## Every torsion module over a principal ideal domain is the direct sum of its p-primary components

Let $R$ be a principal ideal domain and let $N$ be a torsion $R$-module. Prove that the $p$-primary component of $N$ is a submodule for every prime $p \in R$. Prove that $N$ is the direct sum of its $p$-primary components.

We proved this in a previous exercise.