Reduced rank and normalizing elements
type: Item
instance of: scholarly article
title: Reduced rank and normalizing elements
zbMATH Open document ID: 0644.16022
DOI: 10.1016/0021-8693(88)90169-X
author: E. A. Whelan
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: The author calls a ring R a BPC ring provided every ideal has a finite normalizing set of generators. The paper has three main objectives. Firstly, the author proves that if N is a semiprime ideal of a BPC ring R then the ring R/N contains an Ore set T of regular elements such that the ring of quotients of R/N with respect to T is a finite direct sum of simple rings. Secondly, this fact permits the definition of a bimodule reduced rank analogous to the usual Goldie reduced rank of Noetherian rings. Thirdly an analogue of the Invertible Ideal Theorem of \textit{A. W. Chatters}, \textit{A. W. Goldie}, \textit{C. R. Hajarnavis} and \textit{T. H. Lenagan} [J. Algebra 61, 582-589 (1979; Zbl 0436.16009)] is proved, namely, given a BPC ring R, invertible ideal \(X\neq R\), prime ideal P minimal over X then height \(P\leq 1\). It follows that BPC rings satisfy the descending chain condition on prime ideals.
reviewed by: Patrick F. Smith
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16P60
zbMATH DE Number: 4049821
zbMATH Keywords: normalizing set of generators
zbMATH Keywords: semiprime ideal
zbMATH Keywords: Ore set
zbMATH Keywords: ring of quotients
zbMATH Keywords: direct sum of simple rings
zbMATH Keywords: bimodule reduced rank
zbMATH Keywords: Invertible Ideal Theorem
zbMATH Keywords: BPC rings
zbMATH Keywords: descending chain condition on prime ideals
MaRDI profile type: Publication
cites work: Reduced rank in Noetherian rings
cites work: Q3890834
cites work: Non-Commutative Unique Factorisation Rings
cites work: On Additive Ideal Theory in General Rings
cites work: Normalizing extensions: Prime ideals and incomparability
cites work: Normalizing extensions: Nilpotency
cites work: Relative krull dimension and prime ideals in right noetherian rings
cites work: Principal ideal theorem for Noetherian P.I.rings
cites work: Q5603302
cites work: Localization and the AR Property
cites work: An infinite construction in ring theory
cites work: QUASI-COMMUTATIVE PRINCIPAL IDEAL RINGS
cites work: Finite subnormalizing extensions of rings
Recommended article: Rank Reducing Matrix Norms
Recommended article: Normalization of row reduced matrices
Recommended article: Rank reduction, factorization and conjugation
Recommended article: Rank reduction, normal forms and Stokes matrices
Recommended article: Q4301041
Recommended article: Rank reduction and conjugation
Recommended article: On reduced matrices
Recommended article: The rank of reduced dispersion matrices
Recommended article: Q4330707
Recommended article: A Rank–One Reduction Formula and Its Applications to Matrix Factorizations
description: scientific article; zbMATH DE number 4049821