Corollary 1

If an A-module $M$ is the direct sum of a family $(M_i)_{i \in I}$ of submodules, then $\operatorname{Ass}(M) = \bigcup_{i \in I} \operatorname{Ass}(M_i)$.

It may be reduced to the case where $I$ is finite by means of (1), then to the case where $\operatorname{Card}(I) = 2$ by induction on $\operatorname{Card}(I)$. Then let $I = \{i, j\}$, where $i \neq j$; as $M/M_i$ is isomorphic to $M_j$, $\operatorname{Ass}(M) \subset \operatorname{Ass}(M_i) \cup \operatorname{Ass}(M_j)$ (Proposition 3); moreover, $\operatorname{Ass}(M_i)$ and $\operatorname{Ass}(M_j)$ are contained in $\operatorname{Ass}(M)$ (Proposition 3), whence the result.