|Title||Approximation Theorems in Commutative Algebra. Classical and Categorical Methods|
|Publisher||Kluwer Academic Publishers|
|Year of publication||1992|
|Reviewed by||Andrei Marcus|
Approximation theorems occur frequently in commutative algebra and they also have applications in many other fields. The volume under review is divided into two parts. Part I is dedicated to the investigation of approximation theorems from a classical point of view and deals with three algebraic systoms: fields and rings, partially ordered groups, and multivalued commutative systems such as multirings and d-groups. The reader will find various approximation theorems for valuations on fields (Ribbenboim's theorem and generalizations due to Griffin) with applications to topological rings, approximation theorems for Manis valuations on arbitrary commutative rings (mainly due to Gräter), for lattice-ordered groups, for groups having the divisor theory, introductory notions and results about multirings and d-groups and approximation thoorems for these systems.
In Part II approximation theorems are investigated from the categorical point of view. The main tool for this approach is a first-order many-sorted logic. It is shown that there is a strong connection between approximation theorems and some modern topics like theory of sheaves, topos theory and model theory.
This is a valuable monograph written by experts who also organize very well a large amount of information. Some facts appear in book form for the first time. The second part is essentially self-contained and requires only the knowledge of elements of category theory and first order logic. A large bibliographical list and a good index increase the value of the book.
This volume is entirely accessible to graduate students and it is recommended to specialists and to anyone who is interested in commutative algebra, category theory and applications of logic.