Generalised composition factors Let $A$ be a semiprimary ring.
A simple module $L$ is said to be a generalised composition factor of $M$ if there are $M'$ and $M''$, $M'' \subset M'$, submodules of $M$, such that $M'/M'' \cong L$.
Suppose $L$ is a generalised composition factor of $M$. Is it possible to have a submodule $N$ of $M$ such that $L$ is neither a generalised composition factor of $N$ nor a generalised composition factor of $M/N$?

I feel very foolish for just noticing this now...
The answer to the question is no.
Let $P$ be the projective cover of the simple module $L$ (and let $Q$ be its injective hull). It is easy to see that $L$ is a generalised composition factor of a module $M$ if and only if $\operatorname{Hom}_A(P,M)\neq 0$ (if and only if $\operatorname{Hom}_A(M,Q)\neq 0$).
Consider the exact sequence
$$0 \longrightarrow N \longrightarrow M \longrightarrow M/N \longrightarrow 0.$$
Because $\operatorname{Hom}_A(P,-)$ is exact (in fact we only need the functor to be half exact) then $\operatorname{Hom}_A(P,M)\neq 0$ if and only if $\operatorname{Hom}_A(P,N)\neq 0$ or $\operatorname{Hom}_A(P,M/N) \neq 0$.
