Sophie Frisch: Ideal-theoretic properties of polynomial rings suggested by polynomial functions

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

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