Finite generation is preserved by module extensions ($N,M/N\text{ finitely generated}⟹M\text{ finitely generated}$)

If a submodule $N\subseteq M$ and the quotient module $M/N$ are finitely generated modules, then $M$ is a finitely generated module. Choose finitely many generators of $N$ and lift finitely many generators of $M/N$. For any $m\in M$, subtract an $R$-linear combination of the lifts to obtain an element of $N$, then express this using the generators of $N$. The union of the two finite lists generates $M$.