Two Prüfer domain counterexamples
type: Item
instance of: scholarly article
title: Two Prüfer domain counterexamples
zbMATH Open document ID: 0946.13016
published in: Journal of Algebra
publication date: 2000-10-18T00:00:00Z
review text: In this paper examples are constructed to show (1) a Prüfer domain \(D\) whose maximal ideals are principal but \(D\) itself is not a Bezout domain, (2) a three-dimensional Prüfer domain \(D\) which is the intersection of a collection \(V_i\) of valuation subrings of \(\mathbb{Q}(x,y)\), the field of rational functions in two variables over \(\mathbb{Q}\), with \(V_1\) a three-dimensional, and \(V_i\), \(i>1\), one-dimensional valuation rings. Denoting by \(M_x\) and \(M_{(x,y)}\) the height 1 and height 2 prime ideal contained in the contraction \(M_1\) of \(D\) in \(V_1\), the author shows that \(M_{(x,y)}\) is divisorial while \(M_x\) is not.
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 1400112
zbMATH Keywords: Bezout domain
zbMATH Keywords: Prüfer domain
author: K. Alan Loper
reviewed by: N. Sankaran
MaRDI profile type: Publication
Wikidata QID: Q124855978
full work available at URL: jabr.1999.8010
OpenAlex ID: W1989722953
cites work: Priifer domains with class group generated by the classes of the invertible maximal ideals
cites work: Q5687544
cites work: Q3137901
cites work: Irredundant intersections of valuation rings
cites work: On Prüfer non-\(D\)-rings
DOI: 10.1006/JABR.1999.8010
description: scientific article; zbMATH DE number 1400112