Some results on pseudo-valuation domains
type: Item
instance of: scholarly article
title: Some results on pseudo-valuation domains
zbMATH Open document ID: 0558.13001
author: Akira Okabe
published in: Tsukuba Journal of Mathematics
publication date: 1984-01-01T00:00:00Z
review text: Let D be a domain, with quotient field K. An ideal P of D is strongly prime whenever: if x, \(y\in K\) and xy\(\in P\) then \(x\in P\) or \(y\in P\). If every prime ideal of D is strongly prime, D is called a pseudo- valuation domain (PVD). Every valuation domain is a PVD, every PVD D has only one maximal ideal M and \((D:M)_ K=\{x\in K| xM\subseteq D\}\) is a valuation domain V, \(V\supseteq D\), with maximal ideal M. This definition and results were given by Hedstrom and Houston. In the present paper, the author indicates more properties of PVD's. For example: (1) if \(P\subseteq I\) are ideals of a PVD D and P is prime then P is also a prime ideal of \((I:I)_ K\); (2) P is the unique maximal ideal of \((P:P)_ K\); (3) if \(P\neq M\) then \((P:P)_ K=D_ P\) and \(P=(D:D_ P)_ D=\{d\in D| dD_ P\subseteq D\};\) (4) if P is properly contained in the ideal \(I\neq D\) then \(P\subseteq \cap^{\infty}_{n=1}I^ n;\) (5) the mapping \(D'\to (P:D')_ D=\{d\in D| dD'\subseteq P\}\) establishes a bijection between the set of all overrings of D properly containing \((P:P)_ K\) and the set of all prime ideals P' of D, properly contained in P. An example shows that (5) is not true if D is not assumed to be a PVD.
Mathematics Subject Classification ID: 14E22
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 13B30
Mathematics Subject Classification ID: 13F99
zbMATH DE Number: 3889674
zbMATH Keywords: conductor overring
zbMATH Keywords: pseudo-valuation domain
zbMATH Keywords: PVD
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496160045
OpenAlex ID: W311550302
DOI: 10.21099/TKBJM/1496160045
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description: scientific article; zbMATH DE number 3889674