# Quasivarieties of commutative rings

## DOI:

https://doi.org/10.26577/jmmcs-2018-1-485## Keywords:

quasi-equational theory, undecidable theory, quasi-identity, quasivariety, quasiequational basis, independent quasi-equational bases, recursive independent quasi-equational basis, commutative ring with unit## Abstract

The work is devoted to the problem of undecidability of quasi-equational theories and to the

problem of finite axiomatizability. In 1966, Tarsky poded the following problem: Is there an algorithm

deciding if the equational theory of a finite set of finite structures is finitely axiomatizabile?

In 1986, Maltsev asked the following question: Are there finitely based semigroups, groups, and

rings with the undecidable equational theory? Nurakunov A.M. (Nurakunov, 2012) established

the existence of continuum many quasivarieties of unars, for which the quasi-equational theory is

undecidable and the finite membership problem is also undecidable. In in the paper (Basheyeva,

2017), some results in this direction are obtained for graphs, differential gruppoids, and pointed

Abelian groups. In the present paper, we prove analogous results for the variety of commutative

rings with unit. We prove also that the quasivariety of commutative rings with unit contains continuum

many of subquasivarieties with the undecidable quasi-equational theoryб for which the

finite membership problem is also undecidable. Apart from trhat, we prove here that the quasivariety

of commutative rings with unit contains continuum many subquasivarieties which have an

!-independent quasi-equational basis but does not have an indepen

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*Journal of Mathematics, Mechanics and Computer Science*,

*97*(1), 54–66. https://doi.org/10.26577/jmmcs-2018-1-485