Note on the Krull dimension of rings of integer-valued polynomials

Abstract

Let $D$ be an integral domain with quotient field $K,$ $E$ a subset of $K$ and $X$ an indeterminate over $K$. The set $\mathrm{Int}(E,D):=\{f\in K[X];\; f(E)\subseteq D\}$, of integer-valued polynomials on $E$ over $D$, is known to be an integral domain. The purpose of this note is to calculate the Krull dimension of $\mathrm{Int}(E,D)$ across various classes of integral domains $D$ and specific subsets $E$ of $D$. We further extend our study to the ring $\mathrm{Int}_B(E,D):=\{f\in B[X];\; f(E)\subseteq D\},$ where $B$ is an integral domain containing $D$