Relative injectivity and CS-modules (Q1338416)

Let \(R\) be a ring with identity and let \(M\) and \(N\) be unital right \(R\)- modules. Let \(E(M)\) denote the injective hull of \(M\). A module is called CS if every submodule is essential in a direct summand. If \(\text{Hom}(N,E(M)) = 0\) then \(M \oplus N\) is CS if and only if \(M\) and \(N\) are CS and \(N\) is \(M\)-injective. This generalizes a result of the author and \textit{B. J. Mueller} [Osaka J. Math. 25, 531-538 (1988; Zbl 0715.13006)]. Moreover, if \(M\) is nonsingular and \(N\) is \(M\)-injective then \(M\oplus N\) is CS if and only if \(M\) and \(N\) are both CS. It is also proved that any finite direct sum of relatively injective CS-modules is CS. The same result can be found in a paper of \textit{A. Harmanci} and the reviewer [Houston J. Math. 19, 523-532 (1993; Zbl 0802.16006)].