This gives us that v ≥ 1 and that k divides pu − pv + 1 completing the proof. Let us give some illustrations of the use of Theorems 1.8 and 2.1 for producing local endomorphism nearrings of nonabelian groups with nontrivial K from our groups A(m, θ). We cannot have 2 or 3 for m. If m = 2, then n = 2 and we have u = v = 1 in Theorem 2.1. But then k|(−1) and K is trivial. If m = 3, then n = 3. (Keep in mind that n divides m since Zp [λ] is a subﬁeld of F .) This tells us G/Φ(G) is a type 2 R-module and hence i < j by Corollary 1.9. The only such possibilities for i and j are i = 1 and j = 2 in which case k divides p + p2 − 1. We cannot have u = v in Theorem 2.1 with K nontrivial. Consequently, either u = 1 and v = 2 or vice-versa if R is local. In the ﬁrst case, k would divide p − p2 + 1 which when added to p + p2 − 1 gives us k divides 2p. As k is odd by Theorem 1.7, k = p which is impossible. A similar argument yields k dividing p2 when u = 2 and v = 1, which is again impossible. Thus the ﬁrst possible occurrence of a local endomorphism nearring is when m = 4. As remarked earlier, we have an example of such an occurrence in [9] with p = 2 and m = 4. In this example, i = 1, j = 2, k = 5, n = 4, u = 1, and v = 3. Indeed, once we reach m = 4 numerous examples begin to come forth. For instance, we obtain a local endomorphism nearring when m = 4 for p = 2, i = 1, j = 1, k = 3, n = 2, u = 2, and v = 1; another occurs when p = 3, i = 2, j = 3, k = 5, n = 4, u = 1, and v = 2; yet another when p = 5, i = 1, j = 1, k = 3, n = 2, u = 2, and v = 1. If m = 5, then n = 5 and there are no possible divisors k of 25 − 1 = 31 that divide the possibilities for pi + pj − 1 when p = 2; however, a local endomorphism nearring is obtained with m = 5 by using p = 3, i = 1, j = 2, k = 11, u = 1, and v = 4 and also when p = 5, i = 2, j = 4, k = 11, u = 2, and v = 3. We also can use Theorem 2.1 to give examples showing the converse of Theorem 1.8 does not hold. To obtain one, modify the example of [9] with p = 2, m = 4, and k = 5 by changing u = 1 to u = 2. We continue to have 5 dividing 2i + 2j − 1 with i = 1 and j = 2 as in [9], but there is no v with 5 dividing 2u − 2v + 1 and hence the nearring is not local. The groups A(m, θ) are not the only nonabelian p-groups that can be used to obtain local endomorphism nearrings. Peter Mayr has used the

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matrix groups ⎫ ⎧⎡ ⎤% ⎬ ⎨ 1 a c %% G = ⎣ 0 1 b ⎦%% a, b, c ∈ GF (2n ) ⎭ ⎩ 0 0 1 % in [6] to obtain local endomorphism nearrings with K nontrivial. He also obtains a local endomorphism nearring using the group A(m, θ) with pm = 218 , k = 57, n = 18, u = 6, and v = 3 in our notation. Comparing his example with Theorems 1.8 and 2.1, we immediately have k dividing 2u − 2v + 1 and it divides 2i + 2j − 1 with i = 1 and j = 9. It is interesting to note that since Theorem 2.1 does not depend on Theorem 1.8, Theorem 2.1 in conjunction with Theorem 1.8 gives us the following number theoretic result: Suppose n > 1 is a divisor of m. Also suppose k divides pn − 1, but does not divide ph − 1 for any 1 ≤ h < n (in which case a primitive kth root of unity λ in GF(pm ) satisﬁes |Zp [λ]| = pn ). If k divides pu − pv + 1 for some 1 ≤ u < m and 1 ≤ v < n, then k divides pi + pj − 1 for some 1 ≤ i ≤ j < n. In fact, there is a direct proof of this number theoretic result. Suppose pu − pv + 1 ≡ 0 mod k

(2.2)

with 1 ≤ u < m and 1 ≤ v < n. Let r be the remainder when u is divided by n. Since pn ≡ 1 mod k, congruence (6) becomes pr − pv + 1 ≡ 0 mod k. Notice we cannot have r = v. Multiplying by pn−v , we obtain pr+n−v − 1 + pn−v ≡ 0 mod k.

(2.3)

If r < v, we obtain the congruence pi + pj − 1 ≡ 0 mod k.

(2.4)

with 1 ≤ i ≤ j < n by setting i = n − v and j = r + n − v. If r > v, congruence (7) becomes pr−v − 1 + pn−v ≡ 0 mod k again giving us congruence (8) with i = r − v and j = n − v. We also get a form of a converse. Suppose we have congruence (8) with 1 ≤ i ≤ j < n. If i < j, multiplying congruence (8) by pn−j gives us (2.5) pn+i−j + 1 − pn−j ≡ 0 mod k

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and we have a congruence of the form in (6) with u = n + i − j and v = n − j. If i = j, we do not get a congruence of the form in (6) with u < m when n = m (compare with Corollary 1.9). However, if n < m, congruence (9) does now become a congruence of the form in (6) with u = n and v = n − j. In particular, we then do have the existence of a local endomorphism nearring for each nontrivial divisor of pn − 1 (that does not divide ph − 1 for any 1 ≤ h < n) and pi + pj − 1 even though the converse of Theorem 1.8 does not hold in general.

References [1] B. Huppert, Endliche Gruppen I, Springer-Verlag, 1967 [2]

and N. Blackburn, Finite groups II, Springer-Verlag, 1982

[3]

and

, Finite groups III, Springer-Verlag, 1982

[4] C. Lyons and G. Peterson, Local endomorphism near-rings, Proc. Edinburgh Math. Soc., 31(1988), 409-414 [5] C. Maxson, On local near-rings, Math. Z., 106(1968), 197-205 [6] P. Meyer, Some local endomorphism near-rings from ﬁxed-point-free automorphism groups, preprint [7] J. Meldrum, Near-rings and their links with groups, Pitman, 1985 [8] G. Peterson, On the structure of an endomorphism near-ring, Proc. Edinburgh Math. Soc., 32(1989), 223-229 [9] [10] [11]

, Automorphism groups emitting local endomorphism near-rings, Proc. Amer. Math. Soc., 105(1989), 840-843 , Weakly tame near-rings, Comm. Algebra, 19(1991), 1165-1181 , Endomorphism near-rings of p-groups generated by the automorphism and inner automorphism groups, Proc. Amer. Math. Soc., 119(1993), 1045-1047

[12] S. Scott, Tame near-rings and N -groups, Proc. Edinburgh Math. Soc., 23(1980), 275-296

PLANAR NEAR-RINGS, SANDWICH NEAR-RINGS AND NEAR-RINGS WITH RIGHT IDENTITY Gerhard Wendt∗ Institut f¨ fur Algebra J. Kepler Univ. Linz Austria

Abstract

1.

We show that every near-ring containing a multiplicative right identity can be described as a centralizer near-ring with sandwich multiplication. Using this result we characterize planar near-rings and near-rings solving the equation xa=c in terms of such centralizer near-rings with sandwich multiplication. We also get results on primitive near-rings and on minimal left ideals in primitive near-rings. 2000 Mathematics Subject Classiﬁcation: 16Y30.

Introduction

What concerns the notation in this paper, we are referring to that used in [1], in particular, we are dealing with right near-rings. For a general survey on near-ring theoretical results we are also referring to [1]. We start our work with the following well known result on zero symmetric near-rings with identity: Theorem 1.1 ([4], Theorem 2.8). Let N be a zero symmetric near-ring with identity. Then there exists a group (Γ, +) and a subset S ⊆ End(Γ) such that N is isomorphic to the centralizer near-ring MS (Γ) = {f : Γ −→ Γ | f (0) = 0 and ∀s ∈ S ∀γ ∈ Γ : f (s(γ)) = s(f (γ))}. This is an interesting result, since it tells us that every zero symmetric near-ring with identity is just a centralizer near-ring. Nevertheless, most of the near-rings do not have an identity but still may have a one sided ∗ This

work has been supported by grant P15691 of the Austrian National Science Fund (FWF)

277 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 277–291. c 2005 Springer. Printed in the Neatherlands.

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identity. We especially deal with near-rings having a right identity in this paper. We say that e is a right identity of the near-ring N if for all n ∈ N , ne = n. Since we need it in the next proposition, we recall the following deﬁnition: Let N be a near-ring, then n ∈ N is a right zero divisor if there exists a non-zero element a ∈ N such that an = 0. We mention the following result and include a proof of it: Proposition 1.2 ([1], Remark 1.112). Let N (= {0}) be a zero symmetric near-ring with descending chain condition on N -subgroups which contains an element m which is not a right zero divisor. Then N has a right identity. Proof. Let m be an element of N which is not a right zero divisor. Then clearly mk (k ≥ 1 any natural number) cannot be a right zero divisor. By the descending chain condition on N -subgroups, the chain N m ⊇ N m2 ⊇ N m3 ⊇ ... terminates. So there is some natural number l such that N ml = N ml+1 . Hence, N ml = (N m)ml . Therefore, there exists an e ∈ N such that mml = em(ml ). Consequently, (m − em)ml = 0 and this gives (m−em) = 0, so em = m. Finally, for all k ∈ N , (ke−k)m = 0 and we get ke = k. So e is a right identity. In particular, one can deduce from the proposition above that ﬁnite near-rings have a right identity unless every element is a right zero divisor. This does not mean that they have an identity element! In the next section we generalize Theorem 1.1 to the much larger class of near-rings just having a right identity, but not necessarily having an identity. Some prominent classes of near-rings have a right identity element but not an identity element unless in some very special situations. For example planar near-rings have always a right identity, but they have no identity unless they are near-ﬁelds. Similarily zero symmetric 2-semisimple near-rings with descending chain condition on left ideals have a right identity, but not an identity element in general.

2.

Near-rings with right identity

We will show that every near-ring with right identity is isomorphic to a centralizer near-ring with a slightly changed multiplication, not with the usual multiplication of function composition. By doing so, we get the promised generalization of Theorem 1.1. We have to give some deﬁnitions: Deﬁnition 2.1. Let (N, +) be a group, X ⊆ N a subset of N containing the zero 0 of (N, +) and φ : N −→ X a map such that φ(0) = 0. Deﬁne the following operation ◦ on N X , N X the set of all functions mapping

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from X into N : f ◦ g := f ◦ φ ◦ g for f, g ∈ N X . Then (N X , +, ◦ ) is a near-ring, denoted by M (X, N, φ). Let S ⊆ End(N ), S not empty, be such that ∀s ∈ S ∀n ∈ N : φ ◦ s(n) = s ◦ φ(n) and such that S(X) ⊆ X. Then M0 (X, N, φ, S) := {f : X −→ N | f (0) = 0 and ∀s ∈ S ∀x ∈ X : f (s(x)) = s(f (x))} is a zero symmetric subnear-ring of M (X, N, φ), which we call a sandwich centralizer near-ring. We shortly point out why M0 (X, N, φ, S) is indeed a zero symmetric subnear-ring of M (X, N, φ): Let 0 be the zero function on N . Since S(X) ⊆ X, M0 (X, N, φ, S) is not empty, since the restriction 0 |X of the zero function 0 on N to X is contained in M0 (X, N, φ, S). Let f1 , f2 ∈ M0 (X, N, φ, S). Then, for all s ∈ S and for all x ∈ X, s◦(f1 −ff2 )(x) = s◦f1 (x)−s◦ff2 (x) = f1 ◦s(x)−ff2 ◦s(x) = (f1 −ff2 )◦s(x), by the deﬁnition of M0 (X, N, φ, S) and due to the fact that functions in S are endomorphisms of N . Now for all s ∈ S and for all x ∈ X, s ◦ (f1 ◦ φ ◦ f2 )(x) = (f1 ◦ φ ◦ f2 ) ◦ s(x), since for all n ∈ N and all s ∈ S, φ ◦ s(n) = s ◦ φ(n), and for all x ∈ X and s ∈ S, f1 (s(x)) = s(f1 (x) and f2 (s(x)) = s(ff2 (x)). So, M0 (X, N, φ, S) is a subnear-ring of M (X, N, φ). Note that the zero of M0 (X, N, φ, S) is 0 |X , the restriction of the zero function 0 to X. Since for all x ∈ X, g ◦ φ ◦ 0 |X (x) = g(φ(0)) = g(0) = 0, due to the fact that φ(0) = 0 and g(0) = 0 by assumption, M0 (X, N, φ, S) is a zero symmetric near-ring. Before formulating our main theorem of this section, we remark the following: If X happens to be just the set consisting of zero, then M0 (X, N, φ, S) is just the near-ring consisting of 0 |X . But in that case 0 |X clearly is a right identity. So, the following description explicitely includes the zero near-rings as a class of near-rings having a right identity. Note that we easliy could avoid this by assuming |X| ≥ 2. It is a matter of taste whether zero may be seen as a right identity in case of zero near-rings or not. Finally, here is our main theorem of this section: Theorem 2.2. Let M be a near-ring. Then the following are equivalent: (1) M is a zero symmetric near-ring with right identity. (2) There exists a group (N, +), a subset X of N with 0 ∈ X, there exists a non-empty subset S ⊆ End(N ) with S(X) ⊆ X, and there exists a function φ : N −→ X with φ(0) = 0, φ |X = id and φ ◦ s(n) = s ◦ φ(n) for all s ∈ S and n ∈ N , such that M∼ = M0 (X, N, φ, S). Proof. (2) ⇒ (1): Let M ∼ = M0 (X, N, φ, S). Then, clearly, the function f : X −→ N, x → x is in M0 (X, N, φ, S). Let g ∈ M0 (X, N, φ, S). Then

