Recall that a right $R$-module $M$ is CS if for every submodule $N\subseteq M$ there exists a summand $L \subseteq^{\oplus} M$ such that $N$ is an essential submodule of $L$.
It's well-known that a summand of a CS module is again CS (see, for instance, Continuous and Discrete modules by Mohamed and Müller, Proposition 2.7, page 20).
The class of CS modules was generalized as follows: A right $R$-module $M$ is called GCS (generalized $CS$-module) if for every submodules $A, B \subseteq M$ with $A \cong B$ and $A \cap B = 0$ there exists $K, L \subseteq^{\oplus} M$ such that $A$ is essential in $K$ and $B$ is essential in $L$.
I ask whether a summand of a GCS module is again GCS.
Another question on GCS-modules: Is the following statement true?
If $M$ is GCS and $f \colon M \to M'$ is any homomorphism with $M'$ nonsingular, then $\ker f \subseteq^{\oplus} M$?
Thanks in advance.
