A note on semi-prime rings
type: Item
instance of: scholarly article
title: A note on semi-prime rings
zbMATH Open document ID: 0587.16002
DOI: 10.1007/BF01301657
author: Roger Yue Chi Ming
published in: Monatshefte für Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178272
review text: The following generalization of semi-prime rings is introduced: A is called an SRN ring if the sum of all reduced ideals of A contains no non- zero nilpotent ideal of A. (An ideal is reduced iff it contains no non- zero nilpotent element.) It is proved that A is SRN iff the sum of all reduced ideals of A is a reduced ideal. The unique maximal strongly regular ideal of some SRN rings is then considered. If A is semi-prime, then (a) any complement left ideal which is an ideal is a left annihilator; (b) if A is also left CM [see the author, ibid. 91, 153-166 (1981; Zbl 0452.16007)], then any complement left ideal is a left annihilator. Certain properties of semi-prime right self-injective rings are given. If A is right self-injective (resp. continuous) regular such that for any proper complement left ideal C, either C is an ideal of A or r(C)\(\neq 0\), then A is left self-injective (resp. continuous). Simple Artinian and semi-simple Artinian rings are characterized.
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16D50
Mathematics Subject Classification ID: 16Kxx
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16Dxx
zbMATH DE Number: 3941724
zbMATH Keywords: generalization of semi-prime rings
zbMATH Keywords: reduced ideals
zbMATH Keywords: nilpotent ideal
zbMATH Keywords: strongly regular ideal
zbMATH Keywords: SRN rings
zbMATH Keywords: complement left ideal
zbMATH Keywords: left annihilator
zbMATH Keywords: right self-injective rings
zbMATH Keywords: semi-simple Artinian rings
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3941724