|Title||Approximation Theorems in Commutative Algebra. Classical and Categorical Methods|
|Publisher||Kluwer Academic Publishers|
|Year of publication||1992|
|Reviewed by||Mirela Stefanescu|
The authors deal with approximation theorems from two points of view, based on the valuation theory of fields, on the multiplicative ideal theory, on the theory of partially ordered abelian groups, and even on the theory of multivalued commutative structures in which the addition is not singlevalued; a new point of view, consisting in formulation of such theorems in the most general system like the category.
Part I of the book is entirely devoted to the investigation of approximation theorems in fields and rings, partially ordered group, multirings (hyperrings) and d-groups.
First the authors present classical theorem of Ribenboim and its generalized versions obtained by M. Griffin. Groups with a divisor theory, R-Prüfer rings in connection with Manis valuations, relationships between the theory of rings and the theory of p.o.-groups via multirinys and d-groups constitute the topics studied in the Part I.
In Part II, the authors explain some possible relationships between the approximation theorems in these different systems, by using the categories.
A first-order many-sorted logic and its interpretation in categories make available a description of the compatibility conditions for the approximation theorems in these general systems. The starting point is the theory of toposes (Chapter 5), the variants of the theorems being presented in the last chapter of the book. Part II is self-contained; it requires only rudiments of category theory and first-order logic.
The monography is interesting for researchers and graduate students of commutative algebra, category theory, applications of logic.