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for all x ∈ X, g ◦ φ ◦ f (x) = g ◦ φ(x) = g(x), so f is a right identity. M clearly is zero symmetric. (1) ⇒ (2): Let (M, +, ∗) be a zero symmetric near-ring with right identity 1r . For k ∈ M deﬁne ψk : M −→ M ; m → m ∗ k. Every such function is an M -endomorphism of (M, +). Let S := {ψk | k ∈ M }. Now deﬁne X := 1r ∗ M = {1r ∗ m | m ∈ M } and deﬁne the function φ : M −→ X, m → 1r ∗ m. φ is certainly well deﬁned, φ(0) = 0 and φ |X = id, since 1r is a right identity. It is easy to see that φ commutes with all elements in S: Let ψk ∈ S and n ∈ N . Then, ψk (φ(n)) = φ(n) ∗ k = (1r ∗ n) ∗ k and φ(ψk (n)) = φ(n ∗ k) = 1r ∗ (n ∗ k) = (1r ∗ n) ∗ k. Note that X ⊆ M and S(X) ⊆ X. Hence, the sandwich centralizer near-ring M0 (X, M, φ, S) exists and we now show that M is isomorphic to M0 (X, M, φ, S). For every k ∈ M , let fk be the function fk : M −→ M, m → k ∗ m. Consider the restrictions fk |X . We show that these functions are indeed elements of M0 (X, M, φ, S): Let k ∈ M . Take an arbitrary M endomorphism s ∈ S and an arbitrary element x ∈ X. Then fk |X (s(x)) = k ∗ (s(x)) = s(k ∗ x) = s(ffk |X (x)) and consequently, fk |X ∈ M0 (X, M, φ, S) since clearly fk (0) = 0. Now the function δ : M −→ M0 (X, M, φ, S), k → fk |X is a near-ring isomorphism, as we will show in the following: δ is a near-ring homomorphism: Let j and k be arbitrary elements of M . Then δ(j + k) = f(j+k) |X . By the right distributive law, this is just the function fj |X +ffk |X . Furthermore, we have δ(j ∗ k) = fj ∗k |X . Let x ∈ X be arbitrary. Then fj ∗k |X (x) = (j ∗ k) ∗ x. On the other hand, δ(j) ◦ δ(k) = δ(j) ◦ φ ◦ δ(k) = fj |X ◦φ ◦ fk |X and fj |X ◦φ ◦ fk |X (x) = j ∗ (φ(k ∗ x)) = j ∗ (1r ∗ (k ∗ x)) for all x ∈ X. Since 1r is a right identity, and by associativity of multiplication j ∗ (1r ∗ (k ∗ x)) = (j ∗ k) ∗ x. δ is injective: It suﬃces to show that Kerδ = 0. Suppose there exists an element j ∈ M such that fj |X = 0 |X . But 1r ∈ X by deﬁnition of X. So j ∗ 1r = j = 0. So we can embed M into the sandwich centralizer near-ring M0 (X, M, φ, S). Now we show that δ is surjective: Let f ∈ M0 (X, M, φ, S). Since 1r ∈ X, we may consider the function value f (1r ), f (1r ) = a say. Let x ∈ X. Then, f (x) = f (1r ∗ x) = f (ψx (1r )) = ψx (f (1r )) = ψx (a) = a ∗ x. So, δ(a) = f and this shows the surjectivity. Therefore, δ is a near-ring isomorphism and our theorem is proved.

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Note that if our right identity 1r in the proof above happens to be an identity, then we get back the statement of Theorem 1.1, since then X = M and φ = id, in the notation used in the proof above. One of the reasons why the common centralizer near-rings (where multiplication is just function composition) are so heavily studied is that every near-ring with identity is a centralizer near-ring. With the same motivation one could now try to develop a systematic study of sandwich centralizer near-rings. Certainly, this programm is too general to attack and in the following sections we will only study some very special types of sandwich centralizer near-rings.

3.

Near-rings solving the equation xa=c

For a near-ring N one can deﬁne an equivalence relation ≡ in the following way: Let a, b in N , then a ≡ b if for all n ∈ N , na = nb. a and b are said to be equivalent multipliers. We furthermore deﬁne the sets N # := {n ∈ N | n ≡ 0} and A := {n ∈ N | n ≡ 0}. In this section we will focus on zero symmetric near-rings N having the property that an equation of the form xa = c (a ≡ 0, c ∈ N ) has a unique solution in N . It is the content of the next proposition to show that such near-rings are indeed interesting and arise very naturally. Proposition 3.1. Let {0} = L be a minimal left ideal of a zero symmetric near-ring N which acts 1-primitively on the N -group Γ. Then for every c ∈ L and every 0 ≡ a ∈ L (≡ ( restricted to L), xa = c has a unique solution in L. Proof. Since Γ is faithful, ∃γ ∈ Γ : Lγ = {0}. Since Γ is strongly monogenic, N γ = Γ and hence, by ([1], Proposition 3.4), Lγ N Γ. Therefore, Lγ = Γ. Hence ψ : L −→ Γ, l → lγ is a surjective N endomorphism. Since L is a minimal left ideal and Ker ψ = L ∩ (0 : γ), Ker ψ = {0}. Hence ψ is an N -isomorphism and therefore Γ ∼ =N L which means that N is 1-primitive on L. Therefore, every non-zero N endomorphism of L is an automorphism, by [1, Corollary 4.18]. Now for j ∈ L# , gj : L −→ L, l → lj is an N -automorphism and this proves our theorem. It is also an easy exercise to see that ﬁnite strongly monogenic nearrings N , that are near-rings where (N, +) is a strongly monogenic N group, are near-rings where every equation of the form xa = c has a unique solution for all c ∈ N , as long as a ≡ 0. So the near-rings under study in this section should be interesting in their own right, but work on them will also give us important knowledge on planar near-rings, on which we will focus our consideration in the next section.

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Now we want to use Theorem 2.2 for getting a characterization result in terms of sandwich centralizer near-rings. We ﬁrst prove a lemma on the internal structure of near-rings solving uniquely the equation xa = c. We will see that the internal structure of such near-rings is similar to that of planar near-rings (see [1, Theorem 8.90]). Lemma 3.2. Let (N, +, ∗) be a zero-symmetric near-ring such that |N/ ≡ | ≥ 2 and such that for every a ∈ N # and for every c ∈ N the equation xa = c has a unique solution. For a ∈ N # let 1a be the unique solution of xa = a and let Ba := {x ∈ N # | 1a x = x}. Then the following holds: (a) Each (Ba , ∗) is a group with identity 1a and 1a is a right identity of N . (b) A and the sets of the form Ba (a ( ∈ N # ) form a partition of N . Proof. Let a ∈ N # (such an a exists since |N/ ≡ | ≥ 2). Let b, b1 ∈ Ba . Then 1a (bb1 ) = (1a b)b1 = bb1 . If bb1 ∈ A, this would imply bb1 = 0. Since 0 is also a solution of xb1 = 0, b = 0. This is a contradiction. Hence bb1 ∈ N # and therefore bb1 ∈ Ba . This shows that Ba is closed under multiplication. Since (1a 1a )a = 1a (1a a) = 1a a = a, we have that 1a 1a = 1a . Hence 1a ∈ Ba (1a ∈ N # for the same reasons as above) and 1a is a left identity in Ba . Since for every m ∈ N , m1a and m is a solution of x1a = m1a , we have m = m1a and 1a is a right identity of N. Let b ∈ Ba . Let b be the (unique) solution of xb = 1a . Since (1a b)b = 1a (bb) = 1a , we get 1a b = b and consequently b ∈ Ba . On the other side, bb = b1a b = bbbb and 1a bb = bb and hence bb = 1a . So b is the multiplicative inverse of b in Ba . The proof of part (a) is now complete. For part (b) it suﬃces to show that ∀a, b ∈ N # either Ba ∩ Bb = ∅ or Ba = Bb : Let m ∈ Ba ∩ Bb . Then 1a m = m = 1b m and hence 1a and 1b are solutions of xm = m. So 1a = 1b and Ba = Bb . We now restrict to near-rings N such that |N/ ≡ | ≥ 3, because if |N/ ≡ | = 2, then by Lemma 3.2 there exists a right identity and consequently, every element m ≡ 0 is a right identity. So the multiplication is trivial. Hence we omit this case, since we do not really lose generality, on the other hand we are aiming for a description of planar near-rings in the next section, and planar near-rings have the property |N/ ≡ | ≥ 3. We still have to ﬁx one further notation: Deﬁnition 3.3. An automorphism φ of a group (N, +) is said to have a ﬁxed point n ∈ N if φ(n) = n. If 0 is the only ﬁxed point of φ, then

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φ is called ﬁxedpointfree on N . A subgroup Φ of Aut(N, +) is said to be ﬁxedpointfree on N if every element φ ∈ Φ \ {id} is ﬁxedpointfree. Similarily, let S ⊆ N . Then we say that Φ is ﬁxedpointfree on S if every function φ ∈ Φ \ {id} has no ﬁxed points in S \ {0}. Now we can formulate the main theorem of this section: Theorem 3.4. Let M be a near-ring. Then the following are equivalent: (1) M is a zero symmetric near-ring such that |M/ ≡ | ≥ 3 and such that for every c ∈ M and for every 0 ≡ a ∈ M the equation xa = c has a unique solution. (2) There is a group (N, +) and a subgroup C ≤ Aut(N, +) with | C |> 1. There is an element n ∈ N \ {0} such that C has no ﬁxed points in C(n). There exists a function φ : N −→ C(n) ∪ {0} such that φ |C(n) = id, φ(0) = 0 and φ ◦ c(n1 ) = c ◦ φ(n1 ) for all c ∈ C and n1 ∈ N . Let X := C(n) ∪ {0}. Then M is isomorphic to the sandwich centralizer near-ring M0 (X, N, φ, C). Remark 3.5. Since |C| > 1, we have in fact that φ(0) = 0: φ commutes with the elements of the automorphism group C, so let id = c ∈ C. Such a c exists, since |C| > 1. Then φ(0) = φ(c(0)) = c(φ(0)). Since φ(0) ∈ X and C has no ﬁxed points in X \ {0}, we get that φ(0) = 0. We would not have to assume this explicitely but we have done this for a better readability. Proof. (1) ⇒ (2): Let (M, +, ∗) be a zero symmetric near-ring, such that given a ≡ 0 the equation xa = c has a unique solution for every c ∈ M . By Lemma 3.2, M has a right identity 1r (r ∈ M # ), so we may use the proof ideas developped in the proof of Theorem 2.2, so at some points of the following proof we will not work out all the details again. For k ∈ M deﬁne ψk : M −→ M ; m → m ∗ k. Every such function is an M -endomorphism. If k ≡ 0, then ψk is an automorphism. Let C := {ψk | k ∈ M # }. Since |M/ ≡ | ≥ 3, |C| ≥ 2. Observe that (C, ◦) is a group of automorphisms of (M, +): Clearly C ⊆ AutM (M ). Conversely, let σ ∈ AutM (M ). Then, for all m ∈ M , σ(m) = σ(m ∗ 1r ) = m ∗ σ(1r ) and therefore σ = ψσ(1r ) (note that σ(1r ) ∈ M # ). Hence, AutM (M ) ⊆ C. This shows that AutM (M ) = C and so (C, ◦) is a group of automorphisms of (M, +) (note that even EndM (M ) = AutM (M ) ∪ {0}). Fix an arbitrary n ∈ M # and let 1n be the identity of the group (Bn , ∗). So 1n is a right identity in M by Lemma 3.2. We now show that C has no ﬁxed points in C(1n ). First observe that by deﬁnition of C, C(1n ) = 1n ∗ M # , so 1n ∗ M = C(1n ) ∪ {0}. Let id = c ∈ C:

