Near-Rings and Near-Fields: Proceedings of the Conference on Near-Rings and Near-Fields, Stellenbosch, South Africa, July 9–16, 1997 PDF
Preview Near-Rings and Near-Fields: Proceedings of the Conference on Near-Rings and Near-Fields, Stellenbosch, South Africa, July 9–16, 1997
Near-Rings and Near-Fields Near-Rings and Near-Fields Proceedings of the Conference on Near-Rings and Near-Fields, Stellenbosch, South Africa, July 9-16, 1997 edited by Yuen Fong Department of Mathematics, National Cheng Kung University, Tainan, Taiwan, Republic of China CarI Maxson Department of Mathematics, Texas A & M University, College Station, Texas, U.S.A. John Meldrum Department of Mathematics, University of Edinburgh, Edinburgh, Scotland Giinter Pilz Institute for Mathematics, Johannes Kepler Universităt Linz, Linz, Austria Andries van der Walt and Leon van Wyk Department of Mathematics, University of Stellenbosch, Stellenbosch, South Africa Springer-Science+Business Media, B.V. A c.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-94-010-3802-7 ISBN 978-94-010-0954-6 (eBook) DOI 10.1007/978-94-010-0954-6 Printed an acid-free paper AII Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover l st edition 200 l No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical. including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. CONTENTS Foreword VII Invited talks GerhardBetsch Combinatorialaspects ofnearring theory TothememoryofJAMES RAYCLAY GaryE Birkenmeier Leftself-distributive ringsandnearrings 10 1.J. H.Meyer On thedevelopmentofmatrix nearrings and relatednearringsover the 23 pastdecade Contributedpapers Erhard Aichinger,JiirgenEckerand ChristofNobauer Theuseofcomputersinnear-ringtheory 35 Nurcan Argacand HowardE.Bell Some results onderivationsinnearrings 42 A.Benini,E Moriniand S. Pellegrini Weakly divisible nearrings: genesis. construction and their links with 47 designs Y.Fong, E-K. Huang,W.-E Keand Y.-N. Yeh Onsemi-endomorphismsofabeliangroups 72 R.L.Fray Anoteonpseudo-distributivityingroupnear-rings 79 N.J.Groenewald Thealmostnilpotent radicalfor near rings 84 JaimeGutierrezand CarlosRuiz de Velasco Polynomial near-rings inseveral variables 94 J.E T.Hartneyand S.Mavhungu s-primitiveidealsinmatrix near-rings 103 vi AhmedA.M. Kamal EssentialidealsandR-subgroupsinnear-rings 108 Pu-an Li ConditionsthatM.::I(G) isaring I18 HeatherMcGilvrayand C.J.Maxson Ondependenceandindependenceinnear-rings 122 DorotaNiewieczerzal Onmodulesofhomogeneousmappings 130 ChristofNobauer Thenumberofisomorphismclassesofd.g.near-ringsonthegeneralized 133 quaternion groups Alan Oswald, KirbyC.Smith and Leon vanWyk Whenisacentralizer near-ring isomorphictoamatrixnear-ring?Part2 138 S.D.Scott TopologyandprimaryN-groups 151 Stefan Veldsman Ontheradicalsofcompositionnear-rings 198 FOREWORD This volume contains three invited lectures and sixteen other papers which were pre sented at the 14thInternational Conference on Nearrings and Nearfields held inStellen bosch, South Africa,July 9-161997. It wasalso thefirstnearring conference tobeheldafter theuntimely death ofJames R Clay,whoovertheyears hadbeenaninspirationtomanyalgebraists interested innearring theory.Theoccasion wasmarkedbytheinvitedtalkofGerhardBetsch,whichwasdevoted toanoverview ofClay'scontributionstonearring andnearfieldtheory. Thisbook isaffectionatelydedicated tothememory ofJamesRClay. Allthepapers presented herehavebeen refereed under thesupervisionoftheEditorial Board:Fong Yuen,CarlMaxson, John Meldrum,GUnterPilz,Leon vanWykandAndries vander Walt.Thanks areduetotherefereesandtotheEditorial Board. A special word of thanks isdue toWen-fong Kefor preparing the finalversion of the TEX files, and to Fong Yuenfor his pains inarranging for the publication of the volume withKluwer Academic Publishers. Andries vanderWalt Stellenbosch, August 1999 COMBINATORIAL ASPECTS OF NEARRING THEORY TO THE MEMORY OF JAMES RAYCLAY GERHARDBETSCH AbriefcurriculumvitaeofJames Ray (Jim) Clay BornNovember5,1938atBurley (Idaho). DiedJanuary 16, 1996atTucson (Arizona). Married since 1959toCarolCline BURGE, "a trulybeautiful daughter ofZion" (Dedication ofJim's 1992book). Three daughters, tengrand-children. Trainingandprofessionalcareer: 1956 USNavalAcademy (class 1960);Jim studiedengineering. 