On subcommutative rings
type: Item
instance of: scholarly article
title: On subcommutative rings
zbMATH Open document ID: 0641.16003
author: G. W. S. van Rooyen
published in: Proceedings of the Japan Academy. Series A
publication date: 1987-01-01T00:00:00Z
review text: Subcommutativity of rings was defined by \textit{D. Barbilian} [Arithmetic theory of ideals in noncommutative rings (Romanian), Editura Academiei Republicii Populare Romîne (1956; Zbl 0074.261)] and \textit{J. D. Reid} [Acta Math. Acad. Sci. Hung. 16, 23-26 (1965; Zbl 0131.276)]; reflexivity and complete reflexivity of one-sided ideals were defined by \textit{G. Mason} [Commun. Algebra 9, 1709-1724 (1981; Zbl 0468.16024)]. The present author defines a right ideal I of the ring R to be quasi- reflexive if for all right ideals J and K with JK\(\subseteq I\), KJ\(\subseteq I\) as well; and he calls R strongly subcommutative if every right ideal is quasi-reflexive. After exploring relationships among the various properties, he identifies several conditions equivalent to strong subcommutativity - among them that \(AB=BA\) for all right ideals A and B. He proves that every idempotent e in a strongly subcommutative ring is central, and discusses briefly the ideals eR.
Mathematics Subject Classification ID: 16U80
Mathematics Subject Classification ID: 16W99
Mathematics Subject Classification ID: 16Dxx
zbMATH DE Number: 4044011
zbMATH Keywords: Subcommutativity of rings
zbMATH Keywords: complete reflexivity of one-sided ideals
zbMATH Keywords: strong subcommutativity
MaRDI profile type: Publication
full work available at URL: pjaa.63.268
OpenAlex ID: W2065336336
cites work: Q3239326
cites work: On strongly regular rings
cites work: Centralizers in rings
cites work: Characterizations of strongly regular rings, II
cites work: Reflexive ideals
cites work: On subcommutative rings
cites work: On Duo Rings
DOI: 10.3792/PJAA.63.268
description: scientific article; zbMATH DE number 4044011