(untitled) (Journal of Mathematics Mechanics and Computer Science, 2018)
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