Ore extensions of Jacobson rings.
type: Item
instance of: scholarly article
title: Ore extensions of Jacobson rings.
zbMATH Open document ID: 1307.16025
published in: Journal of Algebra
publication date: 2014-09-04T00:00:00Z
reviewed by: Stefan Veldsman
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16P40
zbMATH DE Number: 6339512
zbMATH Keywords: Jacobson rings
zbMATH Keywords: Ore extensions
author: Alireza Nasr-Isfahani
author: A. Moussavi
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2014.06.011
OpenAlex ID: W2040716096
cites work: Radicals of crossed products of enveloping algebras
cites work: Q3957097
cites work: A Note on Jacobson Rings and Polynomial Rings
cites work: Ore Extensions and Polycyclic Group Rings
cites work: Skew Polynomial Extensions of Commutative Noetherian Jacobson Rings
cites work: Q3154382
cites work: Prime ideals of Ore extensions over commutative rings
cites work: Generic flatness and the nullstellensatz for ore extensions
cites work: Noetherian Ore Extensions and Jacobson Rings
cites work: Q4146866
cites work: The Nullstellensatz and Jacobson Properties for Rings of Differential Operators
cites work: On the semiprimitivity of skew polynomial rings
cites work: Q4027067
cites work: A skew polynomial ring over a jacobson ring need not be a jacobson ring
cites work: Skew Polynimials and Jacobson Rings
cites work: Polynomial extensions of Jacobson rings
DOI: 10.1016/J.JALGEBRA.2014.06.011
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review text: For a Jacobson ring \(R\), it is well-known that \(R[x]\) is a Jacobson ring, but this is not necessarily true for the skew polynomial extension \(R[x;\alpha]\). For an automorphism \(\alpha\) conditions are known when it will be the case. In this paper the authors consider this problem and add to the complexity by requiring that \(\alpha\) is only a monomorphism and not necessarily surjective.NEWLINENEWLINE Fix a monomorphism \(\alpha\colon R\to R\). An ideal \(I\) of \(R\) is called strongly \(\alpha\)-prime if \(\alpha^{-1}(I)=I\) and for all ideals \(B\) and \(C\) of \(R\) with \(\alpha(C)\subseteq C\) and \(BC\subseteq I\), it follows that \(B\subseteq I\) or \(C\subseteq I\). The ring \(R\) is called strongly \(\alpha\)-prime if \(\{0\}\) is a strongly \(\alpha\)-prime ideal of \(R\).NEWLINENEWLINE A ring \(R\) may satisfy the following conditions:NEWLINENEWLINE \((A_1)\) For each strongly \(\alpha\)-prime ideal \(I\) of \(R\), \((R/I)[x;\alpha]\) is semiprimitive.NEWLINENEWLINE \((A_2)\) For each strongly \(\alpha\)-prime ideal \(I\) of \(R\), \(R/I\) is semiprime.NEWLINENEWLINE \((A_3)\) For each prime ideal \(P\) of \(R[x;\alpha]\) with \(x\notin P\), \(x+P\) is a regular element of \(R[x;\alpha]/P\).NEWLINENEWLINE The main result shows that if a Jacobson ring \(R\) satisfies these three conditions, then \(R[x;\alpha]\) is a Jacobson ring. This result covers the known cases for this conclusion. It is also shown that any left Noetherian ring fulfills these three conditions.
description: scientific article; zbMATH DE number 6339512