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Then c = ψk for an element k ∈ M # and k ≡ 1n , so k is not a right identity. Suppose there is a non-zero element 1n ∗ m, m ∈ M # , such that c(1n ∗ m) = 1n ∗ m, so (1n ∗ m) ∗ k = 1n ∗ m. Since 1n is a right identity of the near-ring, this means (n ∗ m) ∗ k = n ∗ m for all n ∈ M . Since m ∈ M # , the equation xm = c has a solution for all c ∈ M and consequently we get u ∗ k = u for all u ∈ M . This is a contradiction to k not being a right identity, hence C acts without ﬁxed points on C(1n ). Now we deﬁne a function φ with the required properties. Recall that 1n ∗ M = C(1n ) ∪ {0}, so we can deﬁne φ in a very natural way. Let φ : M −→ C(1n ) ∪ {0}, m → 1n ∗ m. φ is clearly well deﬁned and φ(0) = 0. Moreover, φ |C(1n ) = id: Let 1n ∗ m in C(1n ). Since 1n is a right identity, it is clear that φ(1n ∗ m) = 1n ∗ (1n ∗ m) = 1n ∗ m. Furthermore, for all c ∈ C, c ◦ φ = φ ◦ c: Take m ∈ M and c ∈ C. Then c(φ(m)) = c(1n ∗ m) = 1n ∗ c(m) = φ(c(m), since c ∈ AutM (M ). Let X := C(1n ) ∪ {0}. Since φ commutes with all functions in C, φ(0) = 0 and C(X) ⊆ X ⊆ M , we can construct the sandwich centralizer near-ring M0 (X, M, φ, C) with input parameters fulﬁlling all the assumptions of (2). Now we show that M ∼ = M0 (X, M, φ, C): For k ∈ M let the function fk be deﬁned as fk : M −→ M, m → k ∗ m. As in the proof of Theorem 2.2 we see that the restriction fk |X ∈ M0 (X, N, φ, C). Deﬁne δ : M −→ M0 (X, M, φ, C), k → fk |X . As in Theorem 2.2 one now shows that δ is a near-ring monomorphism. δ is surjective since let f ∈ M0 (X, M, φ, C) and suppose that f (1n ) = a (remember that 1n ∈ X): Let x ∈ X. Suppose x = 0, then f (x) = f (0) = 0 = fa |X (0). If x = 0, then there exists k ∈ M # such that x = 1n ∗ k, so x = ψk (1n ), ψk ∈ C. Consequently, f (x) = f (ψk (1n )) = ψk (f (1n )) = ψk (a) = a ∗ k = (a ∗ 1n ) ∗ k = a ∗ (1n ∗ k) = a ∗ x. So, δ(a) = fa |X = f and this shows the surjectivity of δ and ﬁnally one direction of theorem is proved. (2) ⇒ (1): Suppose a near-ring P is (isomorphic to) a sandwich centralizer near-ring M0 (X, N, φ, C) = {f : X −→ N | f (0) = 0 and ∀c ∈ C ∀x ∈ X : f (c(x)) = c(f (x))}, with input parameters fulﬁlling all the properties of (2). Note that the zero of M0 (X, N, φ, C) is the function 0 |X . We identify M0 (X, N, φ, C)/ ≡ | ≥ 3 and P with M0 (X, N, φ, C) and show that |M that for every 0 |X ≡ f1 ∈ M0 (X, N, φ, C) and f2 ∈ M0 (X, N, φ, C) the equation x ◦ f1 = f2 has a unique solution. Let 0 = z ∈ X and ﬁx n ∈ N . Then there is a function f ∈ M0 (X, N, φ, C) such that f (z) = n: Since X \ {0} = C(z) (this holds since X is just an orbit of C adjoined with a zero) and C has no ﬁxed points in X \ {0}, every non-zero element in X can be written as c(z) for

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some unique c ∈ C. Consider the function f : X −→ N , which is deﬁned as: f (0) = 0 and f (x) = c(n) if x = 0 and x = c(z). This function is well deﬁned since C has no ﬁxed points in X \ {0} and furthermore, f ∈ M0 (X, N, φ, C), which is easily checked. Now f (z) = n. Next we show that |M M0 (X, N, φ, C)/ ≡ | ≥ 3: Since |X| ≥ 3, there are at least two diﬀerent non-zero elements k, d in X ⊆ N . Take a non-zero element y ∈ X. Then there exist two functions f1 and f2 in M0 (X, N, φ, C), such that f1 (y) = k and f2 (y) = d. We claim that f1 and f2 are both not equivalent to the zero 0 |X of M0 (X, N, φ, C): Suppose for example that f1 ≡ 0 |X , hence ∀f ∈ M0 (X, N, φ, C) : 0 = f ◦ φ ◦ 0 |X (y) = f ◦ φ ◦ f1 (y) = f (φ(k)), since φ(0) = 0 and f (0) = 0 for any f ∈ M0 (X, N, φ, C). Since k ∈ X, φ(k) = k, so f (k) = 0 for all f ∈ M0 (X, N, φ, C) . But a function f ∈ M0 (X, N, φ, C), mapping an element of X \ {0} to zero, has to be the zero function 0 |X : Suppose f (j) = 0 for some j ∈ X \ {0}. This means that for all c ∈ C, 0 = c(f (j)) = f (c(j)). But C(j) = X \ {0} and so f = 0 |X is the zero function. Since we have f (k) = 0 for all f ∈ M0 (X, N, φ, C) and for k ∈ X \ {0} we get that M0 (X, N, φ, C) = {0 |X }, which clearly is a contradiction, so f1 ≡ 0 |X . Suppose f1 ≡ f2 . Then, for all f ∈ M0 (X, N, φ, C): f ◦ φ ◦ f1 = f ◦ φ ◦ f2 . In particular, this gives f ◦ φ(k) = f ◦ φ(d) for every function f ∈ M0 (X, N, φ, C). Now d and k are two elements of X, so φ(k) = k and φ(d) = d. Hence we get f (d) = f (k) for any function f ∈ M0 (X, N, φ, C). Since k = c(d) for some non-identity function c ∈ C, we have f (d) = f (c(d)) = c(f (d)), for all f ∈ M0 (X, N, φ, C). Choose f ∈ M0 (X, N, φ, C) such that f (d) = d. Such a function f exists in M0 (X, N, φ, C) by the statements above. Hence, d = f (d) = f (c(d)) = c(f (d)) = c(d). Since id = c and d ∈ X \ {0}, this is a contradiction to the fact that C has no ﬁxed points in X \ {0}. M0 (X, N, φ, C)/ ≡ | ≥ 3. Therefore f1 ≡ f2 , which ﬁnally shows |M Now we show that for all 0 |X ≡ f1 ∈ M0 (X, N, φ, C) and f2 ∈ M0 (X, N, φ, C) the equation x ◦ f1 = f2 has a unique solution inside M0 (X, N, φ, C): Since f1 ≡ 0 |X , there is an element 0 = z ∈ X such that φ(f1 (z)) ∈ X \ {0}. Now consider the following equation: x ◦ φ ◦ f1 (z) = f2 (z). Let k := φ ◦ f1 (z) = φ(f1 (z)). So the equation can be written as x(k) = f2 (z). As outlined above, there exists an f ∈ M0 (X, N, φ, C) such that f (k) = f2 (z), since k ∈ X \ {0}. Suppose there is a second function µ ∈ M0 (X, N, φ, C), such that µ(k) = f (k) = f2 (z). Then for every c ∈ C we have µ(c(k)) = c(µ(k)) = c(f (k)) = f (c(k)) and since C(k) = X \ {0} and µ(0) = f (0) = 0 we get µ = f . So, f is the unique function

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in M0 (X, N, φ, C) with the property f (k) = f2 (z). Now we go on to show that f is indeed the solution of the equation x ◦ f1 = f2 : Clearly f is a solution of x ◦ φ ◦ f1 (0) = f2 (0), since functions in M0 (X, N, φ, C) are zero preserving and φ(0) = 0. Moreover, f solves uniquely the equation x ◦ φ ◦ f1 (z) = f2 (z), by the above. Take an arbitrary non-zero element w ∈ X. Then there exists a c ∈ C such that w = c(z). Then c−1 (f ◦ φ ◦ f1 (w)) = f ◦ φ ◦ f1 (z) = f2 (z) = c−1 (ff2 (w)), since all the functions in M0 (X, N, φ, C) commute with all the functions in C and φ commutes with the functions in C, by assumption. This gives f ◦ φ ◦ f1 (w) = f2 (w), since C is a group of automorphisms and therefore the function f is the unique solution of the equation x ◦ f1 = f2 .

4.

Planar near-rings and sandwich near-rings

Now we will turn our attention to planar near-rings. A planar nearring N is a near-ring fulﬁlling the following properties: |N/ ≡ | ≥ 3 and for all a, b, c ∈ N with a ≡ b the equation xa = xb + c has a unique solution. By Lemma 3.2, we see that a planar near-ring has a right identity and planar near-rings in particular fall into the class of nearrings described in the last section. So the results obtained so far show us that planar near-rings can be described as sandwich centralizer nearrings. So there is not too much work left, we only have to make some speciﬁcations. Note that a planar near-ring has no identity unless it is a near-ﬁeld (see [1, Corollary 8.91]). So planar near-rings are a typical class of near-rings having a multiplicative right identity but not having an identity in general. First we need some additional deﬁnitions and facts about planar nearrings (recall Deﬁnition 3.3 in this context): Deﬁnition 4.1 ([2], Deﬁnition 4.12). A Ferrero pair is a pair of groups (N, Φ), where {id} = Φ ≤ Aut(N, +) is a ﬁxedpointfree automorphism group on (N, +) where each id = φ ∈ Φ has the property that −φ + id is surjective. Note that if (N, +) is ﬁnite and Φ = {id} is a ﬁxedpointfree automorphism group on (N, +), then (N, Φ) is a Ferrero pair. Furthermore, given a Ferrero pair (N, Φ) and id = φ ∈ Φ, then the functions −φ + id are bijections on N . Theorem 4.2 ([2], Theorem 4.13). Let N be a planar near-ring. For a ∈ N # consider the functions ψa : N −→ N ; n → na. Then G := {ψa | a ∈ N # } is a ﬁxedpointfree automorphism group on (N, +), G = {id} and (N, G) is a Ferrero pair. The next lemma is easy to prove:

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Lemma 4.3 ([1], Proposition 8.87). A planar near-ring is zero symmetric. The next lemma will be of use in the proof of the main Theorem of this section (in fact we have already proved a similar statement implicitely in one of our arguments of the last theorem). As usual, 0 denotes the zero function: Lemma 4.4. Let N be a planar near-ring. Then EndN (N )\{0} = {ψa | a ∈ N # } = AutN (N ). Proof. Clearly, every element in G := {ψa | a ∈ N # } is a non-zero N -endomorphism of N . Conversely, let φ be an arbitrary non-zero N endomorphism of N . Since N is planar, there exists a right identity 1r . It is clear that φ(1r ) ∈ N # and consequently, ψφ(1r ) ∈ G. But for all n ∈ N , ψφ(1r ) (n) = nφ(1r ) = φ(n1r ) = φ(n) and therefore φ = ψφ(1r ) . According to the results of the last section, the main theorem of this section does not come as a surprise. Theorem 4.5. Let M be a near-ring. Then the following are equivalent: (1) M is planar. (2) M ∼ = M0 (X, N, φ, C), where (N, C) is a Ferrero pair, X = C(n) ∪ {0} for an element n ∈ N \ {0}, φ : N −→ X is a function with φ |C(n) = id, φ(0) = 0 and φ ◦ c(n1 ) = c ◦ φ(n1 ) for all c ∈ C and n1 ∈ N . Proof. (1) ⇒ (2): Suppose (M, +, ∗) is a planar near-ring. Then M is zero symmetric, |M/ ≡ | ≥ 3 and every equation of the form xa = c (a ≡ 0) has a unique solution for all c ∈ M . Hence, by Theorem 3.4, M is isomorphic to a sandwich centralizer near-ring M0 (X, M, φ, C) where C = AutM (M ) = {ψa | a ∈ M # }, X := 1n ∗ M = C(1n ) ∪ {0} where n ∈ M # is arbitrary but ﬁxed (1n is the identity of (Bn , ∗)) and φ is the function φ : M −→ X, m → 1n ∗ m. Since M is planar, (M, C) is a Ferrero pair by Lemma 4.4. Certainly, φ(0) = 0, φ |C(1n ) = id and φ ◦ c(n1 ) = c ◦ φ(n1 ) for all c ∈ C and n1 ∈ N . So, one direction of the theorem is proved. (2) ⇒ (1): Suppose a near-ring P is (isomorphic to) a sandwich centralizer near-ring M0 (X, N, φ, C), where (N, C) is a Ferrero pair and which fulﬁlles all the other assumptions of (2). We identify P with M0 (X, N, φ, C) and show that M0 (X, N, φ, C) is a planar near-ring. Certainly, |M M0 (X, N, φ, C)/≡| ≥ 3 by Theorem 3.4.

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It remains to show that for all f1 , f2 , f3 ∈ M0 (X, N, φ, C) with f1 ≡ f2 the equation x◦ f1 = x◦ f2 +ff3 has a unique solution in M0 (X, N, φ, C): Since f1 ≡ f2 , there is an element 0 = z ∈ X such that φ(f1 (z)) = φ(ff2 (z)). We ﬁrst consider the following equation: x ◦ φ ◦ f1 (z) = x ◦ φ ◦ f2 (z) + f3 (z). Let k = φ ◦ f1 (z) and b = φ ◦ f2 (z). Suppose ﬁrst that k = 0 and b = 0. Then there exists a unique function id = c ∈ C such that b = c(k) (since b, k ∈ X \ {0}). So the above equation can be written as x(k) = x(c(k)) + f3 (z). Since we are looking for solutions in M0 (X, N, φ, C), it therefore suﬃces to consider the equation x(k) = c(x(k)) + f3 (z). We substitute x(k) = q and consider the equation (−c + id)(q) = f3 (z). By assumption, (−c + id) is bijective and hence there is a unique q1 ∈ N such that (−c + id)(q1 ) = f3 (z). Now there is a function f ∈ M0 (X, N, φ, C) with f (k) = q1 , as we have shown in the proof of Theorem 3.4 and therefore f ◦ φ ◦ f1 (z) = f ◦ φ ◦ f2 (z) + f3 (z). Suppose there is a second function µ ∈ M0 (X, N, φ, C), such that µ ◦ φ ◦ f1 (z) = µ◦φ◦ff2 (z)+ff3 (z). Then µ(b) = c(µ(b))+ff3 (z) and consequently (−c + id)(µ(b)) = f3 (z). So µ(b) = q1 = f (b) and, as done in the proof of Theorem 3.4, one now can show that µ = f . So f is the unique function in M0 (X, N, φ, C) with the property f ◦ φ ◦ f1 (z) = f ◦ φ ◦ f2 (z) + f3 (z) and clearly f is a solution of x ◦ φ ◦ f1 (0) = x ◦ φ ◦ f2 (0) + f3 (0). Now take an arbitrary non-zero element w ∈ X. Then there exists a function c ∈ C such that w = c(z). Then c−1 (f ◦ φ ◦ f1 (w)) = f ◦ φ ◦ f1 (z) = f ◦ φ ◦ f2 (z) + f3 (z) = c−1 (f ◦ φ ◦ f2 (w)) + c−1 (ff3 (w)) = c−1 (f ◦ φ ◦ f2 (w) + f3 (w)), since all the functions in M0 (X, N, φ, C) commute with all the functions in C, φ commutes with the functions in C and C is a group of automorphisms of (N, +). This gives f ◦φ◦f1 (w) = f ◦ φ ◦ f2 (w) + f3 (w) and therefore the function f is the unique solution of the equation x ◦ f1 = x ◦ f2 + f3 in M0 (X, N, φ, C). Now suppose that k = 0 or b = 0. Without loss of generality, let us assume that b = φ(ff2 (z)) = 0. Then for all c ∈ C, c(φ(ff2 (z))) = φ(ff2 (c(z))) = 0. This means that for all x ∈ X, φ(ff2 (x)) = 0. So we only have to ﬁnd a solution for the equation x ◦ f1 = f3 and we are in a situation considered in Theorem 3.4.

