Lemma 2

Let M be a torsion module (II, p. 313) over an integral domain A. For every finite family $(x_i)_{1 \leq i \leq n}$ of elements of M, there exists an element $\gamma \neq 0$ in A such that the $x_i$ all belong to $M(\gamma)$. Indeed, for each index i there exists an element $\alpha_i \neq 0$ in A which annihilates $x_i$, and the element $\gamma = \prod_{i=1}^n \alpha_i$ will fit the bill.