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

摘要

Let $D$ be an integral domain with quotient field $K$ and $\Omega$ a finitesubset of $D$. McQuillan proved that the ring ${\rm Int}(\Omega,D)$ ofpolynomials in $K[X]$ which are integer-valued over $\Omega$, that is, $f\inK[X]$ 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, wegeneralize 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${\rm Int}_K(S,A)$ of integer-valued polynomials over $S$, that is, polynomialsover $K$ whose image over $S$ is contained in $A$. We show that the integralclosure of ${\rm Int}_K(S,A)$ is equal to the contraction to $K[X]$ of ${\rmInt}(\Omega_S,D_F)$, for some finite subset $\Omega_S$ of integral elementsover $D$ contained in an algebraic closure $\bar K$ of $K$, where $D_F$ is theintegral closure of $D$ in $F=K(\Omega_S)$. Moreover, the integral closure of${\rm Int}_K(S,A)$ is Pr\"ufer if and only if $D$ is Pr\"ufer. The result isobtained 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 integralclosure of such a pullback is equal to the ring of polynomials over $K$ whichare integral-valued over the set of roots $\Omega_p$ of $p(X)$ in $\bar K$.