Prime Ideals in Infinite Products of Commutative Rings

In this work we present descriptions of prime ideals and in particular of 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 $\prod \mathcal{P}(\max(D_\lambda))$. If every $D_\lambda$ is in a certain class of rings including finite character domains and one-dimensional domains, then this leads to a characterization of the maximal ideals of $R$. If every $D_\lambda$ is a Pr\"ufer domain, we depict all prime ideals of $R$. Moreover, we give an example of a (optionally non-local or local) Pr\"ufer domain such that every non-zero prime ideal is of infinite height