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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?

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I think this question asks: which varieties of commutative rings have the property that every member is a subdirect product of simple rings (= fields). These are the varieties satisfying some identity of the form $x=x^n$ for some $n>1$. They are exactly the congruence distributive varieties of commutative rings. These varieties are identified in

Werner, H. and R. Wille
Charakterisierungen der primitiven Klassen arithmetischer Ringe.
Math. Zeitschr. 115, 197-200 (1970).

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  • $\begingroup$ Yes, that's precisely the result I was looking for. Thank you, Keith! $\endgroup$ Feb 26 '20 at 18:00

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