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 Prüfer domain, we completely characterize all prime ideals of $R$.

to appear in Communications in Contemporary Mathematics