Submodule maximal if and only if quotient module simple?

Question

Why is the case that a submodule $L$ of a module $L$ is maximal, among submodules of $M$ distinct from $M$, if and only if the quotient module $M/L$ is simple?

Answer 1

As I said in the other question. The bijection between submodules containing $L$ and submodules of $M/L$ tells us that if we assume that $L$ is maximal, that there exist no proper submodules that strictly contains $L$, then there exist no proper submodules in $M/L$ as a proper submodule there would correspond to a proper submodule strictly containing $L$ which we have already excluded. Ergo $M/L$ is simple.

Assuming $M/L$ is simple, that means there exists no proper non-trivial submodules. Now through same reasoning as before, the existence of a proper submodule that contains $L$ would correspond to a proper non-trivial submodule in $M/L$ which we already excluded, and as such $L$ must then be maximal.