On direct sums of Baer modules.
type: Item
instance of: scholarly article
title: On direct sums of Baer modules.
zbMATH Open document ID: 1217.16009
author: S. Tariq Rizvi
author: Cosmin S. Roman
published in: Journal of Algebra
publication date: 2009-04-21T00:00:00Z
review text: A ring \(R\) is called Baer if the left annihilator of every nonempty subset of \(R\) is generated by an idempotent. In a general module theoretic setting, the concept of a Baer module was defined by the present authors [in Commun. Algebra 32, No. 1, 103-123 (2004; Zbl 1072.16007)] in terms of its endomorphism ring. Indeed, a right \(R\)-module \(M_R\) (with \(S=\text{End}(M_R)\)) is called Baer if for every submodule \(N\) of \(M\), \(\ell_S(N)=Se\) for some \(e^2=e\in S\), where \(S=\text{End}(M_R)\). By the authors [loc. cit.], a direct sum of Baer modules is not always Baer. Further, [in loc. cit.], it was shown that there is a strong connection between the class of Baer modules and the class of extending modules. Thereby a characterization for a direct sum of Baer modules to be Baer is an interesting result comparing with a solution to the problem of obtaining a characterization for a direct sum of extending modules to be extending. This paper contains many interesting results on Baer modules. The authors provide necessary and sufficient conditions for a direct sum of copies of a Baer module to be Baer and characterize rings \(R\) for which every free (projective) \(R\)-module is Baer. It is shown that every free (projective) right \(R\)-module is Baer if and only if \(R\) is a semiprimary (right) hereditary ring. Further, the authors show that every finitely generated free (projective) right module over a ring \(R\) is a Baer module if and only if \(R\) is right semihereditary and left \(\Pi\)-coherent if and only if \(\text{Mat}_n(R)\) is a Baer ring for every positive integer \(n\). When \(M_R\) is a finitely generated, retractable module, necessary and sufficient conditions for a direct sum of copies of a Baer module to be Baer are provided. Indeed, it is proved that an arbitrary direct sum of a finitely generated, retractable module \(M_R\) is a Baer module if and only if \(S=\text{End}(M_R)\) is semiprimary (right) hereditary. Also some results on direct …
reviewed by: Jae Keol Park
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16S50
Mathematics Subject Classification ID: 16D40
zbMATH DE Number: 5545854
zbMATH Keywords: Baer rings
zbMATH Keywords: Baer modules
zbMATH Keywords: direct sums
zbMATH Keywords: free modules
zbMATH Keywords: semihereditary rings
zbMATH Keywords: extending modules
zbMATH Keywords: coherent rings
zbMATH Keywords: retractable modules
zbMATH Keywords: idempotents
zbMATH Keywords: right annihilators
zbMATH Keywords: direct summands
zbMATH Keywords: quasi-Baer modules
zbMATH Keywords: endomorphism rings
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2008.10.002
OpenAlex ID: W2011320340
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DOI: 10.1016/J.JALGEBRA.2008.10.002
description: scientific article; zbMATH DE number 5545854