Non-triviality Conditions for Integer-valued Polynomial Rings on Algebras

Abstract

Let $D$ be a commutative domain with field of fractions $K$ and let $A$ be a torsion-free $D$-algebra such that $A \cap K = D$. The ring of integer-valued polynomials on $A$ with coefficients in $K$ is ${\rm Int}_K(A) = \{f \in K[X] \mid f(A) \subseteq A\}$, which generalizes the classic ring ${\rm Int}(D) = \{f \in K[X] \mid f(D) \subseteq D\}$ of integer-valued polynomials on $D$. The condition on $A \cap K$ implies that $D[X] \subseteq {\rm Int}_K(A) \subseteq {\rm Int}(D)$, and we say that ${\rm Int}_K(A)$ is nontrivial if ${\rm Int}_K(A) \ne D[X]$. For any integral domain $D$, we prove that if $A$ is finitely generated as a $D$-module, then ${\rm Int}_K(A)$ is nontrivial if and only if ${\rm Int}(D)$ is nontrivial. When $A$ is not necessarily finitely generated but $D$ is Dedekind, we provide necessary and sufficient conditions for ${\rm Int}_K(A)$ to be nontrivial. These conditions also allow us to prove that, for $D$ Dedekind, the domain ${\rm Int}_K(A)$ has Krull dimension 2.