On divisorial prime ideals in Prüfer v-multiplication domains
The author extends the characterization of the divisorial prime ideals in a Prüfer domain obtained by \textit{M. Fontana}, \textit{J. A. Huckaba} and \textit{I. J. Papick} in Can. Math. Bull. 27, 324--328 (1984; Zbl 0548.13011) and in J. Pure Appl. Algebra 39, 95--103 (1986; Zbl 0578.13002)] to Prüfer v-multiplication domains (PVMD). A domain is called PVMD if the finite type v-ideals of \(R\) form a group under the v-multiplication. The main results of the paper are: Proposition 0.1. Let \(R\) be an integral domain. Then the following statements are equivalent: (1) \(R\) is a PVMD; (2) \(R_ m\) is a valuation domain for each maximal \(t\)-prime ideal (= prime \(t\)-ideal) \(M\); (3) \(R\) is integrally closed and each nonzero prime ideal \(Q\) of \(R[X]\) with \(Q\cap R=0\) contains an element \(f\) with \(c_ R(f)_ v=R\). (Here \(c_ R(f)_ v\) denotes the ideal of \(R\) generated by the coefficients of \(f\); \(I_ v=(I^{-1})^{-1}\) and \(I_ t=\cup \{J_ v\mid J= \text{finitely generated subideal of}\;I\}\); an ideal is called \(I_ v\)-ideal (resp. \(I_ t\)-ideal) if \(I_ v=I\) (resp. \(I_ t=I\)).) Proposition 1.6. Let \(A\) be an integrally closed overring of \(R\) and assume that whenever \(I\) is a finitely generated ideal of \(R\) with \(I^{-1}=R\), then the inverse of \(IA\) with respect to \(A\) is \(A\). Then \(A\) is a PVMD, and, in particular, if \(P\) is a nonzero nonmaximal \(t\)-prime of \(R\), then \(P^{-1}\) is a PVMD. Equivalent conditions for the maximal \(t\)-prime ideal for being divisorial are studied in the last section of the paper.
scientific article; zbMATH DE number 3977114
scientific article