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Module Theory: An Approach to Linear Algebra by T.S. Blyth

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Module Theory: An Approach to Linear Algebra written by T.S. Blyth . Published by Oxford University Press.
Many branches of algebra are linked by the theory of modules since the notion of a module is obtained essentially by modest generalization of that of a vector space, it is surprising that it plays an important role in the theory of linear algebra. Modules are also of great importance in the higher reaches of group theory and ring theory and fundamental to the study of advanced topics such as homological algebra, category theory, and algebraic topology. The aim of this text is to develop the basic properties of modules and to show their importance in the theory of linear algebra.
The first part, which comprises Section 1 to 11 is intended as a self contained course for first year honours students. The second part of the text, comprising Sections 12 to 17, may be considered as such a course.

Module Theory: An Approach to Linear Algebra written by T.S. Blyth cover the following topics.

  • Part I Module and Vector Spaces
    1. Moduels, Vector Spaces, Algebra
    2. Submodules, Intersections and Sums
    3. Morphisms, Axact Sequences
    4. Quotient Modules, Isomorphism Theorems
    5. Chain Conditions, Jorden Holder Towers
    6. Poducts and Coproducts
    7. Free Modules, Bases
    8. Groups of Morphisms; Projective Modules
    9. Quality, Transposition
    10. Matrices; Linear Equations
    11. Inner Prodcut Spaces

  • Part II Advaced Linear Algebra
    12. Injective Modules
    13. Tensor Products, Flat Modules
    14. Tensor Products (Commutative Ground Ring); Tensor Algebras
    15. Exterior Powers, exterior Algebras, Determinants
    16. Modules Over a Principal Ideal Domain, Finitely Generated Abelian Groups
    17. Vector Spaces Decomposition Theorms, Canonical Forms Under Similarites

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