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

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 ${\rm Int}_K(A) = {f \in K[X] \mid f(A) \subseteq A}$,whichgeneralizestheclassicring${\rm Int}(D) = {f \in K[X] \mid f(D) \subseteq D}$ofinteger-valuedpolynomialson$D$. The condition on $A\cap K$ implies that $D[X] \subseteq {\rm Int}_K(A) \subseteq {\rm Int}(D)$,andwesaythat${\rm Int}_K(A)$isnontrivialif${\rm Int}_K(A) \ne D[X]$\mathrm{.}Foranyintegraldomain$D$,weprovethatif$A$ is finitely generated as a $D$-module, then ${\mathrm{I}\mathrm{n}\mathrm{t}}_K(A)$ is nontrivial if and only if $\mathrm{I}\mathrm{n}\mathrm{t}(D)$ is nontrivial. When $A$ is not necessarily finitely generated but $D$ is Dedekind, we provide necessary and sufficient conditions for ${\mathrm{I}\mathrm{n}\mathrm{t}}_K(A)$ to be nontrivial. These conditions also allow us to prove that, for $D$ Dedekind, the domain ${\mathrm{I}\mathrm{n}\mathrm{t}}_K(A)$ has Krull dimension 2.