Model Theoretic Algebra (with particular emphasis on Fields, Rings, Modules) written by
Christian U. Jensen , University of Copenhagen, Denmark and
Helmut Lenzing , University of Paderborn, FRG. Published by
Gordon and Breach Science Publishers S .A. It is the purpose of these notes to present some subjects from ring theory, field theory and module theory from a model theoretic point of view, basically, by making a semantic (first order) analysis of the corresponding algebraic concepts. Many non-trivial questions hereby arise, which may be of independent interest.

Model Theoretic Algebra (with particular emphasis on Fields, Rings, Modules) written by
Christian U. Jensen, Helmut Lenzing
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
13. Problems
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
Bibliography