Prüfer v-multiplication domains and the ring \(R[X]_{N_ v}\)
type: Item
instance of: scholarly article
title: Prüfer v-multiplication domains and the ring \(R[X]_{N_ v}\)
zbMATH Open document ID: 0668.13002
DOI: 10.1016/0021-8693(89)90040-9
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 13A05
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 4093601
zbMATH Keywords: Prüfer v-multiplication domain
zbMATH Keywords: fractional ideals
zbMATH Keywords: *-operation
zbMATH Keywords: multiplicative ideal theory
zbMATH Keywords: PVMD
zbMATH Keywords: almost Dedekind domain
author: B. G. Kang
reviewed by: David Kirby
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90040-9
OpenAlex ID: W2042638261
cites work: Multiplication Ideals, Multiplication Rings, and the Ring R(X)
cites work: Q4149551
cites work: On the Ideal Theory of the Kronecker Function Ring and the Domain D(X)
cites work: On flat overrings, ideal transforms and generalized transforms of a commutative ring
cites work: Q5661989
cites work: Some Results on v-Multiplication Rings
cites work: Some remarks on star-operations
cites work: On divisorial prime ideals in Prüfer v-multiplication domains
cites work: Characterizations of *-Multiplication Domains
cites work: A Localization of R[x]
cites work: Q3283972
cites work: Q5603302
cites work: On a question posed by Huckaba-Papick
cites work: Generalizations of Multiplicative Ideal Theory to Commutative Rings with Zerodivisors
cites work: On Pruefer v-multiplication domains
cites work: Q5623808
cites work: Idéaux divisoriels d'un anneau de polynômes
cites work: Some polynomial characterizations of Prüfer v-multiplication domains
description: scientific article; zbMATH DE number 4093601