Prime ideals in infinite products of commutative rings

Abstract

We describe the prime ideals and, in particular, the maximal ideals in products $R = \prod D_\lambda$ of families $(D_\lambda)_{\lambda \in \Lambda}$ of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra $\prod \mathcal{P}(\max(D_\lambda))$, where $\max(D_\lambda)$ is the spectrum of maximal ideals of $D_\lambda$, and $\mathcal{P}$ denotes the power set. If every $D_\lambda$ is in a certain class of rings including finite character domains and one-dimensional domains, we completely characterize the maximal ideals of $R$. If every $D_\lambda$ is a Pr\"ufer domain, we completely characterize all prime ideals of $R$.