## Journal of Commutative Algebra

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

Giulio Peruginelli

#### 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

#### 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

#### References

• M. Bhargava, On $P$-orderings, rings of integer-valued polynomials and ultrametric analysis, J. Amer. Math. Soc. 22 (2009), 963–993.
• N. Bourbaki, Commutative algebra, Addison-Wesley Publishing Co., Reading, MA, 1972.
• ––––, Algebra, P.M. Cohn and J. Howie, eds., in Elements of mathematics, Springer-Verlag, Berlin, 1990.
• J.G. Boynton, Pullbacks of arithmetical rings, Comm. Alg. 35 (2007), 2671–2684.
• J.G. Boynton and S. Sather-Wagstaff, Regular pullbacks, Progr. Comm. Alg. 2, 145–169, Walter de Gruyter, Berlin, 2012.
• J.-P. Cahen and J.-L. Chabert, Integer-valued polynomials, Amer. Math. Soc. Surv. Mono. 48, Providence, 1997.
• S. Evrard, Y. Fares and K. Johnson, Integer valued polynomials on lower triangular integer matrices, Monats. Math. 170 (2013), 147–160.
• M. Fontana, J.A. Huckaba and I.J. Papick, Prüfer domains, Mono. Text. Pure Appl. Math. 203, Marcel Dekker, Inc., New York, 1997.
• S. Gabelli and E. Houston, Ideal theory in pullbacks, in Non-Noetherian commutative ring theory, Kluwer Academic Publishers, Dordrecht, 2000.
• R. Gilmer, Multiplicative ideal theory, Corrected reprint of the 1972 edition, Queen's University, Kingston, Ontario, 1992.
• M. Griffin, Prüfer rings with zero divisors, J. reine angew. Math. 239/240 (1969), 55–67.
• S. McAdam, Unique factorization of monic polynomials, Comm. Alg. 29 (2001), 4341–4343.
• D.L. McQuillan, Rings of integer-valued polynomials determined by finite sets, Proc. Roy. Irish Acad. 85 (1985), 177–184.
• M. Nagata, Local rings, Intersci. Tracts Pure Appl. Math. 13, John Wiley & Sons, New York, 1962.
• G. Peruginelli, Integral-valued polynomials over sets of algebraic integers of bounded degree, J. Num. Theor. 137 (2014), 241–255.
• ––––, Integer-valued polynomials over matrices and divided differences, Monatsh. Math. 173 (2014), 559–571.
• G. Peruginelli and N. Werner, Integral closure of rings of integer-valued polynomials on algebras, in Commutative algebra: Recent advances in commutative rings, integer-valued polynomials, and polynomial functions, M. Fontana, S. Frisch and S. Glaz, eds., Springer, 2014.
• J.F. Steffensen, Note on divided differences, Danske Selsk. Math.-Fys. Medd. 17 (1939).
• B.L. van der Waerden, Algebra, Volume I., Springer-Verlag, New York, 1991.