Consider the following problem: given a variety of algebras (a class of algebras in a given algebraic signature defined by some set of equations), describe its semisimple subvarieties. That is, describe its subvarieties $\mathsf{K}$ such that each algebra $\mathbf{A} \in \mathsf{K}$ is semisimple (the intersection of the maximal congruences of $\mathbf{A}$ is the identity relation).

For example, a subvariety $\mathsf{K}$ of the variety of MV-algebras is semisimple if and only if $\mathsf{K} \subseteq \mathsf{E}_{n}$ for some $n \in \omega$, where $\mathsf{E}_{n}$ is the variety of all MV-algebras satisfying the equation $x^{n} = x^{n+1}$.

For commutative rings the universal algebraic notion of semisimplicity corresponds to the ring theoretic notion of being semiprimitive, or Jacobson semisimple. Boolean rings provide an example of a Jacobson semisimple variety of commutative rings. My question is now: is there an analogous description of the Jacobson semisimple varieties of commutative rings? Failing that, is there *some* intelligible description of the Jacobson semisimple varieties of commutative rings?