An Introduction to Module Theory

Description

Module theory is a fundamental area of algebra, taught in most universities at the graduate level. This textbook, written by two experienced teachers and researchers in the area, is based on courses given in their respective universities over the last thirty years. It is an accessible and modern account of module theory, meant as a textbook for graduate or advanced undergraduate students, though it can also be used for self-study. It is aimed at students in algebra, or students who need algebraic tools in their work. Following the recent trends in the area, the general approach stresses from the start the use of categorical and homological techniques. The book includes self-contained introductions to category theory and homological algebra with applications to Module theory, and also contains an introduction to representations of quivers. It includes a very large number of examples of all kinds worked out in detail, mostly of abelian groups, modules over matrix algebras, polynomial algebras, or algebras given by bound quivers. In order to help visualise and analyse examples, it includes many figures. Each section is followed by exercises of all levels of difficulty, both computational and theoretical, with hints provided to some of them. 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