A class of generalized radical rings
type: Item
instance of: scholarly article
title: A class of generalized radical rings
zbMATH Open document ID: 0776.16004
author: Zimei Jiang
published in: Chinese Science Bulletin
publication date: 1993-04-01T00:00:00Z
review text: For the ring \(R\) and \(A,B\subseteq R\), \(a\in R\) are denoted: \(A-B=\{a- b\mid a\in A\), \(b\in B\}\), \(G^{(a)}=\{x\in R\mid xa=ax=x\}\), \(E=\{e\in R\mid e^ 2=e\neq 0\}\). \(R\) is called DJ-ring if for each \(a\in R\) the set \(a-G^{(a)}\) contains at least one quasi-regular element and at most one two-sided nonquasi-regular element. The following statements on \(R\) are equivalent: (i) \(R\) is DJ-ring, \(R\neq J(R)\) and there is an \(e\in E\) such that \(E-e\in J(R)\); (ii) \(R=eRe+eR(1-e)+(1-e)Re\), where \(0\neq e=e^ 2\in R\), \(eR(1-e)+(1-e)Re\) is a subring of \(R\) and \(eRe\) is a division ring. A lattice isomorphism between the lattices \(\Sigma=\{S \vartriangleleft R\mid S\nsubseteq J(R)\}\) and \(\Sigma'=\{T\vartriangleleft R\mid T=J(R)\), \(\exists u\in E\), \(uJ(R)\subseteq T\), \(J(R)u\subseteq T\}\) is established.
Mathematics Subject Classification ID: 16N80
Mathematics Subject Classification ID: 16N20
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 148465
zbMATH Keywords: idempotents
zbMATH Keywords: Jacobson radical
zbMATH Keywords: DJ-ring
zbMATH Keywords: quasi-regular element
zbMATH Keywords: division ring
zbMATH Keywords: lattice isomorphism
reviewed by: A. I. Kashu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 148465