Reduced right ideals which are strongly essential in direct summands
type: Item
instance of: scholarly article
title: Reduced right ideals which are strongly essential in direct summands
zbMATH Open document ID: 0636.16016
DOI: 10.1007/BF01194383
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: Let R denote an associative ring with unity and X and Y are right ideals of R. We say X is strongly essential in Y if Y is an essential extension of X and X is an essential extension of an isomorphic copy of Y. R is said to be right GFC if every faithful cyclic right R-module generates Mod-R. In this paper we show that if R satisfies any of the following conditions, then every finitely generated reduced (i.e., no nonzero nilpotent elements) right ideal is strongly essential in a direct summand of R: (i) every finitely generated reduced right ideal is essential in a direct summand of R; (ii) R is right p.p. right GFC, (iii) R is semiprime quasi-Baer right GFC. Examples of such rings are: regular rings, right CS rings, and semiprime right FPF rings.
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 4036020
zbMATH Keywords: strongly essential
zbMATH Keywords: essential extension
zbMATH Keywords: faithful cyclic right R-module
zbMATH Keywords: finitely generated reduced right ideal
zbMATH Keywords: direct summand
zbMATH Keywords: right p.p. right GFC
zbMATH Keywords: semiprime quasi-Baer right GFC
zbMATH Keywords: right CS rings
zbMATH Keywords: semiprime right FPF rings
author: G. F. Birkenmeier
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4036020