  Let $D$ be a domain and $M$ a maximal ideal of $D$. The ring of
integer-valued polynomials on a subset $E$ of $D$, as well as more general
rings of functions from $E$ to $D$, can be viewed as subrings of the product
$D^E=\prod_{e\in E}D$. We investigate which maximal ideals of $Int(E,D)$ (or
any other subring of $D^E$) come from ultrapowers of the residue class ring
$D/M$.
