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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The Skolem property in rings
of integer-valued polynomials


Authors: Jean-Luc Chabert, Scott T. Chapman and William W. Smith
Journal: Proc. Amer. Math. Soc. 126 (1998), 3151-3159
MSC (1991): Primary 13B25; Secondary 11S05, 12J10, 13E05, 13G05
MathSciNet review: 1452796
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $D$ be an integral domain with quotient field $K$ and $E\subseteq D$. We investigate the relationship between the Skolem and completely integrally closed properties in the ring of integer-valued polynomials

\begin{displaymath}\mathrm{Int}(E,D)= \{f(X) \mid f(X) \in K[X] \text{ and } f(a)\in D \text{ for every } a\in E\}. \end{displaymath}

Among other things, we show for the case $D=\mathbb{Z}$ and $\vert E \vert = \infty$ that the following are equivalent: (1) $\text{Int}(E,\mathbb{Z})$ is strongly Skolem, (2) $ \text{Int}(E,\mathbb{Z})$ is completely integrally closed, and (3) $ \text{Int}(E,\mathbb{Z})= \text{Int}(E\backslash\{a\}, \mathbb{Z})$ for every $a\in E$.


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Additional Information

Jean-Luc Chabert
Affiliation: Faculté de Mathématiques et d’Informatique, Université de Picardie, 33 rue Saint Leu, 80 039 Amiens, France
Email: jlchaber@worldnet.fr

Scott T. Chapman
Affiliation: Department of Mathematics, Trinity University, 715 Stadium Drive, San Antonio, Texas 78212-7200
Email: schapman@trinity.edu

William W. Smith
Affiliation: Department of Mathematics, The University of North Carolina at Chapel Hill, North Carolina 27599-3250
Email: wwsmith@math.unc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04376-7
PII: S 0002-9939(98)04376-7
Keywords: Integer-valued polynomial, Skolem property, Pr\"{u}fer domain
Received by editor(s): October 10, 1996
Received by editor(s) in revised form: March 25, 1997
Additional Notes: Part of this work was completed while the third author was on leave visiting Trinity University.
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1998 American Mathematical Society