I asked the following question here.
Why is the case that a submodule $L$ of a module $M$ is maximal, among submodules of $M$ distinct from $M$, if and only if the quotient module $M/L$ is simple?
Mariano Suárez-Álvarez commented the following.
It is because there is bijection between the submodules of $M$ that contain $L$ and the submodules of $M/L$.
How do I establish this bijection?
The bijection is fairly easy, let $K$ be a submodule in $M/L$ and $L\subset N$ a submodule of $M$ that contains $L$. We can easily see that $\pi(N)$ is a submodule, where $\pi$ is the quotient homomorphism. So it sends any submodule containing $L$ to a submodule of $M/L$. For the inverse, $\pi^{-1}(K)$. Let $x',y'\in K$ be given, they come in some form of $x+L$ and $y+L$, as it is a submodule we have that $x'+y'=x+L+y+L=x+y+L\in K$ and as such we have $x+y\in\pi^{-1}(K)$. For module production we show the same and establish that $\pi^{-1}(K)$ is a submodule, to see that it contains $L$ we notice that $K$ must contain the $0$ element which is the set $L$ for us in the quotient.
From there we see there is a bijection.