Pseudovarieties of associative rings: Congruence-linearity and decidability (Q1910271)

A pseudovariety, i.e. a class of finite universal algebras closed with respect to finite direct products, subalgebras, and homomorphic images, is said to be recursive if there is an algorithm for any finite algebra deciding whether the algebra belongs to the pseudovariety or not. A class of universal algebras is said to be congruence-linear if the congruence lattice of any SI-member is a chain. The author gives a number of conditions characterizing congruence-linearity on recursive pseudovarieties of associative rings. Namely, it is shown that congruence-linearity is equivalent to decidability of the theory of finite rings for recursive pseudovarieties of associative rings.