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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The Skolem property in rings of integer-valued polynomials
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by Jean-Luc Chabert, Scott T. Chapman and William W. Smith PDF
Proc. Amer. Math. Soc. 126 (1998), 3151-3159 Request permission


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 \[ \mathrm {Int}(E, D) = \{f(X) \mid f(X) \in K[X] \text { and } f(a)\in D \text { for every } a\in E\}. \] 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:
  • Scott T. Chapman
  • Affiliation: Department of Mathematics, Trinity University, 715 Stadium Drive, San Antonio, Texas 78212-7200
  • MR Author ID: 47470
  • Email:
  • William W. Smith
  • Affiliation: Department of Mathematics, The University of North Carolina at Chapel Hill, North Carolina 27599-3250
  • Email:
  • 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
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 3151-3159
  • MSC (1991): Primary 13B25; Secondary 11S05, 12J10, 13E05, 13G05
  • DOI:
  • MathSciNet review: 1452796