This may seem like a really obvious question, but I am doubting myself. Suppose $M$ is an $R$-module, for some ring $R$. If $N$ is a submodule then obviously we have the quotient module $M/N$. Now suppose $L$ is some non-zero cyclic submodule of $M/N$, i.e. generated by a single element that is not in $N$. Then can we construct a surjective homomorphism $f:M/N\to L$?
I would imagine that we would perhaps need to impose some restrictions onto $R$.
No. Take $R = \mathbb Z$, $M = \mathbb Q$, $N = 0$, and $L = \mathbb Z$ (the usual subset of $\mathbb Q$).