Prüfer v-multiplication domains and the ring \(R[X]_{N_ v}\)
For the set of fractional ideals A of an integral domain \(R\subseteq K\), the quotient field, a *-operation is a map \(A\mapsto A_*\) with \(R=R_*\), \((aA)_*=aA_*\), \(A\subseteq A_*\), \(A\subseteq B\) implies \(A_*\subseteq B_*\) and \((A_*)_*=A_*\). With any *-operation is associated a second one, \(*_ s\), given by \(A_{*_ s}=\cup \{B_*| \quad A\supseteq B\) is finitely generated\(\}\). For \(f=\sum^{n}_{i=0}a_ iX^ i \in K[X]\), the fractional ideal \(\sum Ra_ i \) is denoted by \(A_ f\) and \(\{f\in R[X]| \quad (A_ f)_*=R\}\) by \(N_*(R)\). The general theme of this note is to relate the multiplicative ideal theory of R to that of \((R[X])_{N_*}\) especially when the *-operation is v given by \(A_ v=(A^{-1})^{-1}\). It is shown, for example, that (i) when \(*=*_ s\), an ideal I of R is *-invertible if, and only if, I R[X]\({}_{N_*}\) is invertible, and (ii) for all *, the invertible ideals of \(R[X]_{N_*}\) are all principal. When \(*=v\) several criteria for R to be a PVMD are given in terms of \(R[X]_{N_ v}\). The subintersections of a PVMD R are characterized as those overrings D with \(N_ v(R)\subseteq N_ v(D).\) In the final section the notion of a t-almost Dedekind domain is introduced, where \(t=v_ s\), as a domain R such that \(R_ M\) is a DVR for every maximal t-ideal M. It is shown that R is t-almost Dedekind if, and only if, \(R[X]_{N_ v}\) is almost Dedekind; and the equivalence to t-almost Dedekind is established for each of five other properties.
scientific article; zbMATH DE number 4093601
scientific article