Direct Sum Decompositions of Torsion-Free Finite Rank Groups

Direct Sum Decompositions of Torsion-Free Finite Rank Groups

  • Electronic book text
By (author) 

List price: US$132.95

Currently unavailable

We can notify you when this item is back in stock

Add to wishlist

AbeBooks may have this title (opens in new window).

Try AbeBooks


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 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