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

 

A classification of all $D$
such that $Int(D)$ is a Prüfer domain


Author: K. Alan Loper
Journal: Proc. Amer. Math. Soc. 126 (1998), 657-660
MSC (1991): Primary 13F05, 13F20; Secondary 13B25, 11C08
MathSciNet review: 1459137
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Abstract: Let $D$ be an integral domain with quotient field $K$. The ring of integer-valued polynomials $Int(D)$ over $D$ is defined by $Int(D) = \{f(x) \in K[x] \mid f(D) \subseteq D\}$. It is known that if $Int(D)$ is a Prüfer domain, then $D$ is an almost Dedekind domain with all residue fields finite. This condition is necessary and sufficient if $D$ is Noetherian, but has been shown to not be sufficient if $D$ is not Noetherian. Several authors have come close to a complete characterization by imposing bounds on orders of residue fields of $D$ and on normalized values of particular elements of $D$. In this note we give a double-boundedness condition which provides a complete charaterization of all integral domains $D$ such that $Int(D)$ is a Prüfer domain.


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Additional Information

K. Alan Loper
Affiliation: Department of Mathematics, Ohio State University-Newark, Newark, Ohio 43055
Email: lopera@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04459-1
PII: S 0002-9939(98)04459-1
Received by editor(s): August 26, 1996
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1998 American Mathematical Society