Prove module $M$ is finitely generated if $N$ and $M/N$ are finitely generated

Question

Let $R$ be a ring with $1$ and $N$ be a submodule of an $R$-module $M$.

Prove that $M$ is finitely generated if $N$ and $M/N$ are finitely generated.

There are two definitions of "finitely generated" for me to use:

1. $M$ is finitely generated if $\exists k\in \mathbb{Z}^+, m_1,m_2,\cdots,m_k\in M$ such that $M=Rm_1+Rm_2+\cdots+Rm_k$.
2. $M$ is finitely generated if there exists a surjective $R$-module homomorphism $\phi : \delta\rightarrow M$, where $\delta$ is free of finite rank.

Any suggestion on which definition will make the proof easier?

Answer 1

It's really a preference. If this is your first exposure to modules, and you like doing things concretely, 1. would be better.

If you know about short exact sequences, then 2. will give you a more "hands off" solution.