Prüfer domains and rings of integer-valued polynomials (Q581604)

Let \(R\) be an integral domain with quotient field \(K\), and let \(\text{Int}(D)\) be the ring of integer-valued polynomials on \(D\), i.e. \(\text{Int}(D)=\{f(X)\in K[X]: f(D)\subseteq D\}\). The special case where \(D\) is the ring of integers in a finite algebraic number field \(K\) was originally considered by \textit{G. Pólya} [J. Reine Angew. Math. 149, 97-116 (1919; JFM 47.0163.04)] and \textit{A. Ostrowski} [ibid. 117-124 (1919; JFM 47.0163.05)]. More recently \textit{D. Brizolis} [Commun. Algebra 7, 1065-1077 (1979; Zbl 0422.13011)] showed that in the finite number field case, \(\text{Int}(D)\) is a Prüfer domain, and he asked for equivalent conditions under which \(\text{Int}(D)\) is a Prüfer domain for an arbitrary domain D. For D Noetherian, \textit{J.-L. Chabert} [J. Algebra 107, 1-16 (1987; Zbl 0635.13004)] has shown that \(\text{Int}(D)\) is a Prüfer domain if and only if \(D\) is a Dedekind domainwith finite residue fields; the same result can be obtained from work of \textit{D. L. McQuillan} [J. Reine Angew. Math. 358, 162-178 (1985; Zbl 0568.13003)]. In the general case Chabert has observed that if \(\text{Int}(D)\) is a Prüfer domain, then D is an almost Dedekind domain with finite residue fields. Examples of non- Noetherian almost Dedekind domains are known. Three specific examples [of \textit{N. Nakano}, J. Sci. Hiroshima Univ., Ser. A 16, 425-439 (1953; Zbl 0052.27502), of \textit{W. Heinzer} and \textit{J. Ohm}, Trans. Am. Math. Soc. 158, 273-284 (1971; Zbl 0223.13017), and of \textit{A. Grams}, Proc. Camb. Philos. Soc. 75, 321-329 (1974; Zbl 0287.13002)] and three general constructions of such integral domains, which involve integral closure in an infinite algebraic field extension, Kronecker function rings, and monoid rings, are considered in section 2 of the present paper. It is shown that \(Int(D)=D[X]\), and hence \(\text{Int}(D)\) is not a Prüfer domain if \(D\) is the integral domain in any of the mentioned constructions (theorem 2, proposition 4 and theorem 5). None of the non-Noetherian almost Dedekind domains considered in this section is with finite residue fields. An example of a non-Noetherian almost Dedekind domain with finite residue fields is obtained in section 3 by a construction involving valuation domains (construction K and theorem 10), which stems from work of \textit{J. T. Arnold} and \textit{R. Gilmer} [J. Sci. Hiroshima Univ., Ser. A-I 31, 131-145 (1967; Zbl 0268.13001)]. Using theorem 10, the author gives sufficient conditions for \(\text{Int}(D)\) to be a Prüfer domain, where \(D\) is the integral closure of a semilocal PID in an infinite algebraic extension of its quotient field (theorem 12). The author also shows, that under appropriate hypotheses, unboundedness of the set \(\{| D/M|: M\in \text{MaxSpec}(R)\}\) implies that \(\text{Int}(D)\) is not a Prüfer domain (theorem 13). Using theorem 13, the author produces an almost Dedekind domain \(D\) with finite residue fields such that \(\text{Int}(D)\) is not a Prüfer domain(example 14). The paper ends with two open questions ((Q4) and (Q5)). The first one is a quite general question: is \(\text{Int}(D)\) a Prüfer domain, if \(D\) is almost Dedekind domain such that \(\{| D/M|: M\in \text{MaxSpec}(D)\}\) is bounded, and the second relates to the construction of almost Dedekind domains considered in this section.