On direct sums of Baer modules.
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 sum decompositions of Baer modules are included. Moreover, using the quasi-retractability defined in this paper, the authors give a characterization for a module \(M_R\) to be Baer in terms of its endomorphism ring \(S=\text{End}(M_R)\), which is, \(M_R\) is Baer if and only if \(S\) is a Baer ring and \(M_R\) is quasi-rectractable. When \(n\) is a positive integer, it is shown that if a ring \(R\) is an \(n\)-fir, then \(R^{(n)}_R\) is a Baer module (hence \(\text{Mat}_n(R)\) is a Baer ring). From this result, the authors obtain an interesting example of a module \(M_R\) such that \(M^{(n)}_R\) is Baer, but \(M^{(n+1)}_R\) is not Baer.
scientific article; zbMATH DE number 5545854
scientific article