Faticoni Direct Sum Decompositions of Torsion-Free Finite Rank Groups

1. Auflage 2010
ISBN: 978-1-58488-727-0
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 344 Seiten

Reihe: Chapman & Hall/CRC 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.

Zielgruppe

Pure and applied mathematicians working with abelian groups, rings, and modules.

Autoren/Hrsg.

Faticoni, Theodore G.

Fachgebiete

• Mathematik | Informatik Mathematik Algebra Algebraische Strukturen, Gruppentheorie

Inhaltsverzeichnis

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