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 $\Int_K(A)=\{f\in K[X] \mid f(A)\subseteq A\}$ is a \Pruf{} domain. If $D$ is a semiprimitive domain, then we prove that $\Int_K(A)$ is Pr\"ufer 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.