
An Introduction to Module Theory
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Content
- Cover
- Title page
- Copyright page
- Contents
- Introduction
- I Rings and Algebras
- I.1 Introduction
- I.2 Rings and modules
- I.3 Algebras
- I.4 Algebra morphisms
- I.5 Principal ideal domains
- II Modules
- II.1 Introduction
- II.2 Modules and submodules
- II.3 Module morphisms
- II.4 The isomorphism theorems
- III Categories and functors
- III.1 Introduction
- III.2 Categories and functors
- III.3 Products and coproducts of modules
- III.4 Free modules
- IV Abelian categories
- IV.1 Introduction
- IV.2 Linear and abelian categories
- IV.3 Fibered products and amalgamated sums
- IV.4 Equivalences and dualities of categories
- V Modules over principal ideal domains
- V.1 Introduction
- V.2 Free modules and torsion
- V.3 The structure theorems
- V.4 An application: the Jordan form of a matrix
- VI Functors between modules
- VI.1 Introduction
- VI.2 The tensor product of modules
- VI.3 Exact functors
- VI.4 Projectives, injectives and flats
- VII The chain conditions
- VII.1 Introduction
- VII.2 Artinian and noetherian modules and algebras
- VII.3 Decompositions of algebras
- VII.4 Composition series
- VII.5 Semisimple modules and algebras
- VIII Radicals
- VIII.1 Introduction
- VIII.2 Radical and socle of a module
- VIII.3 Radicals of algebras
- VIII.4 Indecomposability
- VIII.5 The radical of a module category
- IX Projectives and quivers
- IX.1 Introduction
- IX.2 Projective modules over artinian algebras
- IX.3 Morita equivalence
- IX.4 Bound quiver algebras
- X Homology
- X.1 Introduction
- X.2 Homology and cohomology
- X.3 Derived functors
- XI Extension and torsion
- XI.1 Introduction
- XI.2 The extension and torsion functors
- XI.3 Exact sequences and extensions
- XII Homological dimensions
- XII.1 Introduction
- XII.2 Homological dimensions of modules
- XII.3 Homological dimensions of algebras
- XII.4 Classes of algebras
- Bibliography
- Index
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