Isomorphism of subquotient and quotient modules

Question

Is it true that for every submodules $A\subset B$ of $C$ there is some submodule $L\subset C$ such that $C/L = B/A$?

Answer 1

No. Let $A = 0$. Then your question is whether, for every submodule $B \subseteq C$, we can find a submodule $L \subseteq C$ such that $C/L \cong B$. In other words, the question is whether every submodule of $C$ is isomorphic to a quotient module of $C$.

This is true if $B$ is a direct summand, in which case we can take $L$ to be its complement, but not true in general. For example, let $R$ be a commutative ring which has a non-principal ideal $I$. Then $I$ is a submodule of $R$ but not a quotient module of $R$.

Answer 2

No. Consider the Prüfer group $G=\mathbb{Z}(p^{\infty})$; a quotient of a proper subgroup is cyclic, no quotient of $G$ (except $G/G$, of course) is cyclic.