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Model theoretic algebra: with particular emphasis on fields, rings, modules. (English) Zbl 0728.03026
Algebra, Logic and Applications, 2. New York etc.: Gordon and Breach Science Publishers. xiii, 443 p. (1989).
In the preface of their book the authors explain its main aim as follows: “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. Our treatment is more algebraic than model-theoretic; in fact, the model theoretic concepts we consider are the most basic ones, and the model theoretic tools we use are quite modest: mainly ultrapowers (or equivalently the compactness theorem), elementary equivalence, (finite) axiomatizability, elementary substructures, elementarily definable substructures, quantifier elimination for modules and algebraically closed fields. Our notes should by no means be regarded as a textbook, either in model theory or in algebra; in particular, we do not pretend to any kind of completeness. The topics we have treated are selected according to personal taste. A guiding principle has been to omit subjects that have already been treated in textbooks or (well-known) lecture notes. This for instance applies to the Ax-Kochen theory of p-adic fields.” (Preface, p. xi).
Hilbert’s Nullstellensatz, Hilbert’s 17th problem and the Noether- Ostrowski’s irreducibility theorem are observed as applications of some basic principles of model theory in Chapter 1.
In Chapter 2 the authors characterize elementary equivalence of rings, function fields, and power series fields, respectively, and consider axiomatizable and finitely axiomatizable classes, MacLane separability, \(C_ i\)-fields, real closed fields, and the importance of the Pythagoras number of a field.
Chapter 3 deals with elementary definability of rings and its applications to polynomial rings, power series rings, fields of rational functions, quotient fields, as well as with its connection to elementary equivalence.
Peano rings (Peano fields) are defined in Chapter 4 as rings (fields) which are elementarily equivalent to the ring of integers (field of rational numbers). Every Peano field is a quotient field of a uniquely determined Peano subring. There exist rigid Peano fields of arbitrary infinite cardinality. Transcendence degree, Krull dimension, elementary equivalence to rings of integers or polynomials, and cancellation for polynomial rings are considered.
Chapter 5 is dedicated to the class of Hilbert fields as an axiomatizable but not finitely axiomatizable class in connection with Galois groups.
Chapters 6 and 7 deal with the one-sorted first order language of (left) R-modules over a fixed associative ring R (positive primitive formulas, finitely definable subgroups, pure-exact sequences, elementary equivalence of R-modules, localization and globalization of elementary equivalence, elementary and axiomatizable classes) and an analysis of the corresponding notion of algebraic compactness (characterization of algebraically compact modules, pure-injective envelopes, comparison with injective functors, splitting criterion for algebraic compactness, injective resolution, injective ultraproducts).
Decompositions and algebraic compactness is the topic of Chapter 8. For a left R-module M the \(\Sigma\)-algebraic compactness is equivalent to each of 7 conditions (e.g., M satisfies the descending chain condition for definable subgroups). A ring A is left pure-semisimple iff every left R- module is algebraically compact. The spectral category of a Grothendieck category is again a Grothendieck category with some additional properties. Each left R-module M is elementarily equivalent to a direct sum of algebraically compact indecomposable modules. Non-algebraically compact ultrapowers, Krull dimension for mod(R), Krull dimension for a Dedekind domain, indecomposable pure-injective modules over a Dedekind domain, indecomposable pure-injective Kronecker modules, and reduction modulo the radical, respectively, are considered, too.
In Chapter 9 is introduced a two-sorted first order language of modules over unspecified rings. Main points of consideration are modules of finite length, finitely generated and finitely presented modules, flat modules, projective modules, indecomposable modules.
Chapters 10, 11, 12 deal with the first order theory of rings (finitely axiomatizable classes of rings, rings with chain condition, coherent rings, weak global and global dimension, Krull and Gelfand-Kirillov dimension), Pure global dimension and algebraically compact rings, and Representation theory of finite-dimensional algebras (elimination of quantifiers, van den Dries’s test, affine varieties and affine schemes, orders and lattices, finite representations).
In Chapter 13 some problems are formulated. The tables after this chapter summarize many properties of fields, rings, and modules, respectively, and describe their behaviour under the considered constructions.
The book gives an interesting view of some subjects of the theory of rings, fields, and modules in correspondence with model theoretic aspects and is well legible. The headings and subtitles make the orientation easier and many exercises complete the represented theoretical knowledge. The appendices give further information to readers who are not so familiar with the basic notations. The bibliography comprises 213 titles.

03C60 Model-theoretic algebra
03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations
12L12 Model theory of fields
13L05 Applications of logic to commutative algebra
16B70 Applications of logic in associative algebras
03C20 Ultraproducts and related constructions
12E05 Polynomials in general fields (irreducibility, etc.)
12E25 Hilbertian fields; Hilbert’s irreducibility theorem
13E05 Commutative Noetherian rings and modules
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations