Let $A$ be an principal ideal domain, and $M$ an $A$-module. If $p$ is irreducible in $A$, let's define $\mathrm{Tor}_p(M):=\{m\in M\mid p^km=0\text{ for some }k\in\mathbb{N}\}.$
I need to show that if $M$ is finitely generated, then $\mathrm{Tor}(M)$ can be written as a direct sum of the submodules $\mathrm{Tor}_p(M)$.
Through the Decomposition Theorem, I have that $M$ is a direct sum of a finite number of cyclic modules and a free module. Thus, the torsion submodule of M is the direct sum of the torsion submodules of each of these cyclic submodules (the torsion submodule of the free module is zero). But how can I show that $\mathrm{Tor}(C_{p_i^{\alpha_i}})=\mathrm{Tor}_p(M)$ or that $\mathrm{Tor}(C_{p_i^{\alpha_i}})$ is a direct sum of several $\mathrm{Tor}_p(M)$, for some irreducibles $p\in A$, where $p_i^{\alpha_i}$ is the order of the cyclic submodule?