On noncommutative localization
type: Item
instance of: scholarly article
title: On noncommutative localization
zbMATH Open document ID: 0545.16001
DOI: 10.1016/0021-8693(84)90167-4
author: John A. Beachy
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let R be a ring which for convenience will be assumed to be right and left Noetherian although this is not always necessary, and let P, S, N denote respectively a prime ideal, a semi-prime ideal, and the nilpotent radical of R. The two localizations considered are: the local ring Q of R at P as constructed by Goldie; and the inversive localization of R at S which in this review will be denoted by R(S), i.e. R(S) is universal with respect to the property that all square matrices over R which are regular modulo S are invertible over the ring R(S). It is shown that R(S) is Artinian if and only if some power of S is contained in the kernel of the natural map from R to R(S), and this condition is satisfied if \(S=N\). Also Q is isomorphic to R(P)/J' where J' is the intersection of the powers of the Jacobson radical of R(P); and Q is universal with respect to the properties (1) Q/J(Q) is naturally isomorphic to the quotient ring of R/P, and (2) the intersection of the powers of J(Q) is zero.
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16Nxx
zbMATH DE Number: 3867521
zbMATH Keywords: prime ideal
zbMATH Keywords: semi-prime ideal
zbMATH Keywords: nilpotent radical
zbMATH Keywords: localizations
zbMATH Keywords: inversive localization
zbMATH Keywords: Jacobson radical
reviewed by: A. W. Chatters
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90167-4
OpenAlex ID: W2020564152
cites work: Inversive localization at semiprime Goldie ideals
cites work: Rings with finite reduced rank
cites work: Localization at semiprime ideals
cites work: Q3890834
cites work: Q5639812
cites work: Inversive localization in noetherian rings
cites work: Localization in non-commutative Noetherian rings
cites work: A note on noncommutative localization
cites work: Non-commutative principal ideal rings
cites work: Relative krull dimension and prime ideals in right noetherian rings
cites work: Two remarks about Goldie's localization
cites work: Q4049223
description: scientific article; zbMATH DE number 3867521