A Characterization of Finitely Decidable Congruence Modular Varieties

Abstract

For every finitely generated, congruence modular variety V \mathcal {V} of finite type we find a finite family R \mathcal {R} of finite rings such that the variety V \mathcal {V} is finitely decidable if and only if V \mathcal {V} is congruence permutable and residually small, all solvable congruences in finite algebras from V \mathcal {V} are Abelian, each congruence above the centralizer of the monolith of a subdirectly irreducible algebra A \mathbf {A} from V \mathcal {V} is comparable with all congruences of A \mathbf {A} , each homomorphic image of a subdirectly irreducible algebra with a non-Abelian monolith has a non-Abelian monolith, and, for each ring R R from R \mathcal {R} , the variety of R R –modules is finitely decidable.