Is the finite sum of finitely generated submodules finitely generated. Below, I formulate this question precisely and give my answer to it. It is an evident claim but it is nowhere to find in the literature, so I leave it here to discuss if someone disagrees with it.
Let $A$ be a commutative ring. Let $M$ be an $A$-module and let $N_1, \dots, N_n$ be a finite family of finitely generated submodules of $M$. Then the linear sum of submodules $\sum_{i=1}^nN_i:=\mathrm{Span}_A(\bigcup_iN_i)$ is finitely generated over $A$, too.
Proof:
For every element $x\in\sum_{i=1}^nN_i$, by definition, we have $x=\sum_{i=1}^nx_i$ with $x_i\in N_i$. Since for every $i=1, \dots, n$, the submodule $N_i$ is finitely generated with generators $e_1^{(i)}, \dots, e_{m_i}^{(i)}$, we have
$x_i=\sum_{j=1}^{m_i}a_j^{(i)}e_j^{(i)}$ and hence
$$x=\sum_{i=1}^n\sum_{j=1}^{m_i}a_j^{(i)}e_j^{(i)}.$$
Thus, $\sum_{i=1}^nN_i$ is generated by $e_1^{(1)}, \dots, e_{m_1}^{(1)}, \dots, e_{1}^{(n)}, \dots, e_{m_n}^{(n)}$. Hence, $\sum_{i=1}^nN_i$ is finitely generated over $A$. QED