模块理论入门FIRST COURSE IN MODULE THEORY, A

Introduction
1 Rings and Ideals
1.1 Groups
1.2 Rings
1.3 Commutative domains
1.4 Units
1.5 Fields
1.6 Polynomial rings
1.7 Ideals
1.8 Principal ideals
1.9 Sum and intersection
1.10 Residue rings
1.11 Residues of integers
Exercises
2 Euclidean Domains
2.1 The definition
2.2 The integers
2.3 Polynomial rings
2.4 The Gaussian integers
2.5 Units and ideals
2.6 Greatest common divisors
2.7 Euclid's algorithm
2.8 Factorization
2.9 Standard factorizations
2.10 Irreducible elements
2.11 Residue rings of Euclidean domains
2.12 Residue rings of polynomial rings .
2.13 Splitting fields for polynomials
2.14 Further developments
Exercises
3 Modules and Submodules
3.1 The definition
3.2 Additive groups
3.3 Matrix actions
3.4 Actions of scalar matrices
3.5 Submodules
3.6 Sum and intersection
3.7 k-fold sums
3.8 Generators
3.9 Matrix actions again
3.10 Eigenspaces
3.11 Example: a triangular matrix action ~
3.12 Example: a rotation
Exercises
4 Homomorphisms
4.1 The definition
4.2 Sums and products
4.3 Multiplication homomorphisms
4.4 F[X]-modules in general
4.5 F[X]-module homomorphisms
4.6 The matrix interpretation
4.7 Example: p = 1
4.8 Example: a triangular action
4.9 Kernel and image
4.10 Rank &= nullity
4.11 Some calculations
4.12 Isomorphisms
4.13 A submodule correspondence
Exercises
5 Free Modules
5.1 The standard free modules
5.2 Free modules in general
5.3 A running example
5.4 Bases and isomorphisms
5.5 Uniqueness of rank
5.6 Change of basis
5.7 Coordinates
……
6 Quotient Modules and Cyclic Modules
7 Direct Sums of Modules
8 Torsion and The Primary Decomposition
9 Presentations
10 Diagonalizing and Inverting Matrices
11 Fitting Ideals
12 The Decompositn of Moduels
13 Normal Forms for Matrices
14 Projective Modules
Hints and Solutions
Bibliography
Index