A classification of all $D$ such that $\mathrm \{Int\}(D)$ is a Prüfer domain
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- by K. Alan Loper PDF
- Proc. Amer. Math. Soc. 126 (1998), 657-660 Request permission
Abstract:
Let $D$ be an integral domain with quotient field $K$. The ring of integer-valued polynomials $\mathrm {Int}(D)$ over $D$ is defined by $\mathrm {Int}(D) = \{f(x) \in K[x] \mid f(D) \subseteq D\}$. It is known that if $\mathrm {Int}(D)$ 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 $\mathrm {Int}(D)$ is a Prüfer domain.References
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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
- Received by editor(s): August 26, 1996
- Communicated by: Wolmer V. Vasconcelos
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 657-660
- MSC (1991): Primary 13F05, 13F20; Secondary 13B25, 11C08
- DOI: https://doi.org/10.1090/S0002-9939-98-04459-1
- MathSciNet review: 1459137