On the spectrum of rings of functions

For D a domain and E⊆D, we investigate the prime spectrum of rings of functions from E to D, that is, of rings contained in ∏e∈ED and containing D. Among other things, we characterize, when M is a maximal ideal of finite index in D, those prime ideals lying above M which contain the kernel of the canonical map to ∏e∈E(D/M) as being precisely the prime ideals corresponding to ultrafilters on E. We give a sufficient condition for when all primes above M are of this form and thus establish a correspondence to the prime spectra of ultraproducts of residue class rings of D. As a corollary, we obtain a description using ultrafilters, differing from Chabert's original one which uses elements of the M-adic completion, of the prime ideals in the ring of integer-valued polynomials Int(D) lying above a maximal ideal of finite index.