Elementary theory of a module over a local ring
Suppose A is a finite-dimensional local commutative associative k-algebra with unit over a field k of arbitrary characteristic and M is a finitely generated A-module. In this paper we consider the elementary theory of a two-basis model \(M_ A=<M,A;\delta >\), where \(\delta\) is the predicate distinguishing the action of A on M. Besides being of interest in its own right, the study of models of type \(M_ A\) is justified by the fact that many model-theoretic problems for arbitrary finite-dimensional k-algebras and nilpotent k-groups of finite rank can be reduced to them via the technique of bilinear mappings. In the present paper we obtain an algebraic description of the structure of models of the complete theory of \(M_ A\) and give a criterion for its decidability. These results are used in an essential way to classify algebras and nilpotent groups with respect to elementary properties and to characterize among them the objects with a decidable theory.
scientific article; zbMATH DE number 4152353