A First Course in Module Theory (精裝)

書(shū)籍目錄

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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