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Algebra An Approach Via Module Theory

ISBN-10: 0387978399
ISBN-13: 9780387978390
Edition: 1992
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Description: 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  More...

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Book details

Copyright year: 1992
Publisher: Springer
Publication date: 4/23/1999
Binding: Hardcover
Pages: 526
Size: 6.75" wide x 9.75" long x 1.25" tall
Weight: 1.980
Language: English

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
Groups
Definitions and Examples
Subgroups and Cosets
Normal Subgroups, Isomorphism Theorems, and Automorphism Groups
Permutation Representations and the Sylow Theorems
The Symmetric Group and Symmetry Groups
Direct and Semidirect Products
Groups of Low Order
Exercises
Rings
Definitions and Examples
Ideals, Quotient Rings, and Isomorphism Theorems
Quotient Fields and Localization
Polynomial Rings
Principal Ideal Domains and Euclidean Domains
Unique Factorization Domains
Exercises
Modules and Vector Spaces
Definitions and Examples
Submodules and Quotient Modules
Direct Sums, Exact Sequences, and Hom
Free Modules
Projective Modules
Free Modules over a PID
Finitely Generated Modules over PIDs
Complemented Submodules
Exercises
Linear Algebra
Matrix Algebra
Determinants and Linear Equations
Matrix Representation of Homomorphisms
Canonical Form Theory
Computational Examples
Inner Product Spaces and Normal Linear Transformations
Exercises
Matrices over PIDs
Equivalence and Similarity
Hermite Normal Form
Smith Normal Form
Computational Examples
A Rank Criterion for Similarity
Exercises
Bilinear and Quadratic Forms
Duality
Bilinear and Sesquilinear Forms
Quadratic Forms
Exercises
Topics in Module Theory
Simple and Semisimple Rings and Modules
Multilinear Algebra
Exercises
Group Representations
Examples and General Results
Representations of Abelian Groups
Decomposition of the Regular Representation
Characters
Induced Representations
Permutation Representations
Concluding Remarks
Exercises
Appendix
Bibliography
Index of Notation
Index of Terminology

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