Proposition 9

— Let $A$ be a principal ideal domain, let $P$ be a system of representatives of irreducible elements of $A$ and let $M$ be a finitely generated $A$-module. Then there exist positive integers $m(0)$ and $m(p^n)$ ($p \in P, n > 0$), uniquely determined by $M$ and zero except for finitely many of them, such that $M$ is isomorphic to the direct sum of $A^{m(0)}$ and the $(A/(p^n))^{m(p^n)}$ ($p \in P, n > 0$).

The existence of the integers $m(0)$ and $m(p^n)$ ($p \in P, n > 0$) follows from Prop. 8. The integer $m(0)$ is uniquely determined: it is the rank of the free module which is the quotient of $M$ by its torsion submodule. Finally, the $p$-primary component of $M$ is isomorphic to the direct sum of the $(A/(p^n))^{m(p^n)}$; since the family of ideals $(p^n)$ ($n \geq 1$) is totally ordered by inclusion, the uniqueness of the $m(p^n)$ follows from the Cor. to Prop. 2 of VII, p. 16.