$M$ finitely generated if submodule and quotient are finitely generated.

I know this must be easy. I’m new to modules, so perhaps I must be missing something important.

Let $M$ be an $R$-module. Show that if there exists a submodule $N$ such that $N$ and $M/N$ are finitely generated, then $M$ is finitely generated.

If the exact sequence $0 \longrightarrow N \longrightarrow M \longrightarrow M/N \longrightarrow 0 $ (where the second and third arrows are inclusion and projection respectively) splits then $M$ would be isomorphic to the direct sum of $M$ and $M/N$ and the conclusion would follow (I think). But I can’t show that it splits.

Thank you.

Solutions Collecting From Web of "$M$ finitely generated if submodule and quotient are finitely generated."

Take a generating set $\{\bar z_1, \ldots, \bar z_m\}$ for $M/N$. If $x$ is an element of $M$, then there exist $a_1, \ldots, a_n \in R$ such that the image $\bar x$ of $x$ in $M/N$ is
\[
\bar x = a_1\bar z_1 + \cdots + a_n\bar z_n.
\]
Since the map on the right is surjective, you can choose lifts $\{z_1, \ldots, z_m\}$ in $M$ of the aforementioned generating set. Now, the element
\[
x’ = a_1z_1 + \cdots + a_nz_n
\]
has the same image as $x$ in $M/N$. Do you see a way to use the exactness in the middle?

Note that the sequence need not split: consider the $\mathbf Z$-submodule $2\mathbf Z$ of $\mathbf Z$.

Perhaps this works?

Suppose $M/N$ is finitely generated with generators $a_1+N,\dots,a_m+N$, and $N$ is finitely generated with generators $b_1,\dots,b_n$. Take any $m\in M$. Then $m+N=\sum_{i=1}^m r_i(a_i+N)=\left(\sum_{i=1}^m r_ia_i\right)+N$ viewing $M/N$ as an $R$-module.

This implies $m-\sum_{i=1}^m r_ia_i\in N$, and thus $m-\sum_{i=1}^m r_ia_i=\sum_{j=1}^n s_jb_j$ for some $r_i,s_j\in R$. Finally, $m=\sum_{i=1}^m r_ia_i+\sum_{j=1}^n s_jb_j$. So $M$ is generated by $a_i, b_j$ for $i=1,\dots, m$ and $j=1,\dots n$