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)].