Integer-valued polynomials and Prüfer \(v\)-multiplication domains
If \(D\) is a domain with quotient field \(K\) then the ring of integer valued polynomials over \(D\) is \(\text{Int}(D)=:\{f \in K[X] \mid f(D) \subseteq D\}\). This paper is devoted to relating certain properties of \(\text{Int}(D)\) to those of \(D\). A domain, \(D\), is Prüfer if for each prime ideal, \(P\), \(D_P\) is a valuation domain. It is a Prüfer \(v\)-multiplication domain (P\(v\)MD) if \(D_P\) is a valuation domain for each \(t\)-prime ideal, \(P\), i.e., prime ideal which is equal to its \(t\)-closure. \textit{K. A. Loper} [Proc. Am. Math. Soc. 126, No. 3, 657-660 (1998; Zbl 0887.13010)] characterized those domains for which \(\text{Int}(D)\) is Prüfer. The current paper does the same for P\(v\)MD's. The authors partition the collection of prime ideals, \(\text{spec}(D)\), into the set \(\Delta_0\) of int-prime ideals, \(P\), for which \(\text{Int}(D) \not\subseteq D_P[X]\) and \(\Delta_1\), of polynomial prime ideals, \(P\), for which \(\text{Int}(D) \subseteq D_P[X]\). The main result is that \(\text{Int}(D)\) is a P\(v\)MD if and only if \(D\) is a P\(v\)MD, each element of \(\Delta_0\) is a height \(1\) (maximal) ideal and each element of \(\Delta_1\) which is a \(t\)-prime ideal contains a finitely generated ideal not contained in any element of \(\Delta_0\). If we let \(D_0 = \bigcap_{\Delta_0} D_P\) and \(D_1 = \bigcap_{\Delta_1} D_p\) then it is also shown that \(\text{Int}(D) = \text{Int}(D_0) \cap D_1 [X]\) and that if \(\text{Int}(D)\) is a P\(v\)MD, then \(\text{Int}(D_0)\) is Prüfer.
scientific article; zbMATH DE number 1448610
scientific article