Direct Sum Decompositions of Torsion-Free Finite Rank Groups: 1st Edition (Hardback) book cover

Direct Sum Decompositions of Torsion-Free Finite Rank Groups

1st Edition

By Theodore G. Faticoni

Chapman and Hall/CRC

344 pages

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pub: 2007-03-28
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Description

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.

Reviews

"… provides a lot of interesting results not included in other books."

EMS Newsletter, September 2008

"This is a book with the exciting possibility of stimulating research and also drawing in new researchers to abelian groups."

– Frank Okoh, in Mathematical Reviews, 2008f

Table of Contents

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

About the Series

Chapman & Hall/CRC Pure and Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT022000
MATHEMATICS / Number Theory