The failure of the Artin-Rees property for the Jacobson radical in prime Noetherian rings.
type: Item
instance of: scholarly article
title: The failure of the Artin-Rees property for the Jacobson radical in prime Noetherian rings.
zbMATH Open document ID: 1264.16022
author: A. Braun
author: C. R. Hajarnavis
published in: Journal of Algebra
publication date: 2012-04-30T00:00:00Z
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16R20
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 6029095
zbMATH Keywords: noncommutative Noetherian rings
zbMATH Keywords: weak Artin-Rees property
zbMATH Keywords: Jacobson radical
zbMATH Keywords: Noetherian prime PI-rings
zbMATH Keywords: Krull dimension
zbMATH Keywords: trace rings
zbMATH Keywords: Jacobson conjecture
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.09.023
OpenAlex ID: W2040760867
cites work: An additivity principle for p.i. rings
cites work: Q4025942
cites work: Symmetry and localization in Noetherian prime PI rings
cites work: Q3154382
cites work: Q3992979
cites work: Q4040049
cites work: Q3910639
DOI: 10.1016/J.JALGEBRA.2011.09.023
review text: A two-sided ideal \(I\) of a ring \(R\) is said to satisfy the (weak) right Artin-Rees property if for each right ideal \(E\) of \(R\) there exists an integer \(n\geq 1\) such that \(E\cap I^n\subseteq EI\). In contrast to the commutative case, this property does not hold for most ideals in a noncommutative Noetherian ring \(R\), but for a long time it was hoped that it might hold for the Jacobson radical \(J\) of \(R\), in which case Jacobson's Conjecture \(\bigcap_{n=1}^\infty J^n=0\) would also be true as a consequence.NEWLINENEWLINE In the paper under review, the authors put an end to this hope by constructing a Noetherian prime PI-ring of Krull dimension 3 which is a finitely generated module over its centre and whose Jacobson radical does not satisfy the Artin-Rees property. A semiprime homomorphic image of this ring provides a counterexample of Krull dimension 2. Interestingly, the Artin-Rees property is shown to hold for the Jacobson radical of any Noetherian prime PI-ring which has Krull dimension at most 2, or is equal to its trace ring.
description: scientific article; zbMATH DE number 6029095