Corollary

Let M be an A-module and $(M_i)_{i\in I}$ a finite family of submodules of M.

a) If the modules $M_i$ are Artinian (resp. Noetherian), then so is their sum $\sum_{i\in I}M_i$.

b) If the modules $M/M_i$ are Artinian (resp. Noetherian), then so is the module $M/\bigcap_{i\in I}M_i$.

By induction, it suffices to treat the case when $I =\{1,2\}$. The module $M_2/(M_1\cap M_2)$, quotient of $M_2$, is isomorphic to the submodule $(M_1+M_2)/M_1$ of $M/M_1$ (I, §4, No. 6, p. 41, Theorem 4).

In part a), we assume that $M_1$ and $(M_1+ M_2)/M_1$ are Artinian (resp. Noetherian); the same then holds for $M_1+ M_2$ (Proposition 3).

In part b), we assume that $M/M_2$ and $M_2/(M_1\cap M_2)$ are Artinian (resp. Noetherian); the same then holds for $M/(M_1\cap M_2)$ (loc. cit.).