Direct Sum Decompositions of Torsion-Free Finite Rank Groups (Pure and Applied Mathematics)

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 Content: PREFACE NOTATION AND PRELIMINARY RESULTS Abelian Groups Associative Rings Finite Dimensional Q-Algebras Localization in Commutative Rings Local-Global Remainder Integrally Closed Rings Semi-Perfect Rings Exercise MOTIVATION BY EXAMPLE Some Well Behaved Direct Sums Some Badly Behaved Direct Sums Corner's Theorem Arnold-Lady-Murley Theorem Local Isomorphism Exercises Questions for Future Research LOCAL ISOMORPHISM IS ISOMORPHISM Integrally Closed Rings Conductor of an Rtffr Ring Local Correspondence Canonical Decomposition Arnold's Theorem Exercises Questions for Future Research COMMUTING ENGOMORPHISMS Nilpotent Sets Commutative Rtffr Rings E-Properties Square-Free Ranks Refinement and Square-Free Rank Hereditary Endomorphism Rings Exercises Questions for Future Research REFINEMENT REVISITED Counting Isomorphism Classes Integrally Closed Groups Exercises Questions for Future Research BAER SPLITTING PROPERTY Baer's Lemma Splitting of Exact Sequences G-Compressed Projectives Some Examples Exercises Questions for Future Research J-GROUPS, L- GROUPS, AND S- GROUPS Background on Ext Finite Projective Properties Finitely Projective Groups Finitely Faithful S-Groups Isomorphism versus Local Isomorphism Analytic Number Theory Eichler L-Groups Are J-Groups Exercises Questions for Future Research GABRIEL FILTERS Filters of Divisibility Idempotent Ideals Gabriel Filters on Rtffr Rings Gabriel Filters on QEnd(G) Exercises Questions for Future Research ENDOMORPHISM MODULES Additive Structures of Rings E-Properties Homological Dimensions Self-Injective Rings Exercises Questions for Future Research APPENDIX A: Pathological Direct Sums Nonunique Direct Sums APPENDIX B: ACD Groups Example by Corner APPENDIX C: Power Cancellation Failure of Power Cancellation APPENDIX D: Cancellation Failure of Cancellation APPENDIX E: Corner Rings and Modules Topological Preliminaries The Construction of G Endomorphisms of G APPENDIX F: Corner's Theorem Countable Endomorphism Rings APPENDIX G: Torsion Torsion-Free Groups E-Torsion Groups Self-Small Corner Modules APPENDIX H: E-Flat Groups Ubiquity Unfaithful Groups APPENDIX I: Zassenhaus and Butler Statement Proof APPENDIX J: Countable E-Rings Countable Torsion-Free E-Rings APPENDIX K: Dedekind E-Rings Number Theoretic Preliminaries Integrally Closed Rings BIBLIOGRAPHY INDEX

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.