Lemma 1

Let M be a module over a principal ideal domain A; for all $a \in A$ such that $a \neq 0$, let $\alpha = \varepsilon \sum_{i=1}^r \pi_i^{n(i)}$ be a decomposition of $\alpha$ into irreducible factors (VII, p. 4). The submodule $N = M(a)$ of elements of M annihilated by a is the direct sum of the submodules $M(\pi_i^{n(i)})$, and the map which sends each $x \in M(\alpha)$ to its component in $M(\pi_i^{n(i)})$ has the form $x \mapsto \gamma_i x \ (\gamma_i \in A)$. Moreover

Note first that N is annihilated by a, so has a natural $A/(\alpha)$-module structure. By Prop. 4 of VII, p. 3, the canonical homomorphism from $A/(\alpha)$ into the product of the rings $A/(\pi_i^{n(i)})$ is a ring isomorphism; now applying Prop. 1 of VII, p. 6, we deduce that N is the direct sum of the $M(\pi_i^{n(i)})$; the projectors of this decomposition are homotheties, by VII, p. 7, Remark 2. The inclusion $M(\pi_i^{n(i)}) \subset M(\alpha) \cap M_{\pi_i}$ is obvious; conversely, let $x \in M(\alpha) \cap M_{\pi_i}$. Then there exists a power $\pi_i^s$ of $\pi_i$ which annihilates x; we may assume $s \geq n(i)$; by Bezout's identity, there exist $\lambda, \mu \in A$ such that $\pi_i^{n(i)} = \lambda \pi_i^s + pa$, so $\pi_i^{n(i)} x = 0$ and so finally $x \in M(\pi_i^{n(i)})$.