A First Course in Module Theory (精装)

1998-12  

World Scientific Publishing Company (1998年9月1日)  

M. E. Keating  

250  

Introduction1 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 Exercises2 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 Exercises3 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 Exercises4 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 Exercises5 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 Modules7 Direct Sums of Modules8 Torsion and The Primary Decomposition9 Presentations10 Diagonalizing and Inverting Matrices11 Fitting Ideals12 The Decompositn of Moduels13 Normal Forms for Matrices14 Projective ModulesHints and SolutionsBibliographyIndex

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