Noncommutative Algebra / Edition 1

Noncommutative Algebra / Edition 1

by Benson Farb, R. Keith Dennis
     
 

ISBN-10: 038794057X

ISBN-13: 9780387940571

Pub. Date: 08/20/1993

Publisher: Springer New York

This book provides a concise introduction to the field of noncommutative algebra. It covers material essential to all students of algebra, particularly those specializing in ring theory, homological algebra, representation theory and K-theory. The core of the book is suitable for a one-semester graduate course; it is also suitable for self-study. The approach is more…  See more details below

Overview

This book provides a concise introduction to the field of noncommutative algebra. It covers material essential to all students of algebra, particularly those specializing in ring theory, homological algebra, representation theory and K-theory. The core of the book is suitable for a one-semester graduate course; it is also suitable for self-study. The approach is more homological than ring-theoretic and begins with the basics of semisimple modules and rings, including the Wedderburn structure theorem. The Jacobson radical is discussed from several different points of view followed by a development of the theory of central simple algebras, including proofs of the Skolem-Noether and Double Centralizer theorems. This theory is then used to give quick proofs of two famous, classical results: the Wedderburn Theorem on finite division rings and the Frobenius Theorem on the classification of central division algebras over the reals. The first part of the book closes with an introduction to the Brauer group and its relation to cohomology. The remaining chapters consist of special topics: the notion of primitive rings is developed along lines parallel to that of simple rings; the representation theory of finite groups is combined with the Wedderburn Structure Theorem to prove Burnside's Theorem; the global dimension of a ring is studied using Kaplansky's elementary point of view; and the Brauer group of a commutative ring is introduced. In addition to the large number of exercises throughout the book, a set of supplementary problems explores further topics and can serve as a starting point for student projects.

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Product Details

ISBN-13:
9780387940571
Publisher:
Springer New York
Publication date:
08/20/1993
Series:
Graduate Texts in Mathematics Series, #144
Edition description:
1993
Pages:
226
Product dimensions:
6.20(w) x 9.30(h) x 0.80(d)

Related Subjects

Table of Contents

Preface
About Other Books
Acknowledgments
A Word About Conventions
IThe Core Course1
0Background Material3
1Semisimple Modules & Rings and the Wedderburn Structure Theorem29
2The Jacobson Radical57
3Central Simple Algebras81
4The Brauer Group109
IISelected Topics149
5Primitive Rings and the Density Theorem151
6Burnside's Theorem and Representations of Finite Groups161
7The Global Dimension of a Ring177
8The Brauer Group of a Commutative Ring185
IIISupplementary Exercises199
References215
Index219

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