5.

Left ideals in 2-primitive near-rings

The minimal left ideals of 1-primitive near-rings can be described as sandwich centralizer near-rings, as we have seen in a foregoing section. So, in particular, this can be done with minimal left ideals of 2-primitive near-rings too, since 2-primitivity implies 1-primitivity. Using this idea, we are looking for a possible connection between 2-primitive and planar near-rings, using our results obtained so far. First, observe the following

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proposition, whose proof may be found in the proof of ([1], Theorem 4.52): Proposition 5.1 ([1], Theorem 4.52). Let N be a zero symmetric nearring which is 2-primitive on the N -group Γ. Then the non-zero N endomorphisms of Γ form a ﬁxedpointfree automorphism group on Γ. Proof. Every non-zero N -endomorphism φ of Γ must be an automorphism, since otherwise Ker φ, respectively Im φ, would give non-trivial N -subgroups of Γ. Let φ be a non-zero N -endomorphism of Γ. Then M := {γ ∈ Γ | φ(γ) = γ} is an N -subgroup of Γ. Hence M = {0} or M = Γ. So either φ = id or φ is ﬁxedpointfree. The following lemma is already clear: Lemma 5.2 ([1], Theorem 4.56 a). Let L be a minimal left ideal of a zero symmetric near-ring N , which is 2-primitive on the N -group Γ. Then L ∼ =N Γ and hence, N is 2-primitive on L. Proof. See the proof of Theorem 3.1. We need another proposition for our proof of the characterization result on minimal left ideals in 2-primitive near-rings. Proposition 5.3. Let N be a zero symmetric near-ring which is 2primitive on an N -group Γ1 and on an N -group Γ2 . Let G1 := AutN (Γ1 ) and G2 := AutN (Γ2 ). Suppose Γ1 ∼ =N Γ2 and suppose that (Γ1 , G1 ) is a Ferrero pair. Then (Γ2 , G2 ) is a Ferrero pair, too. Proof. Let φ : Γ1 −→ Γ2 be an N -isomorphism between Γ1 and Γ2 . We ﬁrst show that φ ◦ G1 ◦ φ−1 = G2 : Let g1 ∈ G1 . Then φ ◦ g1 ◦ φ−1 ∈ G2 . So φ ◦ G1 ◦ φ−1 ⊆ G2 . Similarily, φ−1 ◦ G2 ◦ φ ⊆ G1 . Consequently, G2 ⊆ φ ◦ G1 ◦ φ−1 and so φ ◦ G1 ◦ φ−1 = G2 . Now we have to show the following: For id = g2 ∈ G2 , −g2 + id is surjective. Let γ2 ∈ Γ2 and let id = g2 ∈ G2 . We have to show that there exists an element γ2 ∈ Γ2 , such that (−g2 + id)(γ2 ) = γ2 . Write g2 as g2 = φ ◦ g1 ◦ φ−1 . This is possible by our considerations above. By assumption, we know that there exists an element γ1 ∈ Γ1 , such that (−g1 + id)(γ1 ) = φ−1 (γ2 ). Then we have that (−g2 + id)(φ(γ1 )) = (−φ ◦ g1 ◦ φ−1 + id)(φ(γ1 )) = −φ ◦ g1 (γ1 ) + φ(γ1 ) = φ(−g1 (γ1 ) + γ1 ) = φ ◦ φ−1 (γ2 ) = γ2 . Hence, if (Γ1 , G1 ) is a Ferrero pair, so is (Γ2 , G2 ). Now we can formulate our main theorem of this section:

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Theorem 5.4. Let N be a zero symmetric near-ring which is 2-primitive on the N -group Γ and let L (= {0}) be a minimal left ideal of N . Let Φ := AutN (Γ). If (Γ, Φ) is a Ferrero pair, then L is a planar near-ring. Proof. Since L is a left ideal and N is zero symmetric, L is a subnearring of N and N L ⊆ L. Since LΓ = {0} there is an element γ ∈ Γ such that Lγ = Γ (see the proof of Proposition 3.1 ). Hence, there exists an element e ∈ L \ {0} such that eγ = γ. Therefore (e2 )γ = eγ. Consequently, −e + e2 ∈ (0 : γ) ∩ L = {0}. Therefore e = e2 (these arguments are taken from the proof of ([1], Theorem 4.56)). By Lemma 5.2, N is 2-primitive on L (with the natural action of N on L via nearring multiplication) and G := AutN (L) is a ﬁxedpointfree automorphism group on L, consisting of the non-zero N -endomorphisms of L. Since Γ∼ =N L, (L, G) is also a Ferrero pair, by Proposition 5.3. Furthermore, e is a right identity in L: Le is a non-zero N -subgroup of L. By 2primitivity of N on L we must have Le = L and since e is idempotent, e is a right identity in L. Let a, b ∈ L. Then we say a ≡L b if and only if for all l ∈ L: la = lb (≡L is just the equivalence relation ≡ restricted to the subnear-ring L of N ). Let g1 , g2 ∈ G, such that g1 = g2 but suppose g1 (e) ≡L g2 (e). This would mean that eg1 (e) = eg2 (e) and since G = AutN (L), g1 (e2 ) = g2 (e2 ). Since e2 = e, we get e = g1−1 g2 (e) and since functions in G are ﬁxedpointfree, either e = 0, which is a contradiction, or g1−1 g2 = id, which is again a contradiction. Since |G| ≥ 2, |L/ ≡L | ≥ 3. Proposition 3.1 enables us to apply Theorem 3.4 which states that L is isomorphic to a sandwich centralizer near-ring M0 (X, L, φ, C). Here, from the proof of Theorem 3.4, C := AutL (L), X = 1l L (l ∈ L# is arbitrary but ﬁxed) and φ : L −→ X, l → 1l l. By Proposition 5.1, we know that AutN (L) is a ﬁxedpointfree automorphism group on L. Clearly, AutN (L) ⊆ AutL (L). On the other hand, take ψ ∈ AutL (L). Since e is a right identity of L, we have for all l ∈ L and for all n ∈ N : ψ(nl) = ψ((nl)e) = (nl)ψ(e) = n(lψ(e)) = nψ(l) (note that nl ∈ L). So, ψ ∈ AutN (L) and ﬁnally, C = AutL (L) = AutN (L) = G. Therefore, (L, C) is a Ferrero pair and M0 (X, L, φ, C) is a sandwich centralizer near-ring as described in Theorem 4.5. It follows that L is a planar near-ring.

Note that by Proposition 5.1 the assumption that (Γ, Φ), where Φ := AutN (Γ), is a Ferrero pair is a very natural one. Especially if Γ is ﬁnite, this does not require more than just the fact that there exists a nonidentity N -automorphism of Γ.

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6.

Conclusive remarks

We have seen in the preceeding section, that planarity of near-rings occurs very naturally within 2-primitive near-rings. I want to use this last section to point out the following: It is well known (see for example ([1], Theorem 4.52) for a reference) that zero symmetric 2-primitive near-rings with identity are just “dense” subnear-rings of centralizer near-rings of the type MG (Γ), where G is a ﬁxedpointfree automorphism group of Γ. Now note the high similarity of this with the description of planar near-rings as special centralizer near-rings M0 (X, N, φ, C) with sandwich multiplication, where C is also a ﬁxedpointfree automorphism group of N . This again points out a very close relation between planar and primitive near-rings. The reader who now might wonder if it is possible to describe primitive near-rings using sandwich near-rings, should take a look at [3]. In this paper a description of 1-primitive near-rings (not necessarily having an identity or a right identity) is given in terms of what the authors call centralizer sandwich near-rings. Although the construction idea for centralizer sandwich near-rings is diﬀerent to the construction of sandwich centralizer near-rings used in our paper, lots of the ideas used in our paper have their origin in [3]. Note that 1-primitive near-rings in general do not have a right identity, so we cannot use our sandwich centralizer near-rings to describe them. But in the ﬁnite case they have and it is therefore also possible to describe 1-primitive near-rings as sandwich centralizer near-rings. It is not within the scope of this article to go into the details here.

References [1] G. Pilz, Near-Rings, North Holland Publishing Company, 1977. [2] J. R. Clay, Nearrings - Geneses and Applications, Oxford University Press, 1992. [3] P. Fuchs, G. Pilz, A new density theorem for primitive near-rings, Near-rings and near-ﬁelds (Oberwolfach, 1989), 68–74, Math. Forschungsinst. Oberwolfach, Schwarzwald, 1995. [4] J. D. P. Meldrum, Near-rings and their links with groups, Pitman Advanced Publishing Programm, 1985.

ON THE f -PRIME RADICAL OF NEARRINGS Satyanarayana Bhavanari∗ Department of Mathematics Acharya Nagarjuna University Nagarjuna Nagar 522510 A. P. India [email protected] n

Richard Wiegandt† A. R´ ´enyi Institute of Mathematics Hungarian Academy of Sciences P.O. Box 127 H-1364 Budapest, Hungary [email protected]

Abstract

The f -prime radical rf of 0-symmetric near-rings is an idempotent Hoehnke radical; either rf (N ) = 0 or rf (N ) = β(N ) the prime radical. The radical classes of rf and β coincide. In a universal class of near-rings, if the f -prime radical is complete then rf = β. 2000 Mathematics Subject Classiﬁcation: 16Y30; 16N80.

Keywords: prime near-ring, prime radical, strong nilpotence, idempotent Hoehnke radical

1.

Introduction

Murata, Kurata and Marubayashi [3] introduced and investigated a generalization of prime rings, called f -prime rings. In [1] Groenewald and Potgieter extended results to f -prime and f -semiprime near-rings. The f -prime radical of a ﬁxed near-ring N was deﬁned in [5] as the intersection of all f -prime ideals of N and it was described as the subset ∗ The ﬁrst author is thankful for the 3 months study leave payed at the A. R´ ´enyi Institute of Mathematics under the auspices of the Indo-Hungarian Cultural Exchange Programme and to the Acharya Nagarjuna University for encouragement. † Research supported by the Hungarian OTKA Grant # T043034

293 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 293–299. c 2005 Springer. Printed in the Neatherlands.

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of N consisting of 0 and all f -strongly nilpotent elements. Note that f -primeness is deﬁned by a mapping fN assigning an ideal fN (a) of N to each element a ∈ N which is subject to certain constraints. This mapping fN is not uniquely determined, so each such mapping fN deﬁnes an f -primeness as well as an f -strong nilpotence. In this note we shall study the properties of the assignment rf belonging to a ﬁxed mapping fN and designating the f -prime radical rf (N ) to each near-ring N . We shall refer to this assignment rf brieﬂy as to the f -radical. Throughout we shall work in the variety of all 0-symmetric right nearrings. For details we refer to Pilz’s book [4]. Sometimes we shall restrict our considerations to a universal class A of near-rings, which is a subclass of near-rings closed under taking ideals and homomorphic images. As usual, I N will mean that I is an ideal of the near-ring N . The principle ideal of N generated by a single element a ∈ N will be denoted by (a)N . Let γ be a mapping which assigns to each near-ring N an ideal γ(N ) of N . Such a mapping γ may satisfy some of the following conditions. (1) ϕ(γ(N )) ⊆ γ(ϕ(N )) for each homomorphism ϕ : N → ϕ(N ), (2) γ(N/γ(N )) = 0 for all N , (3) γ is idempotent: γ(γ(N )) = γ(N ) for all N , (4) γ is complete: γ(I) = I N implies γ(I) ⊆ γ(N ). The mapping γ is said to be a Hoehnke radical, if it satisﬁes (1) and (2); γ is called a Plotkin radical, if it satisﬁes (1), (3) and (4). An idempotent and complete Hoehnke radical is called a Kurosh–Amitsur radical. To any radical γ we associate two classes of near-rings, the radical class Rγ = {N | γ(N ) = N } and the semisimple class Sγ = {N | γ(N ) = 0}. As is well known, each Hoehnke radical γ is determined by its semisimple class Sγ via the assignment γ(N ) = ∩(I N | N/I ∈ Sγ ), and every subclass closed under subdirect sums is the semisimple class of a Hoehnke radical. For Plotkin radicals γ and Rγ determine each another uniquely. The prime radical β(N ) of a near-ring N is the intersection of all prime ideals of N . As proved in [2], in the variety of all 0-symmetric near-rings the prime radical β is an idempotent Hoehnke radical which is hereditary (β(N )∩I ⊆ β(I) for all I N and for all N ), but not complete.

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(For rings the radical β is, of course, Kurosh–Amitsur.) In the sequel we follow [7] for radical theoretical notations and deﬁnitions

2.

