Idempotent generated algebras and Boolean powers of commutative rings

Boolean powers were introduced by Foster [5]. It was noticed by Jónsson in the review of [6], and further elaborated by Banaschewski and Nelson [1], that the Boolean power of an algebra A by a Boolean algebra B can be described as the algebra of continuous functions from the Stone space of B to A, where A has the discrete topology. It follows that a Boolean power of the group Z is an `-group generated by its singular elements; that is, elements g > 0 satisfying h ∧ (g − h) = 0 for all h with 0 ≤ h ≤ g. Conrad [4] called such `-groups Specker `-groups because they arise naturally in the study of the Baer-Specker group—the product of countably many copies of Z. Similarly, a Boolean power of the ring R is an R-algebra generated by its idempotents. In analogy with the `-group case, these algebras were termed Specker R-algebras in [3]. As a common generalization of these two cases, for a commutative ring R, we introduce the notion of a Specker R-algebra and show that Specker R-algebras are Boolean powers of R. For an indecomposable ring R, this yields an equivalence between the category of Specker R-algebras and the category of Boolean algebras. Together with Stone duality this produces a dual equivalence between the category of Specker R-algebras and the category of Stone spaces.