Proposition 2

— If M is a finitely generated torsion module over a principal ideal domain A, then the n-primary components of M are zero except for a finite number of them, and the projectors of M onto these components $M_\pi$ are homotheties.

This follows immediately from Lemma 1, for by Lemma 2 there exists $\alpha \neq 0$ in $\mathscr{A}$ such that $M = M(\alpha)$.