| 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 |
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