Synopses & Reviews
This textbook provides a self-contained course on the basic properties of modules and their importance in the theory of linear algebra. The first 11 chapters introduce the central results and applications of the theory of modules. Subsequent chapters deal with advanced linear algebra, including multilinear and tensor algebra, and explore such topics as 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. Suitable for undergraduate courses, the text now includes a proof of the celebrated Wedderburn-Artin theorem which determines the structure of simple Artinian rings.
Table of Contents
1. Modules, Vector Spaces and Algebras
2. Submodules; Intersections and Sums
3. Morphisms; Exact Sequences
4. Quotient Modules; Basic Isomorphism Theorems
5. Chain Conditions; Jordan-Holder Towers
6. Products and Coproducts
7. Free Modules; Bases
8. Groups of Morphisms; Projective Modules
9. Duality; Transposition
10. Matrices; Linear Equations
11. Inner Product Spaces
12. Injective Modules
13. Simple and Semisimple Modules
14. The Jacobson Radical
15. Tensor Products; Flat Modules; Regular Rings
16. Tensor Products; Tensor Algebras
17. Exterior Algebras, Determinants
18. Modules Over A Principal Ideal Domain; Finitely Generated Abelian Groups
19. Vector Space Decomposition Theorems; Canonical Forms Under Similarity
20. Diagonalisation; Normal Transformations