Models, Modules and Abelian Groups A. L. S. Corner Models, Modules and Abelian Groups In Memory of A. L. S. Corner Editors Rüdiger Göbel Brendan Goldsmith ≥ Walter de Gruyter · Berlin · New York Editors RüdigerGöbel BrendanGoldsmith Fachbereich6,Mathematik SchoolofMathematicalSciences UniversitätDuisburg-Essen DublinInstituteofTechnology 45117Essen KevinStreet,Dublin8 Germany Ireland E-mail:[email protected] E-mail:[email protected] MathematicsSubjectClassification2000:13-06,13-XX,16-06,16-XX,20-06,20-XX Keywords:Abeliangroups,Baer(cid:2)Speckergroup,Butlergroups,completelydecomposablegroups,crq-groups, endomorphismsofgroups,h-localdomains,locallyfreeabeliangroups,nilpotentgroups,p-groups,endomorphism rings,noetherianrings,non-singularrings,(cid:2)-algebrasmodules,serialmodules (cid:2)(cid:2)Printedonacid-freepaperwhichfallswithintheguidelinesofthe ANSItoensurepermanenceanddurability. LibraryofCongressCataloging-in-PublicationData A CIP catalogue record for this book is available from the LibraryofCongress. ISBN 978-3-11-019437-1 BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableintheInternetathttp://dnb.d-nb.de. ”Copyright2008byWalterdeGruyterGmbH&Co.KG,10785Berlin,Germany. Allrightsreserved,includingthoseoftranslationintoforeignlanguages.Nopartofthisbookmaybereprodu- cedortransmittedinanyformorbyanymeans,electronicormechanical,includingphotocopy,recordingor anyinformationstorageandretrievalsystem,withoutpermissioninwritingfromthepublisher. PrintedinGermany. Coverdesign:ThomasBonnie,Hamburg. Typesetusingtheauthors’LaTeXfiles:JonathanRohleder,Berlin. Printingandbinding:Hubert&Co.GmbH&Co.KG,Göttingen. Introduction OnhearingofthesuddendeathofA.L.S.CorneronSeptember3rd2006,manyofhis friends and colleagues felt it appropriate that Tony’s contributions to algebra should be acknowledged in a volume which would reflect his wide mathematical influence. Accordingly, this volume was conceived and its title “Models, Modules and Abelian Groups” reflects Tony’s range of research interests. For many Tony’s name is insep- arable from “Abelian Groups” but his contributions were wider: his techniques from setandmodeltheory,hisinterestsinrepresentationtheoryandnon-commutativegroup theoryandhisenormousinfluenceonthestudyofendomorphismringsareallpartof hislegacyandthisiswellreflectedinthecontributionsherein. Someshorttimepriortohisdeath,theeditorspersuadedTonythathisunpublished workontheclassificationofthefinitegroupswhichcanoccurasautomorphismgroups oftorsion-freeAbeliangroups,shouldbewrittenupforpublication. Tonyagreedand hislecturenotesonthetopic,deliveredattheUniversityofPaduain1988,werebeing processed when he died. He did not see the final version which is produced in this Volume, but we believe it is a faithful record of Tony’s work on the subject. We are thankfultoFedericoMenegazzoforhishelpinthisandparticularlyforhisintriguing AppendixtoTony’spaper. The editors would like to record their thanks to the authors who responded so en- thusiastically to their requests for contributions and to the referees who have ensured suchhighmathematicalstandardsintheacceptedpapers. Itisalsoappropriatetothank RobertPlatoandSimonAlbroscheitofWalterdeGruyterfortheirsupportandhelpin thisundertaking. June2008 RüdigerGöbelandBrendanGoldsmith, Editors Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v B. GOLDSMITH AnthonyLeonardSouthernCorner1934–2006 . . . . . . . . . . . . . . . . . . . 1 A.L.S. CORNER GroupsofunitsofordersinQ-algebras . . . . . . . . . . . . . . . . . . . . . . . 9 A. BLASS, S. SHELAH Basicsubgroupsandfreeness,acounterexample . . . . . . . . . . . . . . . . . . 