Proposition 2

Let M be a module over a Noetherian ring A, p a prime ideal of A and $(\mathscr{Q}_i)_{i \in I}$ a non-empty finite family of submodules of M which are p-primary with respect to M. Then $\bigcap_{i \in I} \mathscr{Q}_i$ is p-primary with respect to M.

$M / (\bigcap_{i \in I} \mathscr{Q}_i)$ is isomorphic to a submodule $\neq 0$ of the direct sum $\bigoplus_{i \in I} (M / \mathscr{Q}_i)$. Now

(§ 1, no. 1, Corollary 1 to Proposition 3). Hence $\operatorname{Ass}\left(M / (\bigcap_{i \in I} \mathscr{Q}_i)\right) = \{p\}$ (§ 1, no. 1, Proposition 3 and Corollary 1 to Proposition 2).