Theorem 1

Let M be a torsion module over a principal ideal domain A; for each irreducible element $\pi$ of A, let $M_\pi$ be the submodule of M consisting of elements annihilated by some power of $\pi$. If P is a system of representatives of irreducible elements of A, then M is the direct sum of its submodules $M_\pi$ for $\pi \in P$. Every element $x \in M$ belongs to the submodule $M(\alpha)$ for some $a \neq 0$, so by Lemma 1 is a sum of a finite number of elements, each of which belongs to some submodule $M_\pi$. On the other hand, if $\sum_{\pi \in P} x_\pi = \sum_{\pi \in P} y_\pi$, where $x_\pi, y_\pi \in M_\pi$ for all $\pi \in P$, and where all but finitely many of the $x_\pi$ and $y_\pi$ are zero, then Lemma 2 shows that there exists $\gamma \neq 0$ in A such that all the $x_\pi$ and $y_\pi$ belong to the same submodule $M(\gamma)$; applying Lemma 1 to $M(\gamma)$ shows that $x_\pi = y_\pi$ for all $\pi \in P$, which completes the proof.

Clearly, if $\pi$ and $\pi'$ are two associate irreducible elements, then $M_\pi = M_{\pi'}$; thus, for a given module M, the submodule $M_\pi$ depends only on the ideal $(\pi)$ of A; it is called the rr-primary component of the module M, and the decomposition of M as a direct sum of the $M_\pi$ is called the canonical decomposition of M as a direct sum of its rr-primary components.