Table Of ContentManfred Droste · László Fuchs
Brendan Goldsmith · Lutz Strüngmann
Editors
Groups, Modules,
and Model Theory -
Surveys and Recent
Developments
In Memory of Rüdiger Göbel
Groups, Modules, and Model Theory - Surveys
and Recent Developments
Manfred Droste • László Fuchs (cid:129) Brendan Goldsmith
Lutz Strüngmann
Editors
Groups, Modules, and Model
Theory - Surveys and Recent
Developments
In Memory of Rüdiger Göbel
123
Editors
ManfredDroste LászlóFuchs
InstitutfürInformatik DepartmentofMathematics
UniversitätLeipzig TulaneUniversity
Leipzig,Germany NewOrleans,LA,USA
BrendanGoldsmith LutzStrüngmann
DublinInstituteofTechnology FacultyforComputerSciences
Dublin,Ireland MannheimUniversityofAppliedSciences
Mannheim,Germany
ISBN978-3-319-51717-9 ISBN978-3-319-51718-6 (eBook)
DOI10.1007/978-3-319-51718-6
LibraryofCongressControlNumber:2017942727
©SpringerInternationalPublishingAG2017
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AyouthfulRüdiger
TothememoryofRüdigerGöbel–abelovedfriendanddistinguishedcolleague
whoinspiredgenerationsofyoungmathematicians
Rüdigerinmorerecentyears
Preface
When Rüdiger Göbel died in July 2014, his many friends and colleagues from
around the world felt it appropriate to acknowledge his contributions to a wide-
ranging area of group theory and model theory by organising a conference in his
honour.OutofthisdevelopedtheconferenceNewPathwaysBetweenGroupTheory
and Model Theory which took place in Mülheim an der Ruhr, Germany, during
February1–4,2016.Theenthusiasticresponsetothisconferenceleddirectlytothis
volume;wearemostthankfultoalltheparticipantsofthatconferencewhohelpedto
makeitanunforgettableeventthatRüdigerwouldcertainlyhaveenjoyed.However,
thematerialappearinginthisbookisnotoftheusualconferenceproceedingstype:
theeditorshavetriedtopresentabalancedmixofsurveypapers,whichwillenable
expertandnon-expertaliketogetagoodoverviewofdevelopmentsacrossarange
ofareasofgroup,moduleandmodeltheory,alongwithresearchpaperspresenting
some of the most recent developments in these same areas. Every effort has been
madetomaketheseresearchpaperseasilyaccessibleintheirintroductorysections.
Wewouldhopethatthematerialisofinteresttobothbeginninggraduatestudents
and experienced researchers alike. The topics covered are, inevitably, just a cross
section of the vast areas of group, module and model theory, but they reflect in
a strong way the areas in which Rüdiger Göbel contributed so much. The book
is divided into two sections, surveys and recent research developments, with each
sectioncontainingmaterialfromalltheareasofthetitle.
Finally, we would like to express our sincere thanks to the colleagues who
contributed papers so enthusiastically, to the many experts who acted as referees
for all the papers, to the professional staff at Springer and in particular to Dimana
Tzvetkova, for their help in producing a volume which we hope is an appropriate
commemoration of our late friend Rüdiger Göbel. A special word of thanks goes
vii
viii Preface
toKatrinLeistnerforherinvaluablehelpduringtheorganisationoftheconference
and also during the preparation of this memorial volume. We would also like to
acknowledgethehelpgivenbyGaborBraunandDanielHerdenatvariousstagesof
theorganisationofboththeconferenceandthisvolume.