1959(February) University ofUtah 1960 B.Sc.University ofUtah 1962 M.Sc.inMathematicsUniversityofWashington inSeattle 1966 Ph.D.inMathematics UniversityofWashingtoninSeattle (Supervisor:RossA.Beaumont) 1965/66 MathematicianfortheCIA, Part-time Instructor USDept. ofAgriculture Part-time Assistant ProfessorGeorge Washington University 1966 Assistant Professor UniversityofArizona inTucson 1969 Associate ProfessorUniversityofArizona 1969-1972AssociateHeadofDepartment 1974 FullProfessorUniversityofArizona Visiting Professorat: Tubingen,London (King's College),Munchen/Munich (TU), Ed inburgh, Stellenbosch, Hamburg (Univ. der Bundeswehr), Tainan (Nat. Cheng Kung University), Linz (Joh.KeplerUniversitat). In addition to visiting professorships: 40 International lectures atUniversities in Bul garia, Hungary,Italy,Ireland, India, Hong Kong,Thailand, Singapore, andChina. Y.Fangetal.(eds.),Near-RingsandNear-Fields.1-9. @2001 KluwerAcademicPublishers. 2 GERHARDBETSCH Award: 197m3 Humboldt Foundation'sDistinguishedSeniorU.S.ScientistAward (Jim wasonly 34atthetime!) • Publications: ThreeBooks, overfiftyarticles injournals. Cf. also the Obituary by C. R. MAXSON in Results in Math. 30 (1996) and a fine picture ofJim Clay inthesame volume ofthisjournal. 1. INTRODUCTION On 3rd August 1995, at the Hamburg Nearring Conference, Jim Clay gave a survey lectureon"RecentDevelopments,Discoveries, andDirectionsforPlanarNearrings"[sic!). Nobody could foresee, thatthisexcellentlecture wasthelastpublicpresentationwhichour friend Jim would deliver to the nearringcommunity. On 16January 1996,Jim died of a sudden death attheageof57, while riding hisbicycle homefrom work.- Jim attended all international nearring meetings so far. Andries van der Walt and I decidedthataproperwaytohonour ourdeceasedfriend andcolleaguewould be togiveasurveyoncombinatorialaspects ofnearringtheory. Jim himselfsubstantiallyand decisively contributed tothose branches of nearring the ory,which havecombinatorialaspects. Hedid likethislineofresearch verymuch. According toPeter DEMBOWSKI (1928-1971),CombinatoricsistheTheory (orenu meration)ofsubsets offinitesets. Iclaim: I) Speakingofcombinatorialaspectsofnearring theoryalmostalways involvesfinite planarnearrings,possibly generalizationsofthesestructures; 2) the central combinatorial concept wehave todeal with inthiscontext isthe con cept of balanced incomplete blockdesigns (BIBDs), possibly partially balanced incompleteblock designs (PBIBDs), orother generalizations. Let megiveyouthedefinition. Definition. Abalancedincompleteblockdesign (BIBD)isapair(P,'B)withthefollowing properties: Pisaset and '11 ~2P, v:= \PI>0,b:= 1'111>O.The elementsofParecalled points, and theelementsof '11arecalled blocks. Weassume thatthefollowing axioms are satisfied: • For anyBE'11,IBI = k. • Every pEPbelongstoexactly rblocks. • Every twodistinctblocks haveA>0points incommon. (The integers randAaresupposedtobeconstants:risindependentofp, Aisindependent ofthechosen blocks.) Examples are abundant: Affineplanes; projective planes; Moebius planes;"good"ex perimentaldesigns, with good symmetries. InordertoexplainhowJimClaycametostudyplanarity,letmegiveyousomehistorical information. COMBINATORIALASPECTSOFNEARRINGTHEORY 2. PREHISTORY OF THE SUBJECT 1905 Leonard Eugene DICKSON constructed and investigated finite near fields (see his paper in TAMS 6). Inparticular, he exhibited a proper near-field with9elements,which isinfactthesmallestpropernear-field. 