| 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. |
