Module theory: an approach to linear algebra

✍ Scribed by T. S. Blyth

• Publisher: OUP
• Year: 1977
• Tongue: English
• Leaves: 410
• Category: Library

✦ Synopsis

The aim of this textbook is to develop the basic properties of modules and to show their importance in the theory of linear algebra. It is intended to provide a self-contained course as well as to indicate how the theory may be developed in a number of more advanced directions. Throughout, numerous exercises will enable readers to consolidate their understanding. Prerequisites are few, being only a familiarity with the basic notions of rings, fields, and groups. The first eleven chapters provide a carefully graded introduction to the central results and applications of the theory of modules. Subsequent chapters deal with advanced linear algebra (including multilinear and tensor algebra) and tackle more advanced topics. These include the exterior product approach to the determinants of matrices, a module-theoretic approach to the structure of finitely generated Abelian groups, canonical forms, and normal transformations. This edition has been revised to accommodate many suggestions to ensure its suitability for undergraduate courses. One advanced topic now covered is a proof of the celebrated Wedderburn-Artin theorem which determines the structure of simple Artinian rings.

✦ Table of Contents

Title page......Page 1
Copyright page......Page 2
Preface......Page 3
Dedication......Page 5
Contents......Page 7
PART I: MODULES AND VECTOR SPACES......Page 9
1. Modules; vector spaces; algebras......Page 11
2. Submodules; intersections and sums......Page 19
3. Morphisms; exact sequences......Page 27
4. Quotient modules; isomorphism theorems......Page 48
5. Chain conditions; Jordan-Holder towers......Page 64
6. Products and coproducts......Page 76
7. Free modules; bases......Page 98
8. Groups of morphisms; projective modules......Page 121
9. Duality; transposition......Page 143
10. Matrices; linear equations......Page 157
11. Inner product spaces......Page 187
PART II: ADVANCED LINEAR ALGEBRA......Page 207
12. Infective modules......Page 209
13. Tensor products; flat modules......Page 221
14. Tensor products (commutative ground ring); tensor algebras......Page 247
15. Exterior powers; exterior algebras; determinants......Page 270
16. Modules over a principal ideal domain; finitely generated abelian groups......Page 303
17. Vector space decomposition theorems; canonical forms under similarity......Page 332
Index......Page 408