Another Prüfer ring of integer-valued polynomials
Let \(D\) be an integral domain with quotient field \(K\) and let \(\text{ Int}(D) = \{f \in K [x] \mid f(D) \subseteq D\}\). If \(D\) is Dedekind and all its residue fields are finite then \(\text{ Int}(D)\) is Prüfer while if \(\text{ Int}(D)\) is Prüfer then \(D\) is almost Dedekind (i.e. for every prime ideal \(P\), \(D_P\) is a Noetherian valuation domain) and its residue fields are all finite. Furthermore Noetherian almost Dedekind domains are Dedekind. In this paper the author constructs an example of a non-Noetherian almost Dedekind domain \(D\) such that \(\text{ Int}(D)\) is Prüfer and for which the quotient field \(K\) is an infinite degree transcendental extension of \(\mathbb{Q}\). Earlier examples of this type [\textit{R. W. Gilmer}, J. Algebra 129, 502-517 (1990; Zbl 0689.13009 ), \textit{J.-L. Chabert}, Proc. Am. Math. Soc. 118, No. 4, 118, 1061-1073 (1993; Zbl 0781.13014)] had employed infinite degree algebraic extensions. For background on this topic the reader may consult the recent book by \textit{P. J. Cahen} and \textit{J.-L. Chabert}, ``Integer valued polynomials'' (Providence R.I. 1997) and for further results, \textit{K. A. Loper}'s paper ``A classification of all \(D\) such that \(\text{ Int}(D)\) is a Prüfer domain'' [Proc. Am. Math. Soc. 126, No. 3, 657-660 (1998)].
scientific article; zbMATH DE number 1014027