Models, Modules and Abelian Groups: In Memory of A. L. S. Corner

509 Pages·2008·3.82 MB·English

by  Rüdiger Göbel

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:[e-mail] E-mail:[e-mail]
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.
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A CIP catalogue record for this book is available from the
LibraryofCongress.
ISBN 978-3-11-019437-1
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TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableintheInternetathttp://dnb.d-nb.de.
”Copyright2008byWalterdeGruyterGmbH&Co.KG,10785Berlin,Germany.
Allrightsreserved,includingthoseoftranslationintoforeignlanguages.Nopartofthisbookmaybereprodu-
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Coverdesign:ThomasBonnie,Hamburg.
Typesetusingtheauthors’LaTeXfiles:JonathanRohleder,Berlin.
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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