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Algebra : An Approach via Module Theory - William A. Adkins

Algebra

An Approach via Module Theory

Hardcover

Published: 23rd April 1999
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This book is designed as a text for a first-year graduate algebra course. The choice of topics is guided by the underlying theme of modules as a basic unifying concept in mathematics. Beginning with standard topics in groups and ring theory, the authors then develop basic module theory, culminating in the fundamental structure theorem for finitely generated modules over a principal ideal domain. They then treat canonical form theory in linear algebra as an application of this fundamental theorem. Module theory is also used in investigating bilinear, sesquilinear, and quadratic forms. The authors develop some multilinear algebra (Hom and tensor product) and the theory of semisimple rings and modules and apply these results in the final chapter to study group represetations by viewing a representation of a group G over a field F as an F(G)-module. The book emphasizes proofs with a maximum of insight and a minimum of computation in order to promote understanding. However, extensive material on computation (for example, computation of canonical forms) is provided.

Preface
Groupsp. 1
Definitions and Examplesp. 1
Subgroups and Cosetsp. 6
Normal Subgroups, Isomorphism Theorems, and Automorphism Groupsp. 15
Permutation Representations and the Sylow Theoremsp. 22
The Symmetric Group and Symmetry Groupsp. 28
Direct and Semidirect Productsp. 34
Groups of Low Orderp. 39
Exercisesp. 45
Ringsp. 49
Definitions and Examplesp. 49
Ideals, Quotient Rings, and Isomorphism Theoremsp. 58
Quotient Fields and Localizationp. 68
Polynomial Ringsp. 72
Principal Ideal Domains and Euclidean Domainsp. 79
Unique Factorization Domainsp. 92
Exercisesp. 98
Modules and Vector Spacesp. 107
Definitions and Examplesp. 107
Submodules and Quotient Modulesp. 112
Direct Sums, Exact Sequences, and Homp. 118
Free Modulesp. 128
Projective Modulesp. 136
Free Modules over a PIDp. 142
Finitely Generated Modules over PIDsp. 156
Complemented Submodulesp. 171
Exercisesp. 174
Linear Algebrap. 182
Matrix Algebrap. 182
Determinants and Linear Equationsp. 194
Matrix Representation of Homomorphismsp. 214
Canonical Form Theoryp. 231
Computational Examplesp. 257
Inner Product Spaces and Normal Linear Transformationsp. 269
Exercisesp. 278
Matrices over PIDsp. 289
Equivalence and Similarityp. 289
Hermite Normal Formp. 296
Smith Normal Formp. 307
Computational Examplesp. 319
A Rank Criterion for Similarityp. 328
Exercisesp. 337
Bilinear and Quadratic Formsp. 341
Dualityp. 341
Bilinear and Sesquilinear Formsp. 350
Quadratic Formsp. 376
Exercisesp. 391
Topics in Module Theoryp. 395
Simple and Semisimple Rings and Modulesp. 395
Multilinear Algebrap. 412
Exercisesp. 434
Group Representationsp. 438
Examples and General Resultsp. 438
Representations of Abelian Groupsp. 451
Decomposition of the Regular Representationp. 453
Charactersp. 462
Induced Representationsp. 479
Permutation Representationsp. 496
Concluding Remarksp. 503
Exercisesp. 505
Appendixp. 507
Bibliographyp. 510
Index of Notationp. 511
Index of Terminologyp. 517
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387978390
ISBN-10: 0387978399
Series: Graduate Texts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 526
Published: 23rd April 1999
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 3.18
Weight (kg): 2.06
Edition Number: 2
Edition Type: Revised