Algebra

An Approach via Module Theory

By: William Adkins, Steven Weintraub

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Hardcover | 1 May 1999 | Edition Number 2

At a Glance

Hardcover
544 Pages

Edition Type
Revised

Dimensions(cm)
23.39 x 15.6 x 3.02

Hardcover

$149.01

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This book is designed as a text for a first-year graduate algebra course. As necessary background we would consider a good undergraduate linear algebra course. An undergraduate abstract algebra course, while helpful, is not necessary (and so an adventurous undergraduate might learn some algebra from this book). Perhaps the principal distinguishing feature of this book is its point of view. Many textbooks tend to be encyclopedic. We have tried to write one that is thematic, with a consistent point of view. The theme, as indicated by our title, is that of modules (though our intention has not been to write a textbook purely on module theory). We begin with some group and ring theory, to set the stage, and then, in the heart of the book, develop module theory. Having developed it, we present some of its applications: canonical forms for linear transformations, bilinear forms, and group representations. Why modules? The answer is that they are a basic unifying concept in mathematics. The reader is probably already familiar with the basic role that vector spaces play in mathematics, and modules are a generaliza tion of vector spaces. (To be precise, modules are to rings as vector spaces are to fields.

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Non-Fiction Mathematics Algebra

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Published: 30th September 2012

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Non-Fiction Mathematics Algebra

Preface
Groups	p. 1
Definitions and Examples	p. 1
Subgroups and Cosets	p. 6
Normal Subgroups, Isomorphism Theorems, and Automorphism Groups	p. 15
Permutation Representations and the Sylow Theorems	p. 22
The Symmetric Group and Symmetry Groups	p. 28
Direct and Semidirect Products	p. 34
Groups of Low Order	p. 39
Exercises	p. 45
Rings	p. 49
Definitions and Examples	p. 49
Ideals, Quotient Rings, and Isomorphism Theorems	p. 58
Quotient Fields and Localization	p. 68
Polynomial Rings	p. 72
Principal Ideal Domains and Euclidean Domains	p. 79
Unique Factorization Domains	p. 92
Exercises	p. 98
Modules and Vector Spaces	p. 107
Definitions and Examples	p. 107
Submodules and Quotient Modules	p. 112
Direct Sums, Exact Sequences, and Hom	p. 118
Free Modules	p. 128
Projective Modules	p. 136
Free Modules over a PID	p. 142
Finitely Generated Modules over PIDs	p. 156
Complemented Submodules	p. 171
Exercises	p. 174
Linear Algebra	p. 182
Matrix Algebra	p. 182
Determinants and Linear Equations	p. 194
Matrix Representation of Homomorphisms	p. 214
Canonical Form Theory	p. 231
Computational Examples	p. 257
Inner Product Spaces and Normal Linear Transformations	p. 269
Exercises	p. 278
Matrices over PIDs	p. 289
Equivalence and Similarity	p. 289
Hermite Normal Form	p. 296
Smith Normal Form	p. 307
Computational Examples	p. 319
A Rank Criterion for Similarity	p. 328
Exercises	p. 337
Bilinear and Quadratic Forms	p. 341
Duality	p. 341
Bilinear and Sesquilinear Forms	p. 350
Quadratic Forms	p. 376
Exercises	p. 391
Topics in Module Theory	p. 395
Simple and Semisimple Rings and Modules	p. 395
Multilinear Algebra	p. 412
Exercises	p. 434
Group Representations	p. 438
Examples and General Results	p. 438
Representations of Abelian Groups	p. 451
Decomposition of the Regular Representation	p. 453
Characters	p. 462
Induced Representations	p. 479
Permutation Representations	p. 496
Concluding Remarks	p. 503
Exercises	p. 505
Appendix	p. 507
Bibliography	p. 510
Index of Notation	p. 511
Index of Terminology	p. 517
Table of Contents provided by Blackwell. All Rights Reserved.

Algebra

An Approach via Module Theory

At a Glance

Hardcover

Shipping

How to return your order

Defective items

You Can Find This Book In

Other Editions and Formats

Paperback

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