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Finite direct sums of CS-modules. (English) Zbl 0802.16006
A module \(M\) is called a CS-module if every submodule of \(M\) is essential in a direct summand of \(M\). Examples of CS-modules include injective, quasi-injective, continuous, quasi-continuous and uniform modules. It is still an open question to find necessary and sufficient conditions for a direct sum of CS-modules to be CS. In this paper the authors investigate the finite direct sum case and provide some sufficient conditions for a finite direct sum of CS-modules to be CS. In particular, they prove that any finite direct sum of CS-modules which are relatively injective is again CS. This result is used to give simple proofs for the known facts about when a finite direct sum of modules is quasi-continuous or continuous.

16D50 Injective modules, self-injective associative rings
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)