The Hoehnke radical rf

The notions of f -primeness, f -nilpotence and f -radical of a near-ring N are based on an ideal mapping fN designating to each element a ∈ N an ideal fN (a) of N such that (i) a ∈ fN (a), (ii) I N and x ∈ fN (a) + I imply fN (x) ⊆ fN (a) + I. An ideal mapping fN is, in fact, uniquely determined by the ideal fN (0) as seen from Lemma 1. An ideal mapping fN satisﬁes condition (i) and (ii) if and only if fN (a) = fN (0) + (a)N for all a ∈ N . Proof. Notice that for I = 0 condition (ii) yields fN (x) ⊆ fN (a) for all x ∈ fN (a). Since a ∈ fN (0) + (a)N by (ii) and (i) we get that fN (a) ⊆ fN (0) + (a)N ⊆ fN (0) + fN (a) ⊆ fN (a). Conversely, suppose that fN (a) = fN (0) + (a)N . Then condition (i) is trivially fulﬁlled. If x ∈ fN (a) + I, then x ∈ fN (0) + (a)N + I. Hence fN (x) = fN (0) + (x)N ⊆ fN (0) + (a)N + I = fN (a) + I, and so also (ii) is satisﬁed. We may deﬁne an ideal mapping fN by assigning to each near-ring N an ideal fN (0) of N . In view of Lemma 1 no relation exists between fN (0) and fM (0) where N and M are arbitrary distinct near-rings. Dealing, however, simultaneously with several near-rings, it is reasonable and not restrictive to demand the following additional requirement: (iii) if N ∼ = fM (0) by the same isomorphism. = M then fN (0) ∼ An element a ∈ N is said to be strongly nilpotent in N , if for any given sequence a0 , a1 , . . . of elements of N with a0 = a, ai = ai−1 a∗i−1 where a∗i−1 ∈ (ai−1 )N there corresponds an integer m such that an = 0 for all n ≥ m. The importance of the notion of strongly nilpotence is shown in Proposition 1 ([6]). The prime radical β(N ) of a near-ring is the set of all strongly nilpotent elements of N . A radical γ is said to be hereditary, if γ(N ) ∩ I ⊆ γ(I) for every I N and all near-rings N . Kaarli and Kriis [2] proved that the prime radical of near-rings is hereditary. One can easily see that Proposition 1 provides an alternative proof for the hereditariness of β.

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An element a ∈ N is said to be f -strongly nilpotent in N , if every element of the ideal fN (a) is strongly nilpotent in N , that is, fN (a) ⊆ β(N ). Clearly, every f -strongly nilpotent element is strongly nilpotent. The converse is not true: let N = β(N ) = 0, and deﬁne fN (0) = N . Then the elements of β(N ) are strongly nilpotent in N but not f -strongly nilpotent in N . Moreover, by deﬁnition, the element 0 ∈ N need not be f -strongly nilpotent in N . The f -radical rf – called f -prime radical – was introduced and characterized in [5]: rf (N ) = {all f -strongly nilpotent elements of N } ∪ {0}. For characterizing the f -radical rf of a near-ring N , we have to recall some deﬁnitions. A subset M of a near-ring N is called an m-system, if either M = ∅ or (a)N (b)N ∩ M = ∅ for all a, b ∈ M (cf. Pilz [4]). A subset F of a near-ring N is called an f -system, if F contains an m-system M such that fN (a) ∩ M = ∅ for all a ∈ F or F = ∅. Notice that every m-system is an f -system. A near-ring N is said to be f -prime, if N \ {0} is an f -system. An ideal I of a near-ring N is called an f -prime ideal, if the near-ring N/I is f -prime. Obviously, every prime near-ring is f -prime. Proposition 2 ([5]). The f -radical rf (N ) of a near-ring N is the intersection rf (N ) = ∩(I N | N/I is f -prime). Lemma 2. Let f be an ideal mapping and N a near-ring. Then rf (N ) = β(N ) if and only if fN (0) ⊆ β(N ). If rf (N ) = 0 then rf (N ) = β(N ). If β(N ) = N then rf (N ) = N . Proof. Suppose that rf (N ) = β(N ) = 0. Then each strongly nilpotent element is f -strongly nilpotent. Hence for any nonzero element a ∈ β(N ) we have fN (0) + (a)N = fN (a) ⊆ β(N ) which implies fN (0) ⊆ β(N ). Assume that rf (N ) ⊂ β(N ), and let a ∈ β(N ) \ rf (N ). Then the element a is not f -strongly nilpotent, and so fN (0) + (a)N = fN (a) ⊆ β(N ). Since (a)N ⊆ β(N ) by a ∈ β(N ), we conclude that fN (0) ⊆ β(N ). If rf (N ) = 0 then there exists an element a = 0 which is f -strongly nilpotent, that is fN (0) + (a)N = fN (a) ⊆ β(N ). Hence fN (0) ⊆ β(N ), and so rf (N ) = β(N ). Let β(N ) = N . Since fN (0) ⊆ N = β(N ) is always true, also the last assertion holds.

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Theorem 4. In a universal class A of 0-symmetric near-rings any f radical rf is an idempotent Hoehnke radical which is complete if and only if the prime radical β is complete in A and then rf = β. For a near-ring N , rf (N ) = β(N ) if and only if fN (0) ⊆ β(N ). If fN (0) ⊆ β(N ) then rf (N ) = 0. The radical classes Rf and Rβ coincide for every f -radical rf . Every near-ring N with rf (N ) = 0 is a subdirect sum of f -prime near-rings. Proof. By Proposition 2 the f -radical rf (N ) of a near-ring N is the intersection rf (N ) = ∩(I N | N/I is f -prime). The semisimple class corresponding to the f -radical rf is Sf = {N | rf (N ) = 0} which is obviously closed under taking subdirect sums. Hence rf is a Hoehnke radical. For the idempotence of rf we have to prove that rf (rf (N )) = rf (N ). By Lemma 2 we have to distinguish two cases. Case rf (N ) = 0. Then rf (rf (N )) = rf (0) = 0 = rf (N ). Case rf (N ) = β(N ). Then rf (rf (N )) = rf (β(N )). Suppose that rf (β(N )) = 0. Now rf (β(N )) = β(β(N )), so by Lemma 2 it follows that fβ (N ) (0) ⊆ β(β(N )) = β(N ), a contradiction. Hence rf (β(N )) = 0, and so again by Lemma 2 we have that rf (β(N )) = β(β(N )) = β(N ) = rf (N ). Thus rf (rf (N )) = rf (N ) for all near-rings N . Assume that rf is a complete Hoehnke radical in the universal class considered. Then rf is a Kurosh–Amitsur radical, and so rf is determined by its radical class Rf = {N | rf (N ) = N }. Since every prime near-ring is f -prime, we have that rf (N ) ⊆ β(N ). This shows us that the radical class Rf is contained in the radical class Rβ of the prime radical β. If 0 = N ∈ Rβ then fN (0) ⊆ N = β(N ), and so by Lemma 2 we have that rf (N ) = β(N ) = N , proving the containment Rβ ⊆ Rf . Hence we get (I N | β(N ) = I) = (I N | rf (N ) = I) = rf (N ) = β(N ). I

I

Thus rf as well as β is complete in the universal class considered. The rest follows from Lemma 2.

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Concerning rf -semisimple near-rings we have obviously the following. Corollary 1. rf (N ) = 0 for a near-ring N if and only if either β(N ) = 0 or N possesses nonzero strongly nilpotent elements one of which is f strongly nilpotent in N . Corollary 2. In the variety of all 0-symmetric near-rings an f -radical rf is never complete, and hence never a Kurosh–Amitsur radical. Proof. Kaarli and Kriis [2] proved that the prime radical β is not complete, so the statement follows directly from the Theorem.

3.

On the hereditariness of rf

We shall give a suﬃcient, but not necessary condition for the hereditariness of an f -radical rf . Proposition 3. If an ideal mapping f satisﬁes fI (0) ⊆ fN (0) for every I N and near-ring N , then the corresponding f -radical rf is hereditary. Proof. Let a ∈ rf (N ) ∩ I be a nonzero element. Then the element a is f -strongly nilpotent in N , so fN (a) ⊆ β(N ) and fI (a) ⊆ fI (a) + (a)I ⊆ (ffN (0) ∩ I) + ((a)N ∩ I) ⊆ ⊆ (ffN (0) + (a)N ) ∩ I = fN (a) ∩ I ⊆ β(N ) ∩ I. Since by [2] the prime radical β is hereditary, we have fI (a) ⊆ β(N )∩I ⊆ β(I), and so a ∈ I is f -strongly nilpotent in I. Thus a ∈ rf (I), proving that rf (N ) ∩ I ⊆ rf (I). We ﬁnish this note by two examples. Example 1. The prime radical β is a hereditary f -radical in the variety of all 0-symmetric near-rings, but fI (0) ⊆ fN (0) ∩ I for some I N . Let us deﬁne fN (0) = β(N ). Then rf (N ) = β(N ) for all N , and f -strong nilpotence coincides with strong nilpotence. As proved in [2], the prime radical β is not complete, so there exists a near-ring N possessing an ideal I such that β(I) = I ⊆ β(N ). Hence, taking into account the hereditariness of β, we have β(I) ⊆ β(N ) ∩ I ⊆ β(I). Therefore, there exists an element a ∈ β(I) \ (β(N ) ∩ I). Now a ∈ fI (0) \ (ffN (0) ∩ I), and so fI (0) ⊆ fN (0) ∩ I. Example 2. In the subvariety of all associative rings the assignment τ : N → τ (N ) where τ (N ) is the ideal of N consisting of all additively

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torsion elements and 0, deﬁnes a hereditary Kurosh–Amitsur radical, called the torsion ideal. We may deﬁne fN (0) = τ (N ) for all rings N . Then by Lemma 2, τ (N ) = fN (0) ⊆ β(N ) if and only if rf (N ) = β(N ). In this case a nonzero element a ∈ N is f -strongly nilpotent in N precisely when the element a as well as each additively torsion element is strongly nilpotent in N . If τ (N ) ⊆ β(N ), that is, if there exists an additively torsion element in N which is not strongly nilpotent, then N does not contain f -strongly nilpotent elements at all.

References [1] N. J. Groenewald and P. C. Potgieter, A generalization of prime ideals in nearrings, Comm. in Algebra 12 (1984), 1835–1853. ¨ Toimetised, 764 [2] K. Kaarli and T. Kriis, Prime radical near-rings, Tartu Riikl. Ul. (1987), 23–29. [3] K. Murata, Y. Kurata and H. Marubayashi, A generalization of prime ideals in rings, Osaka J. Math. 6 (1969), 291–301. [4] G. Pilz, Near-rings, North-Holland, 1983. [5] Y. V. Reddy and Bh. Satyanarayana, The f -prime radical in near-rings, Indian J. pure appl. Math. 17 (1986), 327–330. [6] V. Sambasiva Rao and Bh. Satyanarayana, The prime radical in near-rings, Indian J. pure appl. Math. 15 (1984), 361–364. [7] R. Wiegandt, Radical theory of rings, The Math. Student, 51 (1983), 145–183.

ON FINITE GOLDIE DIMENSION OF Mn (N )-GROUP N n Satyanarayana Bhavanari Department of Mathematics Nagarjuna University Nagarjuna Nagar -522 510, India [email protected]

Syam Prasad Kuncham Department of Mathematics Manipal Institute of Technology Manipal Academy of Higher Education Manipal - 576 119, India [email protected]

Abstract

Let N be a zero-symmetric right nearring. The aim of this paper is to prove that the Goldie dimension of the N -group N is equal to that of Mn (N )-group N n where Mn (N ) is the matrix nearring. 2000 Mathematics Subject Classiﬁcation: 16Y30, 16P60, 16S50.

Keywords: Finite Goldie Dimension, Matrix Nearring, Uniform Ideal, Linearly Independent element, u-linearly Independent element.

Introduction A non-empty set N with two binary operations “+” and “·” is called a nearring if it satisﬁes the following axioms. (i) (N, +) is a group (not necessarily abelian) (ii) (N, ·) is a semi-group; (iii) (a + b)c = ac + bc, for all a, b, c ∈ N . Precisely speaking it is a right nearring. Moreover, if N satisﬁes the property n0 = 0 for all n ∈ N then N is called as a zero-symmetric nearring. 301 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 301–310. c 2005 Springer. Printed in the Neatherlands.

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A group (G, +) is said to be an N -group if there exists a mapping N ×G → G (the image of (n, g) is denoted by ng) satisfying the following two conditions: (i) (n + m)g = ng + mg, (ii) (nm)g = n(mg), for all n, m ∈ N and g ∈ G. Throughout, by a nearring, we mean a zero-symmetric right nearring. N stands for a nearring and G stands for an N -group. For the preliminary deﬁnitions and notations we refer to Pilz [6]. X denotes the ideal generated by X, for a given subset X of G and a denotes {a}. The concept of ﬁnite Goldie Dimension in N -Groups was introduced by Reddy and Satyanarayana [12]. An ideal H of G is said to have ﬁnite Goldie dimension (FGD) if H does not contain an inﬁnite number of non-zero ideals of G whose sum is direct. An ideal A of G is said to be essential in an ideal B of G (denote as A ≤e B) if I is an ideal of G contained in B and A ∩ I = (0) imply I = (0). It can be observed that G has FGD ⇐⇒ for any strict increasing sequence H1 ⊆ H2 ⊆ . . . of ideals of G, there exists a positive integer k such that Hi is essential in Hi+1 for all i ≥ k. An ideal A of G is said to be uniform if every non-zero ideal I of G, which is contained in A, is essential in A. If U is an ideal of G, then a direct veriﬁcation shows that the two following conditions are equivalent: (i) U is uniform, and (ii) 0 = x ∈ U and 0 = y ∈ U = =⇒ x ∩ y = (0). Reddy and Satyanarayana [12], proved that if an ideal H of G has FGD, then there exists ﬁnite number of uniform ideals Ui , 1 ≤ i ≤ k of G whose sum is direct and essential in H. This number k is independent of the choice of Ui ’s and the number k is called the Goldie dimension of H. In this case, we write k = dim H. This paper divided into three sections. In section 1, the authors study the relations between the ideals of the N -group N and the ideals of the Mn (N )-group N n where Mn (N ) is the matrix nearring. In section 2, the authors study the relations between the linearly independent elements of the N -group N and the linearly independent elements of Mn (N )-group N n . In section 3, we prove our main result that the Goldie dimension of the N -group N is equal to that of the Mn (N )-group N n .

