On the decidability of equational theories of varieties of rings (Q1280610)

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)]).