Strongly regular elements of Noetherian rings
type: Item
instance of: scholarly article
title: Strongly regular elements of Noetherian rings
zbMATH Open document ID: 0551.16005
DOI: 10.1016/0021-8693(84)90112-1
author: Alfred Goldie
author: Günter Krause
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let R be a right Noetherian ring. As usual, for any ideal I of \(R,\) C'(I)\(=\{c\in R:\) \(r\in R\), cr\(\in I\) implies \(r\in I\}\), \('C(I)=\{c\in R:\) \(r\in R\), rc\(\in I\) implies \(r\in I\}\) and \(C(I)=C'(I)\cap 'C(I)\). In this paper interest centres on \(S'(0)=\cap \quad C'(I)\) where the intersection is taken over all right annihilator ideals I, \('S(0)=\cap \quad 'C(J)\) where the intersection is taken over all left annihilator ideals J, and \(S(0)=S'(0)\cap 'S(0).\) The elements of C(0) and S(0) are called regular and strongly regular, respectively. Clearly \(S'(0)\subseteq C'(0),\quad 'S(0)\subseteq 'C(0)\) and \(S(0)\subseteq C(0).\) The set S(0) is of interest because it contains any right Ore set of regular elements of R and hence R has a (classical) right quotient ring only if \(C(0)=S(0)\). However an example is given of a Noetherian PI- ring R of Krull dimension 1 which satisfies \(C(0)=S(0)\) but which has neither a left nor a right quotient ring. Investigation of the set S(0) for an arbitrary right Noetherian ring R takes different forms. For example, various necessary conditions on R are given which imply \(C(0)=S(0).\) Next for a Noetherian ring R, \(S(0)=S'(0)='S(0)=\cap\) \(C'(M)=\cap\) C(P), where the first intersection is taken over all middle annihilators M of R and the second over all prime middle annihilators P of R, and S(0) is saturated in the sense that st\(\in S(0)\) implies \(s\in S(0)\) and \(t\in S(0)\). This raises the question whether a Noetherian ring has a finite collection of prime middle annihilators \(P_ i\) (1\(\leq i\leq n)\) such that \(S(0)=\cap^{n}_{i=1}C(P_ i).\) Although this question is not settled in general, various cases are given where it is true, for example homomorphic images of enveloping algebras of finite dimensional Lie algebras or of group algebras of polycyclic-by-finite groups, FBN-rings, rings which are finitely generated modules over their Noetherian centres, and Noetherian rings R such that S(0) …
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16Rxx
Mathematics Subject Classification ID: 16S34
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 3877370
zbMATH Keywords: right Noetherian ring
zbMATH Keywords: right annihilator ideals
zbMATH Keywords: right Ore set of regular elements
zbMATH Keywords: right quotient ring
zbMATH Keywords: Noetherian PI-ring
zbMATH Keywords: Krull dimension
zbMATH Keywords: middle annihilators
zbMATH Keywords: enveloping algebras of finite dimensional Lie algebras
zbMATH Keywords: group algebras of polycyclic-by-finite groups
zbMATH Keywords: FBN-rings
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90112-1
OpenAlex ID: W2094284518
cites work: Q5185036
cites work: Q3717234
cites work: Shorter Notes: Artinian Ideals in Noetherian Rings
cites work: A Note on Non-Commutative Noetherian Rings
cites work: The Quotient Problem for Noetherian Rings
cites work: On some questions in Noetherian rings
cites work: Regularity of Zero Divisors
cites work: Noetherian Full Quotient Rings
description: scientific article; zbMATH DE number 3877370