  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 $R$.