63 R. GÖBEL, J. MATZ AnextensionofButler’stheoremonendomorphismrings . . . . . . . . . . . . . 75 D.M. ARNOLD Locallyfreeabeliangroupsoffiniterank . . . . . . . . . . . . . . . . . . . . . . 83 A. MADER, L.G. NONGXA, M.A. OULD-BEDDI Decompositionsofglobalcrq-groups . . . . . . . . . . . . . . . . . . . . . . . . 99 G. CA˘LUGA˘REANU, P. SCHULTZ Endomorphismsandautomorphismsofsquaresofabeliangroups . . . . . . . . 121 E. BLAGOVESHCHENSKAYA ClassificationofaclassoffiniterankButlergroups . . . . . . . . . . . . . . . . 135 A.A. FOMIN Quotientdivisibleandalmostcompletelydecomposablegroups . . . . . . . . . 147 D. HERDEN, L. STRÜNGMANN Puresubgroupsofcompletelydecomposablegroups–analgorithmicapproach . 169 O. KOLMAN StrongsubgroupchainsandtheBaer–Speckergroup . . . . . . . . . . . . . . . 187 C. DE VIVO, C. METELLI OndirectdecompositionsofButlerB(2)-groups . . . . . . . . . . . . . . . . . 199 A. MADER, O. MUTZBAUER Diagonalequivalenceofmatrices . . . . . . . . . . . . . . . . . . . . . . . . . 219 E.F. CORNELIUS JR., P. SCHULTZ Rootbasesofpolynomialsoverintegraldomains . . . . . . . . . . . . . . . . . 235 viii Contents G. CA˘LUGA˘REANU, K.M. RANGASWAMY AsolutiontoaproblemonlatticeisomorphicAbeliangroups . . . . . . . . . . 249 P. HILL, W. ULLERY AnoteonAxiom3anditsdualforabeliangroups . . . . . . . . . . . . . . . . 257 W. MAY Unitsofmodularp-mixedabeliangroupalgebras . . . . . . . . . . . . . . . . . 267 L. FUCHS, T. OKUYAMA Purecoversinabelianp-groups . . . . . . . . . . . . . . . . . . . . . . . . . . 277 P.W. KEEF PartiallydecomposableprimaryAbeliangroupsandthegeneralized coreclassproperty. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .289 K.R. MCLEAN Theadditivegroupofafinitelocalringinwhicheachidealcanben-generated . 301 M. FLAGG AJacobsonradicalisomorphismtheoremfortorsion-freemodules . . . . . . . . 309 A.L.S. CORNER, B. GOLDSMITH, S.L. WALLUTIS Anti-isomorphismsandthefailureofduality . . . . . . . . . . . . . . . . . . . 315 V.V. BLUDOV, M. GIRAUDET, A.M.W. GLASS, G. SABBAGH Automorphismgroupsofmodelsoffirstordertheories . . . . . . . . . . . . . . 325 S. THOMAS TheclassificationproblemforfiniterankButlergroups . . . . . . . . . . . . . . 329 T.G. FATICONI L-groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .339 J. BUCKNER, M. DUGAS Co-localsubgroupsofnilpotentgroupsofclass2 . . . . . . . . . . . . . . . . . 351 L. SALCE Divisibleenvelopesandcotorsionpairsoverintegraldomains . . . . . . . . . . 359 C.M. RINGEL ThefirstBrauer–Thrallconjecture. . . . . . . . . . . . . . . . . . . . . . . . .369 C. GREITHER, D. HERBERA, J. TRLIFAJ AversionoftheBaersplittingproblemfornoetherianrings . . . . . . . . . . . 375 B. OLBERDING Characterizationsandconstructionsofh-localdomains . . . . . . . . . . . . . . 385 C.U. JENSEN VariationsonWhitehead’sproblemandthestructureofExt . . . . . . . . . . . 407 Contents ix L. BICAN Thelatticeoftorsionfreeprecoverclasses . . . . . . . . . . . . . . . . . . . . . 415 U. ALBRECHT Non-singularringsofinjectivedimension1 . . . . . . . . . . . . . . . . . . . . 421 F. MANTESE, A. TONOLO Onclassesdefiningahomologicaldimension . . . . . . . . . . . . . . . . . . . 431 A. DUGAS, B. HUISGEN-ZIMMERMANN, J. LEARNED Truncatedpathalgebrasarehomologicallytransparent . . . . . . . . . . . . . . 445 A. FACCHINI, P. PRˇÍHODA RepresentationsofthecategoryofserialmodulesoffiniteGoldiedimension . . . 463 L. SALCE, P. ZANARDO Commutativitymodulosmallendomorphismsandendomorphismsof zeroalgebraicentropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487