| 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 \( Int _K(A) = \{f \in KX] \mid f(A) \subseteq A\}\), which generalizes the classic ring \( Int (D) = \{f \in KX] \mid f(D) \subseteq D\}\) of integer-valued polynomials on D. The condition on \(A \cap K\) implies that \(DX] \subseteq Int _K(A) \subseteq Int (D)\), and we say that \( Int _K(A)\) is nontrivial if \( Int _K(A) \ne DX]\). 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. |
