Irredundant intersections of valuation overrings of two-dimensional noetherian domains
type: Item
instance of: scholarly article
title: Irredundant intersections of valuation overrings of two-dimensional noetherian domains
zbMATH Open document ID: 1137.13003
author: Bruce Olberding
published in: Journal of Algebra
publication date: 2008-02-14T00:00:00Z
review text: Throughout the paper \(D\) is a noetherian domain of Krull dimension 2. Any ring \(R\) included between \(D\) and its quotient field is called an overring. Denote two sets of valuation overrings of \(D\) by \(\Sigma\) and \(\Gamma\). The author is interested in investigation when (with various irredundancy conditions) the condition \((\bigcap_{V\in\Sigma}V)\cap R=(\bigcap_{W\in\Gamma}W)\cap R\) implies \(\Sigma=\Gamma\). A number of technical conditions are introduced to this end. Thus it is shown for instance, that if \(R\) is integrally closed, the above intersections are strongly irredundant (none of the \(V\) or \(W\) can be replaced by their proper overrings), and every member of \(\Sigma\cup\Gamma\) is of Krull dimension 2, then \(\Sigma=\Gamma\). The latter condition may be curtailed, if instead, another condition is imposed: that \(\Sigma\) and \(\Gamma\) are noetherian subspaces of the Zariski-Riemann space of the quotient field of \(D\). A number of examples are given to illustrate various points. Among them an example where it is shown that these results cannot be extended to overrings of noetherian domains of dimension 3.
Mathematics Subject Classification ID: 13A18
zbMATH DE Number: 5235097
zbMATH Keywords: valuation theory
zbMATH Keywords: noetherian domain
zbMATH Keywords: overring
zbMATH Keywords: irredundant representation
zbMATH Keywords: strongly irredundant representation
zbMATH Keywords: Zariski-Riemann space
zbMATH Keywords: Krull dimension 2
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2007.07.030
OpenAlex ID: W1984076520
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DOI: 10.1016/J.JALGEBRA.2007.07.030
description: scientific article