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Integer-valued polynomials, Prüfer domains, and localization


Author: Jean-Luc Chabert
Journal: Proc. Amer. Math. Soc. 118 (1993), 1061-1073
MSC: Primary 13F05; Secondary 13F20, 13G05
DOI: https://doi.org/10.1090/S0002-9939-1993-1140666-0
MathSciNet review: 1140666
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Abstract: Let $ A$ be an integral domain with quotient field $ K$ and let $ \operatorname{Int} (A)$ be the ring of integer-valued polynomials on $ A:\{ P \in K[X]\vert P(A) \subset A\} $. We study the rings $ A$ such that $ \operatorname{Int} (A)$ is a Prüfer domain; we know that $ A$ must be an almost Dedekind domain with finite residue fields. First we state necessary conditions, which allow us to prove a negative answer to a question of Gilmer. On the other hand, it is enough that $ \operatorname{Int} (A)$ behaves well under localization; i.e., for each maximal ideal $ \mathfrak{m}$ of $ A$, $ \operatorname{Int} {(A)_\mathfrak{m}}$ is the ring $ \operatorname{Int} ({A_\mathfrak{m}})$ of integer-valued polynomials on $ {A_\mathfrak{m}}$. Thus we characterize this latter condition: it is equivalent to an "immediate subextension property" of the domain $ A$. Finally, by considering domains $ A$ with the immediate subextension property that are obtained as the integral closure of a Dedekind domain in an algebraic extension of its quotient field, we construct several examples such that $ \operatorname{Int} (A)$ is Prüfer.


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

  • [1] D. Brizolis, A theorem on ideals in Prüfer rings of integral-valued polynomials, Comm. Algebra 7 (1979), 1065-1077. MR 533204 (80j:13013)
  • [2] P.-J. Cahen, Polynômes à valeurs entières, Canad. J. Math. 24 (1982), 747-754. MR 0309923 (46:9027)
  • [3] P.-J. Cahen and J.-L. Chabert, Coefficients et valeurs d'un polynôme, Bull. Soc. Math. 95 (1971), 295-304. MR 0296065 (45:5126)
  • [4] J.-L. Chabert, Anneaux de "polynômes à valeurs entières" et anneaux de Fatou, Bull. Soc. Math. France 99 (1971), 273-283. MR 0302636 (46:1780)
  • [5] -, Un anneau de Prüfer, J. Algebra 107 (1987), 1-17. MR 883865 (88g:20115)
  • [6] J.-L. Chabert and G. Gerboud, Polynômes à valeurs entières et binômes de Fermat, Canad. J. Math. (to appear). MR 1200319 (94c:13020)
  • [7] R. Gilmer, Integral domains which are almost Dedekind, Proc. Amer. Math. Soc. 15 (1964), 813-818. MR 0166212 (29:3489)
  • [8] -, Multiplicative ideal theory, Dekker, New York, 1972. MR 0427289 (55:323)
  • [9] -, Prüfer domains and rings of integer-valued polynomials, J. Algebra 129 (1990), 502-517. MR 1040951 (91b:13023)
  • [10] H. Hasse, Zwei Existenztheoreme über algebraische Zahlkörper, Math. Ann. 95 (1925), 229-238.
  • [11] W. Krull, Über einen Existenzsatz der Bewertungstheorie, Abh. Math. Sem. Univ. Hamburg 23 (1959), 29-35. MR 0104653 (21:3406)
  • [12] A. Ostrowski, Über ganzwertige polynome in algebraischen Zahlkörpern, J. Reine Angew. Math. 149 (1919), 117-124.
  • [13] G. Pólya, Über ganzwertige polynome in algebraischen Zahlkörpern, J. Reine Angew. Math. 149 (1919), 97-116.

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DOI: https://doi.org/10.1090/S0002-9939-1993-1140666-0
Article copyright: © Copyright 1993 American Mathematical Society

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