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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