Prime ideals in infinite products of commutative rings

TL;DR

This paper characterizes prime and maximal ideals in infinite products of commutative rings, linking maximal ideals to ultrafilters and providing complete descriptions for specific classes of rings.

Contribution

It offers a new description of prime and maximal ideals in product rings, especially relating maximal ideals to ultrafilters and characterizing ideals for certain ring classes.

Findings

Maximal ideals correspond to ultrafilters on a Boolean algebra.

Complete characterization of maximal ideals for rings with finite character or one-dimensional domains.

Complete description of all prime ideals when each factor is a Pr"ufer domain.

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 $R$.