Example

Let us take $A = \mathbf{Z}$, $M = \mathbf{Z}$, $N = n\mathbf{Z}$ for some integer $n > 0$. If $n = p_1^{\alpha_1} \cdots p_k^{\alpha_k}$ is the decomposition of $n$ into prime factors,

is a primary decomposition of $n\mathbf{Z}$ in $\mathbf{Z}$ by Example 4 of no. 1.

By an abuse of language, the relation $N = \bigcap_{i \in I} \mathscr{Q}_i$ is called a primary decomposition of $N$ in $M$. It amounts to the same to say that $\{0\} = \bigcap_{i \in I} (\mathscr{Q}_i/N)$ is a primary decomposition of $\{0\}$ in $M/N$. If $(\mathscr{Q}_i)_{i \in I}$ is a primary decomposition of $N$ in $M$, the canonical mapping from $M/N$ to $\bigoplus_{i \in I} (M/\mathscr{Q}_i)$ is injective. Conversely let $N$ be a submodule of $M$ and $f$ an injective homomorphism from $M/N$ to a finite direct sum $P = \bigoplus_{i \in I} P_i$, where each set $\operatorname{Ass}(P_i)$ is reduced to a single element $p_i$; let $f_i$ be the homomorphism $M/N \to P_i$ obtained by taking the composition off with the projection $P \to P_i$, and let $Q_i/N$ be the kernel off.; then the $Q_i$ distinct from $M$ are primary with respect to $M$ (no. 1, Definition 1) and $N = \bigcap_{i \in I} Q_i$. Moreover, $\operatorname{Ass}(M/N) \subset \bigcup_{i \in I} \{p_i\}$ by virtue of § 1, no. 1, Proposition 3.