Localization, completion and the AR property in Noetherian P.I. rings
type: Item
instance of: scholarly article
title: Localization, completion and the AR property in Noetherian P.I. rings
zbMATH Open document ID: 0873.16015
DOI: 10.1007/BF02785536
author: A. Braun
published in: Israel Journal of Mathematics
publication date: 1997-10-27T00:00:00Z
review text: Let \(R\) be a Noetherian ring satisfying the second layer condition, (for example, a group algebra of a polycyclic-by-finite group, an enveloping algebra of a solvable Lie algebra, or a P.I. ring), and let \(P\) be a prime ideal of \(R\). This paper addresses the relation between the Ore localization of \(P\) and the Noetherian property of the completion \(\widehat R:=\lim_{\gets i}R/P^i\). The idea that these two conditions should be related can be traced back to \textit{A. W. Goldie} [J. Algebra 5, 89-105 (1967; Zbl 0154.28801)]. The central result of the present paper is the following: Let \(R\) and \(P\) be as above, and suppose that \(P\) is semiprimitive, that it is contained in a localizable semiprime ideal, and that \(\lim_{\gets i}R/P^i\) is Noetherian. Then \(P\) is Ore localizable. The author explores consequences of this for enveloping algebras and for P.I. rings, and considers the extent to which the converse fails to be true.
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16U20
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 997617
zbMATH Keywords: AR property
zbMATH Keywords: PI rings
zbMATH Keywords: Noetherian rings
zbMATH Keywords: second layer condition
zbMATH Keywords: Ore localizations
zbMATH Keywords: completions
zbMATH Keywords: localizable semiprime ideals
zbMATH Keywords: enveloping algebras
reviewed by: Kenneth A. Brown
MaRDI profile type: Publication
cites work: Completions of Noetherian P.I. rings
cites work: Q4025942
cites work: On the Representation Theory of Solvable Lie Algebras
cites work: On the Representation Theory of Solvable Lie Algebras Ii: The Abelian Group Attached to a Prime Ideal
cites work: Q3037624
cites work: Localization in prime Noetherian P.I. rings
cites work: Symmetry and localization in Noetherian prime PI rings
cites work: Localization in non-commutative Noetherian rings
cites work: Q3994888
cites work: Localization and Completion at Primes Generated by Normalizing Sequences in Right Noetherian Rings
cites work: Q3717234
cites work: Normal Elements and Completions of Non-Commutative Noetherian Rings
cites work: Normalizing Sequences and Completions of Non-Commutative Noetherian Rings
cites work: The noetherian property in complete rings and moduls
cites work: Corrigendum: Localisation in Enveloping Rings
cites work: I-adic completions of non-commutative rings
cites work: Q3786586
cites work: Q5735274
cites work: Maximal Orders Over Regular Local Rings
cites work: Q4039952
cites work: Krull and Global Dimensions of Semiprime Noetherian PI-Rings
cites work: Regularity of Zero Divisors
cites work: Localisation and completions of Noetherian PI algebras
description: scientific article; zbMATH DE number 997617