On the spectrum of rings of functions

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 $D/M$.