Pseudo-almost valuation domains are quasilocal going-down domains, but not conversely
type: Item
instance of: scholarly article
title: Pseudo-almost valuation domains are quasilocal going-down domains, but not conversely
zbMATH Open document ID: 1164.13003
author: David E. Dobbs
publication date: 2008-06-25T00:00:00Z
review text: This note proves a property of a pseudo-almost valuation domain (for short, PAVD), a concept defined and explored by \textit{A. Badawi} [Commun. Algebra 35, No. 4, 1167--1181 (2007; Zbl 1113.13001)] as follows. Let \(R\) be a domain with fraction field \(K\). For a prime ideal \(P\) of \(R\) one says that \(P\) is pseudo-strongly prime ideal if whenever \(xy P \subseteq P,\) with \(x,y \in K\), there exists \(n=n(x,y)\) such that \(x^n \in R\) or \(y^n P \subseteq P\). If each prime \(P\) of \(R\) is a pseudo-strongly prime ideal, one says that \(R\) is a pseudo-almost valuation domain (or PAVD). A going-down domain is a domain such that every overring of it satisfies the going-down property. The main result of the paper is that every PAVD is a quasilocal going-down domain. Furthermore, examples are provided to show that the class of quasilocal going-down domains sits strictly between the class of PAVDs and that of domains having linearly ordered primes.
reviewed by: Florian Enescu
Mathematics Subject Classification ID: 13B24
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 5292721
zbMATH Keywords: Pseudo-almost valuation domain
zbMATH Keywords: Prime ideal
zbMATH Keywords: Going down domain
zbMATH Keywords: Divide domain
zbMATH Keywords: Quasilocal
zbMATH Keywords: Valuation overring
zbMATH Keywords: Root extension
zbMATH Keywords: Seminormal
zbMATH Keywords: \(D+M\) construction
zbMATH Keywords: Krull dimension
MaRDI profile type: Publication
full work available at URL: s12215-008-0006-7
OpenAlex ID: W1978521973
cites work: On Pseudo-Almost Valuation Domains
cites work: POWERFUL IDEALS, STRONGLY PRIMARY IDEALS, ALMOST PSEUDO-VALUATION DOMAINS, AND CONDUCIVE DOMAINS
cites work: Inside factorial monoids and integral domains
cites work: On Going Down For Simple Overrings II
cites work: Divided rings and going-down
cites work: Q4168695
cites work: On Going-Down for Simple Overrings. III
cites work: Q4163618
cites work: Q3205350
cites work: Q4049183
DOI: 10.1007/S12215-008-0006-7
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 5292721