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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 $a\in E$.

References [Enhancements On Off] (What's this?)

  • 1. D.D. Anderson, D.F. Anderson and M. Zafrullah, Rings between $D[X]$ and $K[X]$, Houston Math. J. 17 (1991), 109-129. MR 92c:13014
  • 2. D. Brizolis, A theorem on ideals in Prüfer rings of integer-valued polynomials, Comm. Algebra 7 (1979), 1065-1077. MR 80j:13013
  • 3. P.-J. Cahen, Polynômes à valeurs entières, Thèse, Université Paris XI, Orsay, 1973.
  • 4. P.-J. Cahen, Integer-valued Polynomials on a Subset, Proc. Amer. Math. Soc. 117 (1993), 919-929. MR 93e:13011
  • 5. P.-J. Cahen and J.-L. Chabert, Integer-Valued Polynomials, Mathematical Surveys and Monographs, American Mathematical Society, Providence, 48 (1997). CMP 97:04
  • 6. J.-L. Chabert, Anneaux de "polynômes à valeurs entières" et anneaux de Fatou, Bull. Soc. Math. France 99 (1971), 273-283. MR 46:1780
  • 7. J.-L. Chabert, Un anneau de Prüfer, J. Algebra 107 (1987) 1-16. MR 88i:13022
  • 8. J.-L. Chabert, Une caractérisation des polynômes prenant des valeurs entières sur tous les nombres premiers, Canad. Math. Bull. 39 (1996), 402-407. MR 97h:11024
  • 9. J.-L. Chabert, S. Chapman and W. Smith, A Basis for the Ring of Polynomials Integer-Valued on Prime Numbers to appear Factorization in Integral Domains, Marcel Dekker.
  • 10. J.-L. Chabert, S. Chapman and W. Smith, Algebraic Properties of the Ring of Polynomials Integer-Valued on Prime Numbers, Comm. Algebra. 25 (1997), 1945-1959. CMP 97:11
  • 11. R. Gilmer, Sets that determine integer-valued polynomials, J. Number Theory 33 (1989), 95-100. MR 90g:11142
  • 12. R. Gunji and D. McQuillan, On rings with a certain divisibility property, Michigan Math. J. 22 (1975), 289-299. MR 53:405
  • 13. D.L. McQuillan, On Prüfer domains of polynomials, J. Reine Angew. Math. 358 (1985), 162-178. MR 86k:13019
  • 14. D.L. McQuillan, Rings of integer-valued polynomials determined by finite sets, Proc. R. Ir. Acad. 85A (1985), 177-184. MR 87g:13016
  • 15. D.L. McQuillan, On a Theorem of R. Gilmer, J. Number Theory 39 (1991), 245-250. MR 92i:13016
  • 16. A. Ostrowski, Über ganzwertige Polynome in algebraischen Zahlköpern, J. Reine Angew. Math. 149 (1919), 117-124.
  • 17. G. Pólya, Ueber ganzwertige ganze Funktionen, Rend. Circ. Matem. Palermo 40 (1915), 1-16.
  • 18. G. Pólya, Über ganzwertige Polynome in algebraischen Zahlköpern, J. Reine Angew. Math. 149 (1919), 97-116.
  • 19. G. Pólya and G. Szegö, Aufgaben und Lehrsätze aus der Analysis, Vol. II, Springer-Verlag, Berlin, 1925.
  • 20. T. Skolem, Ein Satz über ganzwertige Polynome, Norske Vid. Selsk (Trondheim) 9 (1936), 111-113.

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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

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

William W. Smith
Affiliation: Department of Mathematics, The University of North Carolina at Chapel Hill, North Carolina 27599-3250

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

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