Strong lifting splits.
type: Item
instance of: scholarly article
title: Strong lifting splits.
zbMATH Open document ID: 1254.16004
published in: Journal of Pure and Applied Algebra
publication date: 2011-05-10T00:00:00Z
review text: In this paper the concept of an enabling ideal is introduced. An ideal \(I\) of \(R\) is called an enabling ideal of \(R\) if \(a-e\in I\), \(a\in R\), \(e^2=e\in R\), then \(a-f\in I\) for some \(f^2=f\in Ra\). The authors show that an ideal \(I\) of \(R\) is strongly lifting if and only if \(I\) is lifting and enabling. These ideals are studied and their properties are described. It is shown that a left duo ring is an exchange ring if and only if every ideal is enabling, that Zhou's \(\delta\)-ideal is always enabling, and that the right singular ideal is enabling if and only if it is contained in the Jacobson radical. The notion of a weakly enabling left ideal is defined. They characterize a weakly enabling ideal as a left ideal (right ideal) by left (right) partial summand of the ring. It is shown that a ring is an exchange ring if and only if every left ideal is weakly enabling. They show that if \(R\) is a left SSP ring, then an ideal \(I\) of \(R\) is a weakly enabling ideal of \(R\) if and only if \(I\) is enabling. Two related conditions, interesting in themselves, are investigated: the first gives a new characterization of \(\delta\)-small left ideals and the second characterizes weakly enabling left ideals. As an application (which motivated the authors), let \(M\) be an \(I\)-semiregular left module where \(I\) is an enabling ideal. It is shown that \(m\in M\) is \(I\)-semiregular if and only if \(m-q\in IM\) for some regular element \(q\) of \(M\) and, as a consequence, that if \(M\) is countably generated and \(IM\) is \(\delta\)-small in \(M\), then \(M\cong\bigoplus^\infty_{i=1}Re_i\) where \(e^2_i=e_i\in R\) for each \(i\).
reviewed by: Avanish Kumar Chaturvedi
Mathematics Subject Classification ID: 16D25
Mathematics Subject Classification ID: 16E50
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16U99
zbMATH DE Number: 5886347
zbMATH Keywords: lifting ideals
zbMATH Keywords: enabling ideals
zbMATH Keywords: small ideals
zbMATH Keywords: idempotents
zbMATH Keywords: semiregular modules
zbMATH Keywords: left duo rings
zbMATH Keywords: exchange rings
zbMATH Keywords: weakly enabling left ideals
author: Mustafa Alkan
author: W. Keith Nicholson
author: A. Çiğdem Özcan
Wikidata QID: Q105368758
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.11.001
OpenAlex ID: W2154873605
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cites work: Strong lifting.
cites work: Regular Modules
cites work: Generalizations of perfect, semiperfect, and semiregular rings
DOI: 10.1016/J.JPAA.2010.11.001
description: scientific article; zbMATH DE number 5886347