## Journal of Commutative Algebra

- J. Commut. Algebra
- Volume 8, Number 1 (2016), 113-141.

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

#### Abstract

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 $\overline K$.

#### Article information

**Source**

J. Commut. Algebra, Volume 8, Number 1 (2016), 113-141.

**Dates**

First available in Project Euclid: 28 March 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.jca/1459169548

**Digital Object Identifier**

doi:10.1216/JCA-2016-8-1-113

**Mathematical Reviews number (MathSciNet)**

MR3482349

**Zentralblatt MATH identifier**

1338.13015

**Subjects**

Primary: 13B25: Polynomials over commutative rings [See also 11C08, 11T06, 13F20, 13M10]

Secondary: 13B22: Integral closure of rings and ideals [See also 13A35]; integrally closed rings, related rings (Japanese, etc.) 13F05: Dedekind, Prüfer, Krull and Mori rings and their generalizations 13F20: Polynomial rings and ideals; rings of integer-valued polynomials [See also 11C08, 13B25]

**Keywords**

Pullback integral closure integer-valued polynomial divided differences Prüfer ring

#### Citation

Peruginelli, Giulio. The ring of polynomials integral-valued over a finite set of integral elements. J. Commut. Algebra 8 (2016), no. 1, 113--141. doi:10.1216/JCA-2016-8-1-113. https://projecteuclid.org/euclid.jca/1459169548