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