Theorem 1

Let $M$ be a finitely generated module over a Noetherian ring and let $N$ be a submodule of $M$. There exists a primary decomposition of $N$ of the form

where, for all $p \in \operatorname{Ass}(M/N)$, $Q(p)$ is $p$-primary with respect to $M$.

We may replace $M$ by $M/N$ and therefore suppose that $N = 0$. By § 1, no. 4, Corollary to Theorem 2, $\operatorname{Ass}(M)$ is finite; by § 1, no. 1, Proposition 4, there exists, for each $p \in \operatorname{Ass}(M)$, a submodule $Q(p)$ of $M$ such that $\operatorname{Ass}(M/Q(p)) = \{p\}$ and $\operatorname{Ass}(Q(p)) = \operatorname{Ass}(M) - \{p\}$. Let us write $P = \bigcap_{p \in \operatorname{Ass}(M)} Q(p)$; for all $p \in \operatorname{Ass}(M)$, $\operatorname{Ass}(P) \subset \operatorname{Ass}(Q(p))$ and hence $\operatorname{Ass}(P) = \varnothing$, which implies $P = 0$ (§ 1, no. 1, Corollary 1 to Proposition 2) and therefore proves the theorem.