1.

Matrix Nearrings and the Mn(N )-group N n

Meldrum and Van der Walt [4] introduced the concept of matrix nearring. Later it was studied by the authors like Meldrum and Meyer [3], Meyer [5], Satyanarayana, Lokeswara Rao and Syam Prasad [8], Syam

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Prasad and Satyanarayana [11]. The deﬁnition of a matrix nearring is as follows. We consider N with multiplicative identity 1. N n denotes the direct sum of n-copies of (N, +). For any r ∈ N, 1 ≤ i ≤ n and 1 ≤ j ≤ n, deﬁne fijr : N n → N n as fijr (a1 , a2 , . . . , an ) = (0, ..., raj , ..., 0) (here raj is in the ith place). If f r : N → N deﬁned by f r (x) = rx for all x ∈ N, Ii : N → N n is the canonical monomorphism embedding N in the ith position; and πj : N n → N is the jth projection map, then it is clear that fijr = Ii f r πj and fijr ∈ M (N n ) where M (N n ) is the nearring of all mappings from N n → N n . The sub-nearring Mn (N ) of M (N n ) generated by {ffijr : r ∈ N, 1 ≤ i ≤ n, 1 ≤ j ≤ n} is called the matrix nearring over N . Now N n becomes an Mn (N )-group. If N is an associative ring with identity, then the matrix nearring coincides with the ring of matrices over N . For preliminary deﬁnitions and results on matrix nearrings, we refer [4, 8]. In the rest of the paper we use N N for the N -group N , and N n for the Mn (N )-group N n . Notation 1.1. For any ideal I of N n , we write I ∗ ∗ = {x ∈ N : x = πj A for some A ∈ I, 1 ≤ j ≤ n} where πj is the jth projection map from N n to N . Lemma 1.2. Let I, J be ideals of N n . (i) I ∗ ∗ = {x ∈ N : (x, 0, . . . , 0) ∈ I}; (ii) If I ⊆ J, then I ∗ ∗ ⊆ J ∗ ∗; and (iii) If I = J, then I ∗ ∗ = J ∗ ∗. Proof. (i) Let x ∈ right hand side set. Then (x, 0, . . . , 0) ∈ I. This implies π1 (x, 0, . . . , 0) = x ∈ I ∗ ∗. Take x ∈ I ∗ ∗. Then x = πj A for some A = (x1 , x2 , . . . , xn ) ∈ I. This implies x = xj . Since fijr ∈ Mn (N ) and A ∈ I, by the theorem 1.34 of Pilz [6], (xj , 0, . . . , 0) = fij1 A ∈ I, which implies xj belongs to the right hand side set. (ii) follows from (i). (iii) Since I = J, we can take (x1 , x2 , . . . , x n ) ∈ I \J. If (xi , 0, . . . , 0) ∈ J for all 1 ≤ i ≤ n then (x1 , x2 , . . . , xn ) = ni=1 fi11 (xi , 0, . . . , 0) ∈ J, a contradiction. So (xj , 0, . . . , 0) is not in J for some 1 ≤ j ≤ n. This shows that (xj , 0, . . . , 0) ∈ I \ J and so xj ∈ I ∗ ∗ \ J ∗ ∗. This completes the proof. Lemma 1.3. I ∗ ∗ is a left ideal of N . Proof. Let n ∈ N and x ∈ I ∗∗. Now (n+x−n, 0, . . . , 0) = (n, 0, . . . , 0)+ (x, 0, . . . , 0) − (n, 0, . . . , 0) ∈ I (since I is an ideal of N n ). Therefore

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n + x − n ∈ I ∗ ∗. This shows that I ∗ ∗ is a normal subgroup of N . For n1 , n2 ∈ N and x ∈ I ∗ ∗, we have (n1 (n2 + x) − n1 n2 , 0, . . . , 0) = n1 n1 ((n2 , 0, . . . , 0) + (x, 0, . . . , 0)) − f11 (n2 , 0, . . . , 0) ∈ I, which implies f11 that n1 (n2 + x) − n1 n2 ∈ I ∗ ∗. Hence I ∗ ∗ is a left ideal of N . Theorem 1.4. Suppose L is a subset of N . (i) If Ln is an ideal of N n , then L = (Ln ) ∗ ∗; (ii) L is an ideal of N N if and only if Ln is an ideal of N n ; (iii) If L is an ideal of N N , then L = (Ln ) ∗ ∗. (Note that Ln denotes the direct sum of n copies of L). Proof. (i) Suppose Ln is an ideal of N n . Now x ∈ L ⇐⇒ (x, 0, . . . , 0) ∈ Ln ⇐⇒ x ∈ (Ln ) ∗ ∗ (by lemma 1.2). Therefore L = (Ln ) ∗ ∗. (ii) If Ln is an ideal then by (i), L = (Ln ) ∗ ∗ and by lemma 1.3, L = (Ln ) ∗ ∗ is an ideal of N . The converse follows from 4.1 of [4]. (iii) Follows from (ii) and (i). Lemma 1.5. (i) If I is an ideal of N n , then (I ∗ ∗)n = I; and (ii) Every ideal I of N n is of the form K n for some ideal K of N N . Proof. (i) Take (x1 , x2 , . . . , xn ) ∈ (I ∗ ∗)n . Then xi ∈ I ∗ ∗ for 1 ≤ i ≤ n. By lemma 1.2, we have that (xi , 0, . . . , 0) ∈ I. Since I is an ideal of N n , we have that fi11 (xi , 0, . . . , 0) ∈ I. This means that (0, . . . , xi , . . . , 0) ∈ I (xi in the ith position) for 1 ≤ i ≤ n. Therefore (x1 , x2 , . . . , xn ) = (0, . . . , xi , . . . , 0) ∈ I. i

Hence (I ∗ ∗)n ⊆ I. The other part is clear from the notation 1.1 (ii) Follows from (i) (use K = I ∗ ∗). A straightforward veriﬁcation gives the following remark. Remark 1.6. Suppose I, J are ideals of N N . Then (i) (I ∩ J)n = I n ∩ J n and (ii) I ∩ J = (0) ⇐⇒ (I ∩ J)n = (0) ⇐⇒ I n ∩ J n = (0). Note 1.7. (i) For an element u of N , let ιi (u) = (0, . . . , u, . . . , 0) with u in the ith position. Then ιi (u) = (u, 0, . . . , 0) (ii) If U is an ideal of N N and I is an ideal of N n such that I ⊆ U n , then I ∗ ∗ ⊆ U . (iii) A ≤e N N ⇐⇒ An ≤e N n Proof. (i) Suppose u is an element of N . Then (u, 0, . . . , 0) ∈ N n . Now 1 (ι (u)) ∈ ι (u). Therefore (u, 0, . . . , 0) ⊆ ι (u). (u, 0, . . . , 0) = f1i i i i

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On the other hand, ιi (u) = fi11 (u, 0, . . . , 0) ∈ (u, 0, . . . , 0). Therefore ιi (u) ⊆ (u, 0, . . . , 0). (ii) Suppose U is an ideal of N and I is an ideal of N n such that I ⊆ U n . Let x ∈ I ∗ ∗. By the lemma 1.2, (x, 0, . . . , 0) ∈ I. Since I ⊆ U n , we have that (x, 0, . . . , 0) ∈ U n and so x ∈ U . Therefore I ∗ ∗ ⊆ U. (iii) Suppose A≤eN N . Let I be an ideal of N n such that An ∩ I = (0). Now by lemma 1.5, I = (I ∗ ∗)n . So An ∩ (I ∗ ∗)n = (0). This implies (A ∩ I ∗ ∗)n = (0), by the remark 1.6. Again by remark 1.6 we get that A ∩ I ∗ ∗ = (0). Since A ≤e N , we have that I ∗ ∗ = (0). This means that I = (0). Thus An ≤e N n . Conversely suppose that An ≤e N n . Let I be an ideal of N N such that A ∩ I = (0). Then An ∩ I n = (0) and I n is an ideal of N n =⇒ = I n = (0) = =⇒ I = (0). This shows that A≤eN N . Lemma 1.8. If x, u are elements of N and x ∈ u, then (x, 0, . . . , 0) ∈ (u, 0, . . . , 0). Proof. Now we use notation 0.1 [12]. With this notation we have u = ∞ A where Ai+1 = A∗i ∪ A0i ∪ A+ i+1 i=1 i with A0 = {u}, here A∗i = {m + y − m : m ∈ N, y ∈ Ai }, A0i = {a − b : a, b ∈ Ai } ∪ {a + b : a, b ∈ Ai }, and = {n1 (n2 + a) − n1 n2 : n1 , n2 ∈ N and a ∈ Ai }. A+ i Observe that for n1 = 1 and n2 = m, it is clear that A∗i ⊆ A+ i . Since x ∈ u, we have x ∈ Ak for some k. Now we show (by induction on = (x, 0, . . . , 0) ∈ (u, 0, . . . , 0). Suppose k = 0. Then k) that x ∈ Ak =⇒ x ∈ Ak = A0 = {u}. This implies that x = u. Therefore (x, 0, . . . , 0) ∈ (u, 0, . . . , 0) and so the statement is true for k = 0. = Assume the induction hypothesis for k = i, that is, x ∈ Ai =⇒ . (x, 0, . . . , 0) ∈ (u, 0, . . . , 0). Let x ∈ Ai+1 = A∗i ∪ A0i ∪ A+ i Case (i): Suppose x ∈ A∗i , then x = m + y − m for some m ∈ N, y ∈ Ai . Now y ∈ Ai implies that (y, 0, . . . , 0) ∈ (u, 0, . . . , 0) and so (x, 0, . . . , 0) = (m, 0, . . . , 0) + (y, 0, . . . , 0) − (m, 0, . . . , 0) ∈ (u, 0, . . . , 0). Case (ii): Suppose x ∈ A0i . Without loss of generality we assume that x = a − b for some a, b ∈ Ai . Now (a, 0, . . . , 0), (b, 0, . . . , 0) ∈ (u, 0, . . . , 0) and so (x, 0, . . . , 0) = (a − b, 0, . . . , 0) = (a, 0, . . . , 0) − (b, 0, . . . , 0) ∈ (u, 0, . . . , 0) (by the induction hypothesis). Case (iii): Suppose x ∈ A+ i , then x = n1 (n2 + a) − n1 n2 for some n1 , n2 ∈ N and a ∈ Ai . Now n1 n1 ((n2 , 0, . . . , 0) + (a, 0, . . . , 0)) − f11 (n2 , 0, . . . , 0) (x, 0, . . . , 0) = f11 ∈ (u, 0, . . . , 0).

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Thus by mathematical induction we have (x, 0, . . . , 0) ∈ (u, 0, . . . , 0) for any x ∈ u. Theorem 1.9. If u is an element of N , then un = (u, 0, . . . , 0). Proof. Since (u, 0, . . . , 0) ∈ un , we have (u, 0, . . . , 0) ⊆ un . Take (x1 , x2 , . . . , xn ) ∈ un . This implies xi ∈ u for 1 ≤ i ≤ n. By lemma 1.8, (xi , 0, . . . , 0) ∈ (u, 0, . . . , 0) for all i. Now (x1 , x2 , . . . , xn ) = 1 (x2 , 0, . . . , 0) + · · · + fn11 (xn , 0, . . . , 0) (x1 , 0, . . . , 0) + f21 ∈ (u, 0, . . . , 0) (by 1.34 of Pilz [6]). This completes the proof.

2.

Linearly Independent elements in N -groups

The concepts linearly independent elements and u-linearly independent elements were introduced in Satyanarayana and Syam Prasad [9]. The related deﬁnitions are given here. A subset Xof G is said to be a linearly independent (l.i., in short) set if the sum a∈X a is direct. If {ai : 1 ≤ i ≤ n} is a l.i. set, then we say that the elements ai , 1 ≤ i ≤ n are linearly independent. If X is not a l.i. set then we say that X is linearly dependent (l.d., in short). Lemma 2.1. For any x1 , x2 , . . . , xk ∈ N N we have (x1 + x2 + · · · + xk )n = x1 n + x2 n + · · · + xk n . Proof. xi ⊆ x1 + x2 + · · · + xk , we have xi n ⊆ (x1 + x2 + · · · + xk )n for 1 ≤ i ≤ k. Therefore x1 n + x2 n + · · · + xk n ⊆ (x1 + x2 + · · · + xk )n . Let (a1 , a2 , . . . , an ) ∈ (x1 + x2 + · · · + xk )n . Then ai ∈ x1 + x2 + = ai = ai1 + · · · + aik where aij ∈ xj for · · · + xk for 1 ≤ i ≤ n =⇒ 1 ≤ i ≤ n, 1 ≤ j ≤ k. Now (a1 , a2 , . . . , an ) = (a11 + · · · + a1k , . . . , an1 + · · · + ank ) = (a11 , . . . , an1 ) + · · · + (a1k , . . . , ank ) ∈ x1 n + x2 n + · · · + xk n . Therefore (x1 + x2 + · · · + xk )n ⊆ x1 n + x2 n + · · · + xk n . This completes the proof. Theorem 2.2. x1 , x2 , . . . , xk are linearly independent elements in N N if and only if (x1 , 0, . . . , 0), (x2 , 0, . . . , 0), . . . , (xk , 0, . . . , 0) are linearly independent in N n .

