Let $R$ be a Dedekind domain and $M$ be a nonzero finitely generated torsion $R$-module. In Curtis and Reiner's Methods of Representation Theory it states that $M$ has a composition series and if its factors are given by $R/P_i$ with $P_i$ maximal $R$-ideals, then the ideal $\mathrm{ord}(M)=\prod P_i$ is well defined.
I don't know why a composition series for $M$ exists as I'm not sure if $M$ is an Artinian module, and also it's not clear why $\mathrm{ord}(M)$ is well defined.
I'd be very grateful for some help.
A finitely generated torsion module $M$ over a Dedekind domain $R$ is artinian because it is a finitely generated $(R/\operatorname{Ann}M)$-module, and $\operatorname{Ann}M\ne0$. This shows that $M$ is an $R$-module of finite length.
Moreover, by Jordan-Holder theorem one knows that every two composition series are equivalent. But the quotient modules of a composition series are of the form $R/P$, with $P$ a maximal ideal, and then these maximal ideals are uniquely determined by $M$.