The ring of polynomials integral-valued over a finite set of integral elements

Let $D$ be an integral domain with quotient field $K$ and $\Omega$ a finite subset of $D$. McQuillan proved that the ring $\Int(\Omega,D)$ of polynomials in $K[X]$ which are integer-valued over $\Omega$, that is, $f\in K[X]$ such that $f(\Omega)\subset D$, is a Pr\"ufer domain if and only if $D$ is Pr\"ufer. Under the further assumption that $D$ is integrally closed, we generalize his result by considering a finite set $S$ of a $D$-algebra $A$ which is finitely generated and torsion-free as a $D$-module, and the ring $\Int_K(S,A)$ of integer-valued polynomials over $S$, that is, polynomials over $K$ whose image over $S$ is contained in $A$. We show that the integral closure of $\Int_K(S,A)$ is equal to the contraction to $K[X]$ of $\Int(\Omega_S,D_F)$, for some finite subset $\Omega_S$ of integral elements over $D$ contained in an algebraic closure $\olK$ of $K$, where $D_F$ is the integral closure of $D$ in $F=K(\Omega_S)$. Moreover, the integral closure of $\Int_K(S,A)$ is Pr\"ufer if and only if $D$ is Pr\"ufer. The result is obtained by means of the study of pullbacks of the form $D[X]+p(X)K[X]$, where $p(X)$ is a monic non-constant polynomial over $D$: we prove that the integral closure of such a pullback is equal to the ring of polynomials over $K$ which are integral-valued over the set of roots $\Omega_p$ of $p(X)$ in $\overline K$.