Direct sums and summands of weak CS-modules and continuous modules (Q1806236)

A module \(M\) is called a weak CS-module if every semisimple submodule of \(M\) is essential in a direct summand of \(M\). In this paper, the author studies the question of when direct sums and direct summands of weak CS-modules are weak CS-modules. It is shown, in particular, that a finite direct sum of relatively injective weak CS-modules is again weak CS. The author defines a module \(M\) to be a weak \(C_{11}\)-module if every semisimple submodule has a complement in \(M\) which is a direct summand of \(M\), and he shows that any direct sum of weak CS-modules is a weak \(C_{11}\)-module. Finally, some characterizations of continuous modules are given.