Prime ideals in infinite products of commutative rings

Abstract

We describe the prime ideals and, in particular, the maximal ideals in products [Formula: see text] of families [Formula: see text] of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra [Formula: see text], where [Formula: see text] is the spectrum of maximal ideals of [Formula: see text], and [Formula: see text] denotes the power set. If every [Formula: see text] is in a certain class of rings including finite character domains and one-dimensional domains, we completely characterize the maximal ideals of [Formula: see text]. If every [Formula: see text] is a Prüfer domain, we completely characterize all prime ideals of [Formula: see text].