Proposition 5

Let $A$ be a Noetherian ring, $M$ an $A$-module, $N$ a submodule of $M$, $N = \bigcap_{i \in I} Q_i$ a reduced primary decomposition of $N$ in $M$ and, for all $i \in I$, let $\{p_i\} = \mathrm{Ass}(M/Q_i)$. If $p_i$ is a minimal element of $\mathrm{Ass}(M/N)$, $Q_i$ is equal to the saturation on $N$ with respect to $p_i$ (Chapter 11, § 2, no. 4) (cf. Exercise 2).

We can obviously restrict our attention to the case where $N = 0$, replacing if need be $M$ by $M/N$. If $p_i$ is minimal in $\mathrm{Ass}(M)$, the set of elements of $\mathrm{Ass}(M)$ which do not meet $A - p_i$ reduces to $p_i$; the proposition then follows from formula (3) above and § 1, no. 2, Proposition 6, the kernel of the canonical mapping $M \to M_0$, being equal to the saturation of 0 with respect to $p_i$ (Chapter 11, § 2, no. 4).