Pseudo-valuation rings and \(C(X)\)
type: Item
instance of: scholarly article
title: Pseudo-valuation rings and \(C(X)\)
zbMATH Open document ID: 1440.13096
author: Warren Wm. McGovern
author: Lee Klingler
published in: Journal of Algebra
publication date: 2018-08-10T00:00:00Z
review text: A pseudo-valuation domain (in short a PVD) is a local integral domain \(D\) with maximal ideal \(M\) and field of fractions \(K\) such that the set \(M:M=\{x\in K\; ; \; xM\subseteq M\}\) is a valuation domain. The ring \(M:M\) being a valuation domain is equivalent to the following condition: for every \(a,b\) in \(D\) and \(c\) in \(M\), \(a\) divides \(bc\) or \(b\) divides \(ac\) (note that \(D\) is a valuation domain if, and only if, for every \(a,b\) in \(D\), \(a\) divides \(b\) or \(b\) divides \(a\)). After a survey where they are placed in a more general context, the authors consider the ring \(C(\Omega)\) of real valued continuous functions on a compact Hausdorff space \(\Omega\). Let \(P\) be a prime ideal of \(C(\Omega)\). The aim of this paper is to investigate when \(C(\Omega)/P\) is a valuation domain or a pseudo-valuation domain. Now, \(C(\Omega)/P\) is a bounded real closed domain. So the author study more generally the case of bounded real closed domain. Before to state the main results of this paper, we need to list some definitions. An \(f\)-ring is a lattice-ordered ring which is a subdirect product of totally ordered rings. The set of bounded elements of an \(f\)-ring \(A\), with identity element \(1\), is the set \(A^*=\{a\in A\; ; \; \exists n\in \mathbb{N},\; |a|\leq n\}\). If \(A=A^*\), then \(A\) is said to be bounded. Let \(m\in \mathbb{N}\); an \(f\)-ring \(A\) is \(m\) convex if, for every \(a,b\) in \(A\) such that \(0\leq a\leq b^m\), \(b\) divides \(a\). Now, a real closed domain is a domain \(D\) which satisfies the following conditions. For each prime ideal \(Q\), the domain \(D/Q\) is integrally closed, and its quotient field is a real closed field; the squares of \(D\) form a positive cone of a partial order which makes \(D\) into a \(2\)-convex \(f\)-ring. A previous result showed that, for every \(a,b\) in a real closed domain, there exists \(n\in \mathbb{N}\) such that \(a\) divides \(b\) or \(b\) divides \(a^n\). If \(K\) is…
reviewed by: Gerard Leloup
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 13J25
Mathematics Subject Classification ID: 54C40
zbMATH DE Number: 6915740
zbMATH Keywords: pseudo-valuation domain
zbMATH Keywords: real closed ring
zbMATH Keywords: ring of continuous functions
zbMATH Keywords: compact Hausdorff space
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2018.07.006
OpenAlex ID: W2883349806
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DOI: 10.1016/J.JALGEBRA.2018.07.006
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description: scientific article; zbMATH DE number 6915740