Lemma 2

Let A be a noetherian ring and M an A-module. The canonical mapping of M into the product $\prod_{p \in \mathrm{Ass}(M)} M_p$ is injective.

Let indeed m be a nonzero element of M; then Ann(m) is contained in a prime ideal p of A associated with M (IV, § 1, No. 1, prop. 2), and the image of m in $M_p$ is nonzero (II, § 2, No. 2, prop. 4).