On quasi-reflexive rings (semigroups)
type: Item
instance of: scholarly article
title: On quasi-reflexive rings (semigroups)
zbMATH Open document ID: 0669.16001
author: G. W. S. van Rooyen
published in: Proceedings of the Japan Academy. Series A
publication date: 1988-01-01T00:00:00Z
review text: A right ideal I of a ring [semigroup with 0] R is called right quasi reflexive iff (*) AB\(\subseteq I\) implies BA\(\subseteq I\) for all right ideals A,B of R, and R itself is called right quasi-reflexive iff the above holds for \(I=(0)\). Using these and the dual concepts, the author generalizes several results given by \textit{O. Steinfeld} [Quasi-ideals in rings and semigroups (1978; Zbl 0403.16001)] and himself [Proc. Japan Acad., Ser. A 63, 268-271 (1987; Zbl 0641.16003)], for example: (1) Let R be a right quasi-reflexive ring and \(e=e^ 2\in R\). Then eR is right quasi-reflexive iff eR is an ideal iff e is central in R. (2) Let R be a left and right quasi-reflexive ring [semigroup with 0] and \(0\neq e=e^ 2\in R\). Then Re is a [0-]minimal left ideal of R iff eR is such a right ideal iff eRe is a [0-]minimal quasi-ideal of R. (3) For R as in (2), let Q be a [0-]minimal quasi-ideal of R satisfying \(Q^ 2\neq (0)\). Then Q is the intersection of a [0-]minimal left ideal and a [0-]minimal right ideal of R. Reviewer's remarks: Most of the results of the paper can be sharpened considerably, since I is a right quasi-reflexive right ideal of R iff I is an ideal of R satisfying (*) for ideals A,B of R, and hence also iff I is a left quasi-reflexive left ideal of R. In particular, R is right quasi-reflexive iff it is left quasi-reflexive, which in turn holds iff \(AB=(0)\) implies \(BA=(0)\) for all ideals A,B of R.
Mathematics Subject Classification ID: 16U80
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 20M12
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 16U99
zbMATH DE Number: 4095666
zbMATH Keywords: semigroups
zbMATH Keywords: central idempotents
zbMATH Keywords: right ideals
zbMATH Keywords: right quasi-reflexive ring
zbMATH Keywords: [0-]minimal left ideal
zbMATH Keywords: [0-]minimal quasi-ideal
zbMATH Keywords: right quasi- reflexive right ideal
zbMATH Keywords: left quasi-reflexive left ideal
reviewed by: Hanns Joachim Weinert
MaRDI profile type: Publication
full work available at URL: pjaa.64.337
OpenAlex ID: W2060974188
cites work: Reflexive ideals
cites work: Q4187930
cites work: Q4205595
cites work: On subcommutative rings
cites work: On quasi-ideals in rings
DOI: 10.3792/PJAA.64.337
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description: scientific article; zbMATH DE number 4095666