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About this book :-
Model Theoretic Algebra (with particular emphasis on Fields, Rings, Modules) written by Christian Jensen, Helmut Lenzing.
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.
Book Detail :-
Title: Model Theoretic Algebra (with particular emphasis on Fields, Rings, Modules)
Author(s): Christian U. Jensen, Helmut Lenzing
Publisher: Oxford University Press
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Book Contents :-
Module Theory: An Approach to Linear Algebra written by T.S. Blyth cover the following topics.
1. Introduction. Ultraproducts. Definitions and examples
2. Elementary equivalence. Axiomatizable and finitely axiomatizable classes. Examples and results in field theory
3. Elementary definability. Applications to polynomial and series rings and their quotient fields power
4. Peano rings and Peano fields
5. Hilbertian fields and realizations of finite groups as Galois groups
6. The language of modules over a fixed ring
7. Algebraically compact modules
8. Decompositions and algebraic compactness
9. The two-sorted language of modules over unspecified rings
10. The first order theory of rings
11. Pure global dimension and algebraically compact rings
12. Representation theory of finite dimensional algebras
A. Basic notions and definitions from homological algebra
B. Functor categories on finitely presented modules
C. Glossary of some basic notions in ring and module theory
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