Module theory : an approach to linear algebra

Author(s)

Bibliographic Information

Module theory : an approach to linear algebra

T.S. Blyth

Clarendon Press , Oxford University Press, 1990

2nd ed

  • pbk.

Available at  / 16 libraries

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Note

Includes bibliographical references

Description and Table of Contents

Volume

ISBN 9780198532934

Description

The aim of this textbook is to develop the basic properties of modules and to show their importance in the theory of linear algebra. It is intended to provide a self-contained course as well as to indicate how the theory may be developed in a number of more advanced directions. Throughout, numerous exercises will enable readers to consolidate their understanding. Prerequisites are few, being only a familiarity with the basic notions of rings, fields, and groups. The first eleven chapters provide a carefully graded introduction to the central results and applications of the theory of modules. Subsequent chapters deal with advanced linear algebra (including multilinear and tensor algebra) and tackle more advanced topics. These include the exterior product approach to the determinants of matrices, a module-theoretic approach to the structure of finitely generated Abelian groups, canonical forms, and normal transformations. This edition has been revised to accommodate many suggestions to ensure its suitability for undergraduate courses. One advanced topic now covered is a proof of the celebrated Wedderburn-Artin theorem which determines the structure of simple Artinian rings.

Table of Contents

  • Modules, vector spaces, and algebras
  • Submodules
  • intersections and sums
  • Morphisms
  • exact sequences
  • Quotient modules
  • basic isomorhism theorems
  • Chain conditions
  • Jordan-H "older towers
  • Products and coproducts
  • Free modules
  • bases
  • Groups of morphisms
  • projective modules
  • Duality
  • transposition
  • Matrices
  • linear equations
  • Inner product spaces
  • Injective modules
  • Ssimple and semisimple modules
  • The Jacobson radical
  • Tensor products
  • flat modules
  • regular rings
  • Tensor products
  • tensor algebras
  • Exterior algebras, determinants
  • Modules over a principal ideal domain
  • finitely generated abelian groups
  • Vector space decomposition theorems
  • canonical forms under similarity
  • Diagonalisation
  • normal transformations.
Volume

pbk. ISBN 9780198533894

Description

The aim of this textbook is to develop the basic properties of modules and to show their importance in the theory of linear algebra. It is intended to provide a self-contained course as well as to indicate how the theory may be developed in a number of more advanced directions. Throughout, numerous exercises will enable readers to consolidate their understanding. Prerequisites are few, being only a familiarity with the basic notions of rings, fields, and groups. The first 11 chapters provide a carefully graded introduction to the central results and applications of the theory of modules. Subsequent chapters deal with advanced linear algebra (including multilinear and tensor algebra) and tackle more advanced topics. These include the exterior product approach to the determinants of matrices, a module-theoretic approach to the structure of finitely generated Abelian groups, canonical forms, and normal transformations. This edition has been revised to accommodate many suggestions to ensure its suitability for undergraduate courses. One advanced topic now covered is a proof of the celebrated Wedderburn-Artin theorem which determines the structure of simple Artinian rings.

Table of Contents

  • Modules, vector spaces, and algebras
  • Submodules
  • intersections and sums
  • Morphisms
  • exact sequences
  • Quotient modules
  • basic isomorphism theorems
  • Chain conditions
  • Jordan-H "older towers
  • Products and coproducts
  • Free modules
  • bases
  • Groups of morphisms
  • projective modules
  • Duality
  • transposition
  • Matrices
  • linear equations
  • Inner product spaces
  • Injective modules
  • Simple and semisimple modules
  • The Jacobson radical
  • Tensor products
  • flat modules
  • regular rings
  • Tensor products
  • tensor algebras
  • Exterior algebras, determinants
  • Modules over a principal ideal domain
  • finitely generated abelian groups
  • Vector space decomposition theorems
  • canonical forms under similarity
  • Diagonalisation
  • normal transformations.

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