Rings and Homology

by James P. Jans

Overview

This concise text is geared toward students of mathematics who have completed a basic college course in algebra. Combining material on ring structure and homological algebra, the treatment offers advanced undergraduate and graduate students practice in the techniques of both areas.
After a brief review of basic concepts, the text proceeds to an examination of ring structure, with particular attention to the structure of semisimple rings with minimum condition. Subsequent chapters develop certain elementary homological theories, introducing the functor Ext and exploring the various projective dimensions, global dimension, and duality theory. Each chapter concludes with a set of exercises.

Product Details

ISBN-13:	9780486789972
Publisher:	Dover Publications
Publication date:	01/14/2015
Series:	Dover Books on Mathematics
Pages:	96
Product dimensions:	5.90(w) x 8.70(h) x 0.30(d)

About the Author

James P. Jans was Professor of Mathematics at the University of Washington.

Table of Contents

Preface v

1 Basic Concepts 1

Schur's Lemma 4

Another Version of Schur's Lemma 4

2 Everybody Splits 12

Structure Theorem for Semisimple Rings with Minimum Condition 12

First Uniqueness Theorem 19

Second Uniqueness Theorem 21

3 Complexes, Homology, and Ext 27

The Exact Sequence of Homology Theorem 28

Theorem (Splitting of Ext) 43

4 Various Dimensions 47

The Shifting Theorem 47

The Dimension Theorem 48

Injective Producing Lemma 52

Shifting Theorem for Infectives 54

The Injective Dimension Theorem 55

Global Dimension Theorem 55

5 Duality and Quasi-Frobenius Rings 65

Third Dual Theorem 67

Some Properties of Torsionless Modules 67

Torsionless and Dual Theorem 69

References 83

Index 87