Polynomial rings over pseudovaluation rings.
type: Item
instance of: scholarly article
title: Polynomial rings over pseudovaluation rings.
zbMATH Open document ID: 1167.16021
DOI: 10.1155/2007/20138
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2008-06-03T00:00:00Z
full work available at URL: https://eudml.org/doc/55814
review text: Let \(R\) be a ring with unit element, and let \(\sigma\) be an automorphism of \(R\). Then \(R\) is called a pseudo-valuation ring (PVR) if every prime ideal \(P\) of \(R\) is strongly prime, that is, if \(aP\subseteq bR\) or \(bR\subseteq aP\) for all \(a,b\in R\); \(R\) is called \(\sigma\)-divided (divided) if every prime ideal of \(R\) is comparable, under inclusion, to every \(\sigma\)-stable ideal (ideal). Suppose that \(x\not\in P\) for every prime ideal \(P\) of the skew polynomial ring \(R[x,\sigma]\). Then \(R[x,\sigma]\) is a PVR or \(\sigma\)-divided whenever \(R\) is a commutative PVR or \(\sigma\)-divided, respectively. Similar results hold for the Ore extension \(R[x,\delta]\) of a commutative Noetherian \(\mathbb{Q}\)-algebra \(R\) with derivation \(\delta\): if \(R\) is a PVR or divided, then so is \(R[x,\delta]\). The paper concludes with an open question: If \(R\) is a commutative PVR with an automorphism \(\sigma\) and a \(\sigma\)-derivation \(\delta\), does it follow that \(R[x,\sigma,\delta]\) is a PVR?
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 5282523
zbMATH Keywords: skew polynomial rings
zbMATH Keywords: pseudovaluation rings
zbMATH Keywords: divided rings
zbMATH Keywords: strongly prime ideals
author: V. K. Bhat
MaRDI profile type: Publication
OpenAlex ID: W2002699960
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description: scientific article; zbMATH DE number 5282523