Let $R$ be a Dedekind domain and $M$ a torsion module over $R$. Then $M$ is a finitely generated module over the ring $S = R/ \operatorname{Ann}_R(M)$, and any composition series of $M$ over $S$ is a composition series over $R$. Note that $\operatorname{Ann}_R(M)$ is non-zero because $M$ is torsion, and according to Wikipedia it follows that $S$ is a (principal) Artinian ring. As described on the same page, it follows from Hopkins' theorem that $M$ is of finite length over $S$, and therefore over $R$.
