Elementary theory of a module over a local ring
type: Item
instance of: scholarly article
title: Elementary theory of a module over a local ring
zbMATH Open document ID: 0703.03016
DOI: 10.1007/BF00971494
published in: Siberian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 13L05
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 20F18
zbMATH DE Number: 4152353
zbMATH Keywords: elementary theory
zbMATH Keywords: two-basis model
zbMATH Keywords: finite-dimensional k-algebras
zbMATH Keywords: nilpotent k-groups
zbMATH Keywords: structure of models
zbMATH Keywords: decidability
author: Alexei Myasnikov
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00971494
OpenAlex ID: W2005221828
description: scientific article; zbMATH DE number 4152353