Int-decomposable algebras

Let D be a Dedekind domain with fraction field k. Let A be a D-algebra that, as a D-module, is free of finite rank. Let B be the extension of A to a k-algebra. The set of integer-valued polynomials over A   is defined to be Int(A)={f∈B[x]|f(A)⊆A}Int(A)={f∈B[x]|f(A)⊆A}. Restricting the coefficients to elements of k  , we obtain the commutative ring Intk(A)={f∈k[x]|f(A)⊆A}Intk(A)={f∈k[x]|f(A)⊆A}; this makes Int(A)Int(A) a left Intk(A)Intk(A)-module. Previous researchers have noted instances when a D-module basis for A   is also an Intk(A)Intk(A)-basis for Int(A)Int(A). We classify all the D-algebras A   with this property. Along the way, we prove results regarding Int(A)Int(A), its localizations at primes of D, and finite residue rings of A.