\(\aleph\)-QF 3 rings with zero singular ideal
type: Item
instance of: scholarly article
title: \(\aleph\)-QF 3 rings with zero singular ideal
zbMATH Open document ID: 0604.16014
DOI: 10.1016/0021-8693(85)90019-5
author: Giuseppe Baccella
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: A ring R is called right \(\aleph\)-QF 3 if there is a family \(\{e_{\lambda}\}_{\lambda \in \Lambda}\) of nonisomorphic orthogonal idempotents \(e_{\lambda}\) of R with \(Card(\Lambda)=\aleph\) such that each \(e_{\lambda}R\) is the injective envelope of a minimal right ideal and \(\sum e_{\lambda}R\) is faithful. The first main theorem consists of the following equivalent statements: (a) R is right nonsingular and right \(\aleph\)-QF 3; (b) R is right nonsingular, \(E(R_ R)\) is torsionless, and R has essential right socle; (c) R has essential right socle and an R-module is torsionless iff it is nonsingular; (d) R is a two-sided essential product (in the sense of \textit{K. R. Goodearl} [''Ring theory. Nonsingular rings and modules'' (1976; Zbl 0336.16001)]) of right nonsingular right QF 3 rings. The second main theorem consists of several conditions which are equivalent to R being right nonsingular and right and left \(\aleph\)-QF 3. One of these conditions is that R has a two-sided maximal ring of quotients which is a product of simple artinian rings and has essential left and right socles.
Mathematics Subject Classification ID: 16L60
Mathematics Subject Classification ID: 16U99
Mathematics Subject Classification ID: 16P20
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16S90
zbMATH DE Number: 3975142
zbMATH Keywords: right \(\aleph \)-QF 3
zbMATH Keywords: orthogonal idempotents
zbMATH Keywords: injective envelope
zbMATH Keywords: minimal right ideal
zbMATH Keywords: essential right socle
zbMATH Keywords: torsionless
zbMATH Keywords: nonsingular
zbMATH Keywords: essential product
zbMATH Keywords: right nonsingular right QF 3 rings
zbMATH Keywords: maximal ring of quotients
zbMATH Keywords: simple artinian rings
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90019-5
OpenAlex ID: W2057113266
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description: scientific article; zbMATH DE number 3975142