On \(\Sigma\)-extending modules. (English) Zbl 0885.16005
A module \(M\) is called an extending module (or \(CS\)-module) if every submodule of \(M\) is essential in a direct summand of \(M\). Generalizing the well-known concept of \(\Sigma\)-injectivity, a module \(M\) is defined to be \(\Sigma\)-extending if every direct sum of copies of \(M\) is extending. The rings \(R\) which are \(\Sigma\)-extending as right modules over themselves are also called Harada rings (in honour of M. Harada who first introduced them). Harada rings are known to be Artinian \(QF\)-\(3\) and generalize both quasi-Frobenius rings and Nakayama rings. However it is still an open problem whether Harada rings have a selfduality. In a sense, the study of the structure of Harada rings has motivated the study of \(\Sigma\)-extending modules, in general. In the paper under review, it is shown that an indecomposable module is \(\Sigma\)-extending if and only if it is \(\Sigma\)-quasi-injective. Based on this fact, the authors obtain a characterization of \(\Sigma\)-extending modules which are direct sums of indecomposable modules (however, it is still unknown whether \(\Sigma\)-extending modules always have indecomposable decompositions). The main results in this paper refine and extend some results in the reviewer’s recent paper [J. Pure Appl. Algebra 119, No. 2, 139-153 (1997; Zbl 0878.16005)].
Reviewer: Nguyen Viet Dung (Bo Ho)
MSC:
16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
16D50 | Injective modules, self-injective associative rings |
16L60 | Quasi-Frobenius rings |
Keywords:
\(CS\)-modules; \(\Sigma\)-injectivity; \(\Sigma\)-extending modules; \(\Sigma\)-quasi-injective modules; extending modules; direct summands; direct sums; Harada rings; quasi-Frobenius rings; Nakayama rings; indecomposable modules; indecomposable decompositionsCitations:
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