190511907 O. VEBLEN and 1. H. MACLAGAN-WEDDERBURN, in their pa per on "Non-desarguesian and non-pascalian geometries" (TAMS 8 (1907)), applied DICKSON's finite near-fields to construct finite pro jectiveplanes,which are non-desarguesianand non-pascalian(todaywe would say"Pappian" instead of"pascalian"). It follows from atheorem by G. HESSENBERG of 1905(!), that a non-desarguesian plane has to be automatically non-pascalian (Beweisdes Desarguesschen Satzesaus dem Pascalschen. Math. Ann. 61 (1905)). Using DICKSON's proper near-field oforder9,VEBLEN andWEDDERBURNconstructedinpar ticulartwo non-desarguesian and non-pascalian projectiveplanesofor der9. 1931 CARMICHAELdiscovered that the finitesharply 2-transitive permuta tiongroupsareprecisely thegroupsofaffinetransformations xf---+ax+b (a=1:0) ofasuitable finitenear-field into itself(Amer.J.Math. 53(1931), 631 644). Hence,from agroup-theoretical point ofview,itwas importantto determineall finitenear-fields. 1934 Inhis dissertation, Hans ZASSENHAUS determinedall (finite) sharply 3-transitive permutation groups. Of course, the characterization in volved near-fields. (Thedissertation was publishedin Abh.Math.Sem. Univ.Hamburg11(1936), 17-40). 1936 In a famous paper, ZASSENHAUS determined all finite near-fields (Abh.Math.Sem.Univ.Hamburg 11(1936),132-145). He proved: Up to sevenexceptional near-fields(which he described precisely), any fi nitenear-field may bederived fromafinitefieldbyDICKSON'smethod of1905. 1943 Marshall HALLJr.establishedthecoordinatization of projectiveplanes by a ternary ring. This involved the introduction of Planarity: It was requiredthat inthe ternary ringofcoordinatestheequation ax= bx+ chasaunique solution,ifa=1:b. The geometrical meaning isclear: Weaim at the unique point ofinter section oftwodistinct lines. Allfinitenear-fields areautomaticallyplanar.Thisisdue to thefact that any injective map of a finite set into itself is bijective. But what about theinfinite case? 4 GERHARDBETSCH 1959 Inhisbook'TheTheoryofGroups," MarshalIHalIJr.discussesingreat detailthecorrespondencebetween"DoublyTransitiveGroups andNear Fields" (Section 20.7). In constructing a near-field from a strictly 2 transitive group G the author needs an additional hypothesis ("which maynotbenecessary butisrequiredforourproof',page382). This isa transitivitycondition (3) orthecondition (3'),thatGisfinite. 1964 J. L. ZEMMERconstructed infinitenon-planar near-fields (Near-fields. planarandnon-planar.TheMath.Student31(1964), 145-150). At this state of theart,Michael ANSHEL andJim CLAYstarted their investigation of planarity. "In 1967,Iwaswantingtofindsomegeometric applications ofnearrings. Nearfields for which each equation ax= bx+c,a =j:.b,had unique solu tions wereexactly theones whichwereusedsuccessfulIy ingeometry,so Ifocused on this equation for nearrings. Itwas not surprising to various experts at the time [HUGHES, ZEMMERj that nearrings in which each equation ax= bx+c. a =j:. b, had a unique solution would also be near fields. Soadifferentpointofviewwasneeded" (JimCLAY,Theequation ax=bx+c, Preface; see also his contribution to the Oberwolfach 1968 meeting). Nowletmestartasystematic survey. 3. SYSTEMATIC SURVEY Let (N,+,.)bealeftnearring. DefinitionsandRemarks. Ifa,b EN, then a =m b is defined by ax=bx for allx EN. =m Obviously, isanequivalence relation onN. N iscalledplanarif (i) IN/=ml ~ 3,and =m, (ii) ifaisnotequivalent tobunder thenax= bx-s-chasaunique solution. N iscalIedintegral planariff{a Ia=mO} ={O}. There are some MainExamples,duetoMichaelANSHEL, whichJimCLAYreferred toquite frequently: "I keep getting inspiration from these examples" (first sentence of Jim's lectureatFredericton). Take(C,+,.), thefieldofcomplex numbers. Nowdefine ax lal' (i) b:= b; (ii) aeb:»:8(a)-b,where8(a):= lal-I -a, ifa=j:.0,and8(0):=O. Then (C,+,x) and (C,+,.) areintegralplanar nearrings. =m What istheequivalence class ofawithrespect to inthese nearrings ? (i) [zEC Ilzl = lal},thecircle around0withradius lal; (ii) {O} and{zEC Iz= A.a, A.ElitA.>O},the rayfrom0inthedirection ofa.
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