A Characterization of Finitely Decidable Congruence Modular Varieties

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