Direct Sum Decompositions of Torsion-Free Finite Rank Groups

Direct Sum Decompositions of Torsion-Free Finite Rank Groups

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With plenty of new material not found in other books, Direct Sum Decompositions of Torsion-Free Finite Rank Groups explores advanced topics in direct sum decompositions of abelian groups and their consequences. The book illustrates a new way of studying these groups while still honoring the rich history of unique direct sum decompositions of groups.Offering a unified approach to theoretic concepts, this reference covers isomorphism, endomorphism, refinement, the Baer splitting property, Gabriel filters, and endomorphism modules. It shows how to effectively study a group G by considering finitely generated projective right End(G)-modules, the left End(G)-module G, and the ring E(G) = End(G)/N(End(G)). For instance, one of the naturally occurring properties considered is when E(G) is a commutative ring. Modern algebraic number theory provides results concerning the isomorphism of locally isomorphic rtffr groups, finitely faithful S-groups that are J-groups, and each rtffr L-group that is a J-group. The book concludes with useful appendices that contain background material and numerous examples.show more

Product details

  • Electronic book text | 344 pages
  • Taylor & Francis Ltd
  • Chapman & Hall/CRC
  • London, United Kingdom
  • 1584887273
  • 9781584887270

Table of contents

PREFACENOTATION AND PRELIMINARY RESULTS Abelian Groups Associative RingsFinite Dimensional Q-AlgebrasLocalization in Commutative Rings Local-Global Remainder Integrally Closed RingsSemi-Perfect RingsExerciseMOTIVATION BY EXAMPLE Some Well Behaved Direct SumsSome Badly Behaved Direct SumsCorner's TheoremArnold-Lady-Murley TheoremLocal IsomorphismExercisesQuestions for Future Research LOCAL ISOMORPHISM IS ISOMORPHISMIntegrally Closed Rings Conductor of an Rtffr RingLocal Correspondence Canonical DecompositionArnold's Theorem ExercisesQuestions for Future ResearchCOMMUTING ENGOMORPHISMSNilpotent Sets Commutative Rtffr RingsE-Properties Square-Free RanksRefinement and Square-Free Rank Hereditary Endomorphism Rings Exercises Questions for Future ResearchREFINEMENT REVISITEDCounting Isomorphism ClassesIntegrally Closed GroupsExercises Questions for Future Research BAER SPLITTING PROPERTY Baer's LemmaSplitting of Exact Sequences G-Compressed ProjectivesSome ExamplesExercises Questions for Future Research J-GROUPS, L- GROUPS, AND S- GROUPS Background on Ext Finite Projective Properties Finitely Projective GroupsFinitely Faithful S-Groups Isomorphism versus Local IsomorphismAnalytic Number Theory Eichler L-Groups Are J-Groups Exercises Questions for Future ResearchGABRIEL FILTERS Filters of Divisibility Idempotent Ideals Gabriel Filters on Rtffr RingsGabriel Filters on QEnd(G)Exercises Questions for Future Research ENDOMORPHISM MODULES Additive Structures of RingsE-PropertiesHomological DimensionsSelf-Injective Rings Exercises Questions for Future ResearchAPPENDIX A: Pathological Direct Sums Nonunique Direct Sums APPENDIX B: ACD Groups Example by Corner APPENDIX C: Power CancellationFailure of Power CancellationAPPENDIX D: Cancellation Failure of Cancellation APPENDIX E: Corner Rings and Modules Topological Preliminaries The Construction of G Endomorphisms of GAPPENDIX F: Corner's Theorem Countable Endomorphism RingsAPPENDIX G: Torsion Torsion-Free Groups E-Torsion GroupsSelf-Small Corner ModulesAPPENDIX H: E-Flat Groups Ubiquity Unfaithful GroupsAPPENDIX I: Zassenhaus and Butler Statement Proof APPENDIX J: Countable E-Rings Countable Torsion-Free E-Rings APPENDIX K: Dedekind E-Rings Number Theoretic PreliminariesIntegrally Closed RingsBIBLIOGRAPHYINDEXshow more