# Jacobson semisimple varieties of commutative rings

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?

## 1 Answer

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).

• Yes, that's precisely the result I was looking for. Thank you, Keith! Commented Feb 26, 2020 at 18:00