A classification of Prüfer domains of integer-valued polynomials on algebras

Let $D$ be an integrally closed domain with quotient field $K$ and $A$ a torsion-free $D$-algebra that is finitely generated as a $D$-module and such that $A\cap K=D$. We give a complete classification of those $D$ and $A$ for which the ring $In{t}_K\left(A\right)=\{f\in K\left[X\right]\mid f\left(A\right)\subseteq A\}$ is a Prüfer domain. If $D$ is a semiprimitive domain, then we prove that $In{t}_K\left(A\right)$ is Prüfer if and only if $A$ is commutative and isomorphic to a finite direct product of almost Dedekind domains with finite residue fields, each of them satisfying a double boundedness condition on its ramification indices and residue field degrees.