On Finite Goldie Dimension of Mn (N )-Group N n

Proof. x1 , x2 , . . . , xk are linearly independent elements in

307 NN

⇐⇒ the sum x1 + x2 + · · · + xk is direct ⇐⇒ xi ∩ (x1 + · · · + xi−1 + xi+1 + · · · + xk ) = 0 for 1 ≤ i ≤ k. ⇐⇒ [xi ∩ (x1 + · · · + xi−1 + xi+1 + · · · + xk )]n = 0 for 1 ≤ i ≤ k (by remark 1.6(ii)). ⇐⇒ xi n ∩ [(x1 + · · · + xi−1 + xi+1 + · · · + xk )]n = 0 for 1 ≤ i ≤ k (by remark 1.6(ii)). ⇐⇒ xi n ∩ [(x1 n + · · · + xi−1 n + xi+1 n + · · · + xk n ] = 0 for 1 ≤ i ≤ k (by lemma 2. 1). ⇐⇒ x1 n + x2 n + . . . xk n is direct ⇐⇒ (x1 , 0, . . . , 0) + (x2 , 0, . . . , 0) + · · · + (xk , 0, . . . , 0) is direct (by theorem 1.9) ⇐⇒ (x1 , 0, . . . , 0), (x2 , 0, . . . , 0), . . . , (xk , 0, . . . , 0) are linearly independent in N n .

3.

Dimension of the Mn(N )-group N n

The concept “uniformly linearly independent elements” was introduced in Satyanarayana and Syam Prasad [9]. An element u ∈ G is said to be uniform element (u-element, in short) if u is a uniform ideal of G. If ui ∈ G, 1 ≤ i ≤ n are u-elements and are linearly independent, then we say that u1 , u2 , . . . , un are u-linearly independent elements (u.l.i. elements, in short). We say that G is essentially spanned by a subset X of G if x∈X x is essential in G. In this case, we also say that X spans G essentially (or X is an essential spanning set for G). Theorem 3.1 (Satyanarayana and Syam Prasad [9]). If G has FGD, then the following are equivalent: (i) dim G = n; (ii) There exist n uniform ideals Ui , 1 ≤ i ≤ n whose sum is direct and essential in G; (iii) The maximum number of u.l.i. elements in G is n; (iv) n is maximal with respect to the property that for any given {x1 , x2 , . . . , xk } of u.l.i. elements with k < n, there exist xk+1 , . . . , xn such that {x1 , x2 , . . . , xn } are u.l.i. elements; (v) The maximum number of l.i. elements that can span G essentially is n;

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(vi) The maximum number of u.l.i. elements that can span G essentially is n. Before proving the main theorem, we prove two Lemmas. Lemma 3.2. Let u ∈ N N . Then u is a uniform element in only if (u, 0, . . . , 0) is an uniform element in N n .

NN

if and

Proof. Suppose u is a uniform element in N N . In a contrary way, suppose (u, 0, . . . , 0) is not a uniform element in N n . Then (u, 0, . . . , 0) is not a uniform ideal in N n . So there exist two non zero ideals I and J of N n which are contained in (u, 0, . . . , 0) such that I ∩ J = (0). Since I, J are ideals of N n , by lemma 1.5, I = (I ∗ ∗)n and J = (J ∗ ∗)n . Now =⇒ (I ∗ ∗ ∩ J ∗ ∗)n = (0) (by I ∩ J = (0) = =⇒ (I ∗ ∗)n ∩ (J ∗ ∗)n = (0) = remark 1.6) = =⇒ I ∗ ∗ ∩ J ∗ ∗ = (0). Now by note 1.7 (ii) and theorem 1.9, we get that I ∗ ∗ ⊆ u and J ∗ ∗ ⊆ u. So we have that I ∗ ∗ and J ∗ ∗ are two non-zero ideals of N N whose intersection is zero and they are contained in u, a contradiction to the fact that u is a uniform ideal of N N . Conversely, suppose (u, 0, . . . , 0) is a uniform element in N n . This =⇒ un is an uniform means (u, 0, . . . , 0) is an uniform ideal of N n = n ideal of N (by theorem 1.9). If u is not uniform, then there exist two non-zero ideals A and B of N N , contained in u such that A ∩ B = (0). Now An and B n are non-zero ideals of N n which are contained in un . Since A ∩ B = (0), we have (A ∩ B)n = (0) and by remark 1.6, An ∩ B n = (0), a contradiction to the fact that un is uniform. This completes the proof. Lemma 3.3. Suppose x1 , x2 , . . . , xk ∈ N N . Then x1 , x2 , . . . , xk are u.l.i. elements in N N if and only if (x1 , 0, . . . , 0), . . . , (xk , 0, . . . , 0) are u.l.i. elements in N n . Proof. By using theorem 2.2 and lemma 3.2 we get the following. x1 , x2 , . . . , xk are u.l.i. elements in N N ⇐⇒ x1 , x2 , . . . , xk are l.i. elements in N N and each xi is uniform ⇐⇒ (x1 , 0, . . . , 0), . . . , (xk , 0, . . . , 0) are l.i. elements and each (xi , 0, . . . , 0) uniform ⇐⇒ (x1 , 0, . . . , 0), . . . , (xk , 0, . . . , 0) are u.l.i. elements in N n . We now prove our main theorem. Theorem 3.4. dim N N = dim N n . Proof. Suppose dim N N = n. By theorem 3.1, there exists u.l.i. ele= x1 , x2 , . . . , xn ments x1 , x2 , . . . , xn in N N which span N N essentially =⇒ are u.l.i. elements and x1 + x2 + · · · + xn ≤e N N

On Finite Goldie Dimension of Mn (N )-Group N n

309

=⇒ (x1 , 0, . . . , 0), (x2 , 0, . . . , 0), . . . , (xn , 0, . . . , 0) are u.l.i. elements in = N n and (x1 + x2 + · · · + xn )n ≤e N n (by Lemma 3.3 and Note 1.7 (iii)) = =⇒ (x1 , 0, . . . , 0), (x2 , 0, . . . , 0), . . . , (xn , 0, . . . , 0) are u.l.i. elements in N n and x1 n + x2 n + · · · + xn n ≤e N n (by Lemma 2.1) = (x1 , 0, . . . , 0), (x2 , 0, . . . , 0), . . . , (xn , 0, . . . , 0) are u.l.i. elements in =⇒ N n and (x1 , 0, . . . , 0) + (x2 , 0, . . . , 0) + · · · + (xn , 0, . . . , 0) is essen=⇒ dim N n = n (by theorem 3.1). Hence tial in N n (by theorem 1.9) = n dim N N = dim N . Application to Ring Theory: Let R be an associative ring with identity. Consider R as a left R-module, and Rn (the direct sum of n copies of R) as a module over the matrix ring Mn (R). If R has ﬁnite Goldie dimension, then by the theorem 3.4, we have that dim R = dim Rn .

Acknowledgment Part of this paper was done by the authors at the A.Renyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest. The ﬁrst author is thankful to the Hungarian and Indian Governments for selecting and providing him with the ﬁnancial assistance under the IndoHungarian Cultural Exchange Programme (June-Sept, 2003). The second author is thankful to the authorities of Manipal Academy of Higher Education for their encouragement. The authors also wish to express their thanks to the Organizing Committee of the 18th International Conference on Nearrings and Nearﬁelds (2003) for their invitation and hospitality during their stay at Hamburg.

References [1] Anh P.N. and Marki L. ”Left Orders in Regular Rings with Minimum Condition for principal One Sided Ideals”, Math. Proc. Comb. Phil. Soc., 109 (1991) 323 -333. [2] Goldie A.W. ”The Structure of Noetherian Rings” Lectures on Rings and Modules, Lecture notes in Math., Vol. 246 (Springer Verlag, 1972) 213-321. [3] Meldrum J.D.P., and Meyer J.H. ”Intermediate Ideals in Matrix Nearrings”, Comm. Algebra, 24 (5) (1996) 1601-1619. [4] Meldrum J. D. P. and Van der Walt A. P. J. ”Matrix Near Rings”, Arch. Math. 47 (1986), 312-319. [5] Meyer J.H. ”Chains of Intermediate Ideals in Matrix Nearrings”, Arch. Math. 63 (1994) 311-315. [6] Pilz G. ”Nearrings”, North Holland, 1983. [7] Satyanarayana Bh. ”On Essential E-irreducible submodules” Proc. 4th Ramanujan Symposium on Algebra and its Applications, University of Madras, Madras, Feb1-3, 1995, 127-129.

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[8] Satyanarayana Bh., Lokeswara Rao M. B. V. and Syam Prasad K. ”A Note on Primeness in Nearrings and Matrix nearrings”, Indian J. Pure Appl. Math., 27(1996), 227- 234. [9] Satyanarayana Bh. and Syam Prasad K. ”Linearly independent elements in N Groups with FGD”, Bull. Korean Math. Society (to appear). [10] Syam Prasad K. ”Contributions to nearring Theory-II” Doctoral Thesis, Nagarjuna University, 2000. [11] Syam Prasad K. and Satyanarayana Bh ”A Note on IFP N -groups”, Proc. 6th Ramanujan Symposium on Algebra and its Applications, University of Madras, Madras, Feb.24-26, 1999, 62-65. [12] Venkateswara Reddy Y. and Satyanarayana Bh., ”A Note on N -Groups”, Indian J. Pure Appl. Math. 19 (1988) 842-845.

NEAR-RINGS, COHOMOLOGY AND EXTENSIONS Mirela S ¸ tef˘ fanescu Department of Mathematics Ovidius University Bd. Mamaia 124 500127-Constanta Romania [email protected]

Abstract

1.

After historical considerations on the cohomology of groups and nearrings and extensions of near-rings, we analyze some near-rings playing a role ˆ in constructing the cohomology of groups. Then the notion of pseudo-modules does appear naturally and it is presented.

Some history and non only it

In 1960-1962, by a series of papers, A. Fr¨ ohlich (see references) introduced the derived functors and satellites in noncommutative homological algebra. These functors appeared in connection with the notion of group pairs, considered also by A. Fr¨ ohlich. If N is a distributively generated (by S) left near-ring, then an (N, S)group A is a group (A, +) together with a multiplication N × A → A, such that: (x + y)a = xa + ya, (xy)a = x(ya), s(a + b) = sa + sb, for all x, y ∈ N, a, b ∈ A and s ∈ S. Let A be an (N, S)-group and A be a central (N, S)-subgroup of it. Then A/A is called a pair of (N, S)-groups or a group pair. A homomorphism of group pairs, f /f : A/A → B/B , is given by a commutative diagram (1.1)

f

A ↓

−→

A

−→

f

B ↓ B

where the vertical arrows are the inclusions.Usually, f has also the property: f (xa) = xf (a), for all x in N, a in A.We note that f is the restriction of f to A and Im f ⊆ B . 311 H. Kiechle et al. (eds.), Nearrings and Nearﬁelds, 311–319. c 2005 Springer. Printed in the Neatherlands.

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One may consider M AP (A/A , B/B ), the set of all mappings f : A → B with the restriction to A , f : A → B , making commutative a diagram like (1.1). This is a group under pointwise addition, which is a subgroup in M AP (A, B). The set of homomorphisms of pairs is not a subgroup of M AP (A/A , B/B ) but generates one, namely the subMN (A, B), group HOM MN (A/A , B/B ). This is also a subgroup of HOM the subgroup generated by HomN (A, B) in M AP (A, B). In M AP (A, A), the set of all distributive elements, namely End(A), generates a subnear-ring E(A). The same is true for the group B. Then, E(A) acts on the right hand on HOM (A, B) as well as EN (A) MN (A/A , B/B ), while acts on the right hand on HOM MN (A, B) or HOM E(B) acts on the left. Indeed, if we consider α, α ∈ E (A) , β, β ∈ HOM (A, B), then, by taking usual mapping composition βα : A → B, βα (a) := β (α (a)), for all a ∈ A, we have: (βα) α = β (αα ) and (β + β ) α = βα + β α. Now, if γ ∈ E (B), then γβ ∈ HOM (A, B) and this is a right action of E (B) on the group HOM (A, B) . The same will happen for N -groups. MN (A/A , B/B ) are right EN (A)Then both HOM MN (A, B) and HOM groups and left EN (B)-groups. Frohlich ¨ used these algebraic structures for constructing a theory of derived functors and satellites for N -group pairs. In the introduction of the ﬁrst paper [4], A. Fr¨ o¨hlich wrote: ,,In spite of this similarity to the theory for modules, there are natural fundamental diﬀerences. In the ﬁrst place, the sum of homomorphism of noncommutative groups is no longer a homomorphism; this is essentially the reason why the derived functors and satellites fail to be additive though they still retain the property of ,,preserving” null mappings”. In fact, for compensating this, Fr¨ o¨hlich has introduced group pairs. The second attempt to study cohomology of near-rings was made by Hans Lausch ([10] and [11]), who considered extensions of near-rings in connexion with the extension of groups (Baer, Schreier, MacLane) and rings (Everett). More precisely, the extension of a zero-ring by a distributively generated near-ring is built in his two papers. In this respect, the concept of a trimodule over a d.g. near-ring (R, S) is introduced. This is an abelian group with three external operations with ,,scalars” in R, satisfying relatively natural conditions, if we think of the constructions of such extensions for modules over rings, where the concept of bimodule has been used by Eilenberg and MacLane.

