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

 

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
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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1140666-0
PII: S 0002-9939(1993)1140666-0
Article copyright: © Copyright 1993 American Mathematical Society