On the decidability of equational theories of varieties of rings
type: Item
instance of: scholarly article
title: On the decidability of equational theories of varieties of rings
zbMATH Open document ID: 0916.17001
DOI: 10.1007/BF02312771
published in: Mathematical Notes
publication date: 1999-04-11T00:00:00Z
review text: The first examples of finitely based (nonassociative) ring varieties with undecidable equational theory were given by the author in [\textit{V. Yu. Popov}, Algebra Logika 34, No. 3, 347-361 (1995; Zbl 0857.17003)]. In the article under review the author constructs a countable ascending chain of finitely based (nonassociative) ring varieties \({\mathcal X}_1\subset{\mathcal X}_2\subset {\mathcal X}_3\subset\cdots\subset{\mathcal X}_n\subset\cdots\) such that the equational theory of the varieties \({\mathcal X}_1,{\mathcal X}_3,\ldots,{\mathcal X}_{2k+1},\ldots\) is undecidable while the equational theory of the varieties \({\mathcal X}_2,{\mathcal X}_4,\ldots,{\mathcal X}_{2k},\ldots\) is decidable. As a corollary, the author proves that for \(i=1,3,\ldots,2k+1,\ldots\) the theory \(\forall{\mathcal X}_i\) is a unique \(SA\)-critical theory of the variety \({\mathcal X}_i\) (for the definition of \(SA\)-critical theories see \textit{Yu. M. Vazhenin} [Algebra Logika 26, No. 4, 419-434 (1987; Zbl 0647.03006)]).
Mathematics Subject Classification ID: 17A30
Mathematics Subject Classification ID: 03B25
zbMATH DE Number: 1262485
zbMATH Keywords: nonassociative ring variety
zbMATH Keywords: equational theory of a variety
zbMATH Keywords: finitely based variety
zbMATH Keywords: undecidable theory
zbMATH Keywords: critical theory
author: Vladimir Popov
reviewed by: Boris Vernikov
MaRDI profile type: Publication
cites work: Q3958656
cites work: Q3791112
cites work: Q5797093
full work available at URL: https://doi.org/10.1007/bf02312771
OpenAlex ID: W2053320542
description: scientific article; zbMATH DE number 1262485