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Then an extension of the ring A by the d.g. near-ring R, with the set of distributive generators S, is an exact sequence: α

β

0 −→ A −→ E −→ R −→ 0 of near-rings and homomorphisms of near-rings, such that α(A) = Kerβ is distributive in E and for all x ¯ ∈ S, there exists a distributive element u(¯ x) ∈ E, such that β (u (x)) = x ¯. Lausch found also the connexion between the classes of equivalent extensions and the set of cohomology classes of cochains from R with coeﬃcients in A, when A is a zero-ring (Satz 6.3, Lausch [10]). In [15 ], we have constructed an extension of a near-ring A by a nearring R, by using some ideas of Schreier for extensions of groups in the non-abelian case and two multiplications of the types: s : R × R × A → A and w : R × A × A → A, satisfying ﬁve identities. Let A, R, E be zero-symmetric left near-rings and let Φ

Ψ

0 −→ A −→ E −→ R −→ 0 be an extension of A by R. Then Φ(A) is an ideal of E, E/Φ(A) ! R (as near-rings). We identify Φ(A) with A for simplifying the notations. Since there exists the isomorphism R → E/A, we may consider a transversal T in E for the cosets, T = {xα | α ∈ R}, xα being distinct representatives. Denote aα := −xα + a + xα , ab = −b + a + b, for α ∈ R, xα ∈ T, a, b ∈ A. Then we have the following Theorem 1.2. With the above notations, if Φ

Ψ

0 −→ A −→ E −→ R −→ 0 is an extension of near-rings, then there exist the mappings: θ : R → Aut(A, +), m : R × R → A, t : R × R → A, s : R × R × A → A, w : R × R × A → A, such that (i) xα +xβ = xα+β +m(α, β), xα ·xβ = xαβ +t(α, β), for all α, β ∈ R;

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m(α,β) (ii) (aα )β = aα+β , for all α, β ∈ R, a ∈ A; α (Here a = θ(α)(a).) (iii) m(α, β+γ)+m(β+γ) = m(α+β, γ)+(m(α, β))γ , for all α,β, γ ∈ R; (iv) a · b = w(α, a, b) − m(ω, ω) + a, for all a, b ∈ A, α ∈ R, where m(ω, ω) = xω , ω is the zero element in R; (v) s(αβ, γ, s(α, β, a)+w(α, b, a))+w(αβ, c, s(α, β, a)+w(α, b, a)) = = s(α, βγ, a) + w(α, s(β, γ, b), a) + w(α, w(β, c, b), a), for all α, β, γ ∈ R, a, b, c ∈ A; (vi) s(α, β +γ, a)+w(α, t(β, γ)+bγ +c, a) = m(αβ, αγ)+(s(α, β, a))αγ + +(w(α, b, a))αγ + s(α, γ, a) + w(α, c, a), for all α, β, γ ∈ R, a, b, c ∈ A. Proof is tedious, but it uses a natural path. One can see also that: t(ω, ω) = −m(ω, ω) + (m, (ω, ω))2 , t(α, β) = s(α, β, 0), for all α, β ∈ R. We may prove, by verifying the deﬁnition, the following theorem: Theorem 1.3. Given the near-rings A and R and the mappings m, θ, s, w, t satisfying the condition (i)-(vi), the set E = R × A, endowed with the binary compositions: (α, a) + (β, b) := (α + β, m(α, β) + aβ + b), (α, a) · (β, b) := (αβ, s(α, β, a) + w(α, b, a)), is a zero-symmetric left near-ring which is an extension of A by R. The case when R and E are d.g. near-rings and A is a zero-ring remains interesting; here the mappings s and w can be written, for α,β ∈ R, a, b ∈ A, as: s(α, β, a) = t(α, β) + v(a, β), w(α, b, a) = u(α, b), for two ﬁxed mappings u : R × A → A and v : A × R → A. One can see quite easily that −t(α, β) + s(α, β, a) does not depend on α, and w(α, b, a) does not depend on a. But the other relations remain complicated enough in comparison with Everett case of rings.

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Remark 1.4. If we have A a ring and R a left A-module, we may obtain the abstract aﬃne near-ring of Gonshor [7], by taking: m (α, β) = 0, xβ = x, for all α, β ∈ A and x ∈ R, and s (α, β, x) + w (α, β, x) = x + αy, for all α, β ∈ A, x, y ∈ R. Then (α, x) + (β, y) := (α + β, x + y) and (α, x) · (β, y) := (αβ, x + αy), for all (α, x), (β, y) in A × R.

2.

Pseudohomomorphisms and cohomology of groups

In the case of a non-abelian group (G, +), the set of endomorphisms is not a ring, since the sum of two endomorphisms is not an endomorphism of G. The same obstruction appears when we consider the set of homomorphisms Hom(G, H) from G to H, when the group H is not commutative. Which is exactly the obstruction? Let α,β ∈ Hom(G, H). Then we have: (α + β)(x + y) = α(x + y) + β(x + y) = α(x) + β(x) + α(y) + β(y) + {−β(y) + [α(y), β(x)] + β(y)}, where [α(y), β(x)] = −α(y) − β(x) + α(y) + β(x) is the commutator of the two elements. As the commutator subgroup of H, D(H), is normal, −β(y) + [α(y), β(x)] + β(y) ∈ D(H). Therefore the sum of two homomorphisms has the sum property modulo an element of D(H). This fact inspired us the below deﬁnition: Deﬁnition 2.1. A pseudohomomorphism from G to H is a map ϕ : G → H such that: ϕ(x + y) = ϕ(x) + ϕ(y) + dϕ (x, y), with dϕ (x, y) ∈ D(H), for all x, y ∈ G and ϕ(0) = 0. Denote by P (G, H) the set of pseudohomomorphisms from G to H. Then we have the properties in Propositions 1.4, 1.5, 1.6 [17]: 1) P (G, H) is a group with respect to pointwise addition of mappings. 2) HOM (G, H) is a subgroup of P (G, H). 3) P (G, G) is a right 0-symmetric near-ring including as subring EN D(G) and included as a subring of M AP P0 (G). 4) There exists an epimorphism of near-rings Ψ : P (G, G) → End(G/D(G)), Ψ(θ)(x + D(G)) = θ(x) + D(G),

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for θ ∈ P (G, G), x + D(G) ∈ G/D(G). 5) The sequence of near-rings Ψ

0 −→ K −→ P (G, G) −→ End(G/D(G)) −→ 0, with K = KerΨ, is exact. 6) (Proposition 1.5 [17] )The restriction of Ψ to E(G),which is Ψ : E(G) → End(G/D(G)), is surjective if and only if for each α ∈ P (G, G), there exists β ∈ E(G), such that β − α ∈ M AP P0 (G, D(G)). 7) (Proposition 1.6, [17]) α ∈ M AP P0 (G) is in P (G, G) if and only if, for all β, γ ∈ M AP P0 (G), α ◦ γ − α ◦ (β + γ) + α ◦ β ∈ M AP (G, D(G)). 8) If D(G) = G, then P (G, G) = M AP P0 (G). P0 (G, D(G)) ⊆ P (G, G). 9) M AP As we have already seen in the Property 7, P (G, H) is the set of all 1-cochains from G to H, Z 1 (G, H). The n-cochains from G to H can be deﬁned as in Hochschild cohomology, but they are not enough strong to build a cohomology theory on their basis.

3.

Pseudomodules over a near-ring We recall here some deﬁnitions in [19]. Let R be a near-ring (left, 0-symmetric).

Deﬁnition 3.1. Let (H, +) be a group. H is an R-pseudomodule, if there exists a composition R × H → H, (r, x) −→ rx, such that, for all r, r1 , r2 ∈ R and x, x1 , x2 ∈ H, the following conditions are fulﬁlled: (i) (r1 + r2 ) x ≡ r1 · x + r2 · x (modulo D(H)); (ii) (r1 r2 ) · x ≡ r1 · (r2 · x) (modulo D(H)); (iii) r · (x1 + x2 ) ≡ r · x1 + rx2 (modulo D(H)); (iv) r · 0 = 0 · x = 0. Deﬁnition 3.2. Let H, K be two R-pseudomodules. A mapping α ∈ M AP P0 (H, K) is called an R-pseudohomomorphism, if: (i) α ∈ P (H, K); (ii) α(r · x) ≡ r · α(x) (modulo D(K)), for all r ∈ R, x ∈ H. We denote the set of R-pseudohomomorphisms from H to K by PR (H, K); obviously, PR (H, K) ⊆ P (H, K).

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Moreover, the set P Aut(H) = {α ∈ P (H, H)/α(D(H)) = D(H), α is a bijection} is a group with respect to mapping composition. Deﬁnition 3.3. Let (G, ·) and (H, +) be two groups. H is called a left G-pseudomodule, if there exists a mapping ρ : G → SH (the symmetric group on H), such that ρ(e) = 1H , ρ(x)(a) is denoted by x a, for all a ∈ H, x ∈ G, and the following conditions hold: (i)

xy a

≡x (y a) (modulo D(H));

(ii) x (a + b) ≡x a +x b (modulo D(H)); (iii) x 0 = 0; (iv) x D(H) = D(H), for all x, y ∈ G and a, b ∈ H. In the extension theory for arbitrary groups (non-commutative), the extension E of G by H is an exact sequence of groups: i

p

0 −→ H −→ E −→ G −→ 1, where i(H) = Ker (p) , H ! i(H); G ! E/i(H). Then we may deﬁne some group homomorphisms as in the commutative Diagram 3.4. Diagram 3.4: i

0

−→

H Φ ↓

−→

1

−→ Inn(H)

−→

i

E ↓Φ Aut(H)

p

−→ p

−→

G ↓Ψ

−→

1

Out(H)

−→

1

Ψ is called the coupling map of G with H and (H, G, Ψ) is called the abstract kernel of the extension. In this diagram, we have:

1) Φ(e) := θe ∈ Aut(H), for e ∈ E, where θe (y) := i−1 e · i(y) · e−1 , for y ∈ H; 2) Ψ (x) = θe + Inn (H), where x ∈ G, e ∈ E such that p (e) = x, Out (H) being Aut (H) /Inn (H) . Two equivalent extensions have the same coupling map Ψ. The general extension theory is based upon the following theorem: Theorem 3.5.The abstract kernel (H, G, Ψ) is realized as an extension of H by G if and only if there exist a lifting ϕ : G → Aut(H) such that Ψ(x) = ϕ(x) ◦ Inn(H), for all x ∈ G, and a mapping m : G × G → H, such that: (i) ϕ(x) ◦ ϕ(y) = Inn(m(x, y)) ◦ ϕ(xy); (ii) ϕ(x)(m(y, z)) + m(x, yz) = m(x, y) + m(xy, z), for all x, y, z ∈ G.

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Remark 3.6. m in the Theorem 3.5 is a 2-cochain, m ∈ C 2 (G, H). We have got the following result: Proposition 3.7. (Proposition 2.2, [19]) If (H, G, Ψ) is an abstract kernel and there exists a lifting ϕ : G → Aut(H) for Ψ such that the condition (i), in Theorem 3.5 holds with m ∈ C 2 (G, H), then H is a G-pseudomodule. Indeed, the condition for ϕ implies that (ϕ(x) ◦ ϕ(y)) (a) ≡ ϕ(xy)(a) (modulo D(H)), for all x, y ∈ G and a ∈ H. The condition (ii) in Theorem 3.5 means also: (3.8) δ3ϕ (m)(x, y, z) = Inn(m(x, y)) ([m(xy, z), −m(x, yz)]), for all x, y, z ∈ G. Here δ3ϕ : C 2 (G, H) → C 3 (G, H) is deﬁned by: δ3ϕ (m)(x, y, z) = ϕ(x)(α(y, z)) − m(xy, z) + m(x, yz) − m(x, y), for m ∈ C 2 (G, H), x, y, z ∈ G. The condition (3.8) expresses also the fact that δ3ϕ (m)(x, y, z) ∈ D(H), for x, y, z arbitrary in G, hence m ∈ Zϕ2 (G, H). We quote here another two results in connection with the previous notions in [19]: Proposition 3.8. Let E, E be two extensions of H,respectively H , by G, respectively G , a ∈ Hom (H, H ), c ∈ (G, G ) and α ∈ Zϕ2 (G, H), α ∈ Zϕ2 (G , H ) be the 2-cocycles corresponding to E, respectively E . If b ∈ Hom (E, E ), such that (a, b, c) is a homomorphism of extensions. Then: (i) a is a G-homomorphism from H to H , where H is a G-pseudomodule by taking ϕ ◦ c = ϕ , where ϕ : G → Aut (H ) and ϕ : G → Aut (H ); (ii) a∗ (α), c∗ (α ) ∈ Zϕ2 (G, H ) and they belong to the same class in Hϕ2 (G, H ). (Here, a∗ : C 2 (G, H) → C 2 (G, H ), and c∗ : C 2 (G, H ) → C 2 (G, H ) are such that a∗ (α) ∈ Zϕ2 (G, H ) and c∗ (α ) ∈ Zϕ2 (G, H ). We have shown that such mappings exist.) Proposition 3.9. With the same hypotheses and notations as in the previous propositions, if a, c, α, α satisfy (i) and (ii), then there exists a mapping b : E → E such that: (i) b (e1 + e2 ) ≡ b (e1 ) + b (e2 ) (mod D (H )), for all e1 , e2 ∈ E; (ii) b ◦ i = i ◦ a; (iii) p ◦ b = c ◦ p, therefore b is a pseudohomomorphism from E to E.

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319

Acknowledgment I would like to thank the referee for his/her valuable suggestions.

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