If is a Dedekind ring with quotient field , and a ring between and , whose elements induce functions on either or some other -algebra , then ideal theoretic questions about like: what does the spectrum look like, are all radicals of finitely generated ideals intersections of maximal ideals, which co-maximal ideals of remain co-maximal in ? etc. can often be answered by considering ideals of polynomials mapping a fixed element into a given prime ideal of the image of under . For instance, integer-matrix-valued polynomials (just like integer-valued polynomials) satisfy properties similar to Hilbert's Nullstellensatz.
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