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 is an element of K[X] such that f(Omega) subset of D, is a Priffer domain if and only if D is Priffer. 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 (K) over bar 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 Priffer if and only if D is Priffer. 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 (K) over bar.