Leipzig,Germany ManfredDroste
NewOrleans,LA,USA LászlóFuchs
Dublin,Ireland BrendanGoldsmith
Mannheim,Germany LutzStrüngmann
November2016
Contents
PartI SurveyArticles
PropertiesofAbelianGroupsDeterminedbyTheir
EndomorphismRing............................................................. 3
UlrichAlbrecht
TheZero-DivisorGraphofaCommutativeSemigroup:ASurvey......... 23
DavidF.AndersonandAymanBadawi
ARemak-Krull-SchmidtClassofTorsion-FreeAbelianGroups........... 41
DavidM.Arnold,AdolfMader,OttoMutzbauer,andEbruSolak
Rigid@ -FreeAbelianGroupswithPrescribedFactorsandTheir
1
RoleintheTheoryofCellularCovers......................................... 69
GáborBraunandLutzStrüngmann
DefinableValuationsInducedbyDefinableSubgroups...................... 83
KatharinaDupont
GroupsofAutomorphismsofTotallyOrderedSets:Techniques,
ModelTheoryandApplicationstoDecisionProblems ...................... 109
A.M.W.Glass
AlgebraicEntropiesforAbelianGroupswithApplicationstothe
StructureofTheirEndomorphismRings:ASurvey......................... 135
BrendanGoldsmithandLuigiSalce
OnSubsetsandSubgroupsDefinedbyCommutatorsandSome
RelatedQuestions................................................................ 175
Luise-CharlotteKappe,PatriziaLongobardi,andMercedeMaj
RecentProgressinModuleApproximations.................................. 191
JanTrlifaj
ix
x Contents
PartII ResearchArticles
AClassofPureSubgroupsoftheSpeckerGroup............................ 213
A.L.S.Corner
Countable1-TransitiveTrees ................................................... 225
KatieM.ChicotandJohnK.Truss
OnOre’sTheoremandUniversalWordsforPermutations
andInjectionsofInfiniteSets................................................... 269
ManfredDroste
AnExtensionofM.C.R.Butler’sTheoremonEndomorphismRings.... 277
ManfredDugas,DanielHerden,andSaharonShelah
TheJacobsonRadical’sRoleinIsomorphismTheoremsforp-Adic
ModulesExtendstoTopologicalIsomorphism ............................... 285
MaryFlagg
A Note on Hieronymi’s Theorem: Every Definably Complete
StructureIsDefinablyBaire.................................................... 301
AntongiulioFornasiero
Cotorsion and Tor Pairs and Finitistic Dimensions over
CommutativeRings.............................................................. 317
LászlóFuchs
PermutationGroupsWithoutIrreducibleElements......................... 331
A.M.W.GlassandH.DugaldMacpherson
R-HopfianandL-co-HopfianAbelianGroups(withanAppendix
byA.L.S.CorneronNearAutomorphismsofanAbelianGroup) ......... 333
BrendanGoldsmithandKetaoGong
OntheAbelianizationofCertainTopologist’sProducts..................... 351
WolfgangHerfortandWolframHojka
SomeRemarksondp-MinimalGroups........................................ 359
ItayKaplan,EladLevi,andPierreSimon
SquareSubgroupsofDecomposableRankThreeGroups .................. 373
FatemehKarimi
AnInvariantonPrimaryAbelianGroupswithApplications
toTheirProjectiveDimensions................................................. 385
PatrickW.Keef
TheValuationDifferenceRankofaQuasi-OrderedDifferenceField ..... 399
SalmaKuhlmann,MickaëlMatusinski,andFrançoisePoint
TheLatticeofU-SequencesofanAbelianp-Group.......................... 415
K.RobinMcLean
Contents xi
StronglyNon-SingularRingsandMoritaEquivalence...................... 429
BradleyMcQuaig
TheClassof(2,3)-GroupswithHomocyclicRegulatorQuotient
ofExponentp2.................................................................... 435
EbruSolak
UnboundedMonotoneSubgroupsoftheBaer–SpeckerGroup ............ 449
BurkhardWald
ClusterizationofCorrelationFunctions....................................... 459
AlexanderZuevsky
Index............................................................................... 465
Description:This volume focuses on group theory and model theory with a particular emphasis on the interplay of the two areas. The survey papers provide an overview of the developments across group, module, and model theory while the research papers present the most recent study in those same areas. With introd
