Let $M$ be a module and $N$ a submodule of $M$. If $N$ is Noetherian and $M/N$ is Noetherian, so is $M$.
This is usually proven like this: Let $(A_n)$ be an ascending series of submodules of $M$. Then $(A_n\cap N)$ and $((A_n+N)/N)$ are ascending series of submodules of $N$ and $M/N$ respectively, so that they are eventually constant. A little elementary argument shows that, for $A\subset B$ submodules of $M$, $A\cap N=B\cap N$ and $(A+N)/N=(B+N)/N$ together imply $A=B$. So the series $(A_n)$ is eventually constant.
Now suppose we have defined Noetherian as "all submodules are finitely generated" and don't know about the equivalence to the ascending chain condition. There should be a proof of the above statement without the detour via the ascending chain condition, i.e. something like "Let $A$ be a submodule of $M$, $S$ a finite generating set of $A\cap N$ and $T$ a finite generating set of $(A+N)/N$. Then ... is a finite generating set of $A$." I have yet been unable to find such a proof. Can somebody sketch one or link to one?
In case it makes a difference: I am mainly interested in $\mathbf{Z}$-modules.