Torsion module over PID Suppose $p$ is irreducible and $M$ is a tosion module over a PID $R$  that  can be
written as a direct sum of cyclic submodules with annihilators of the form $p^{a_1} | \cdots | p^{a_s}$ and $p^{a_i}|p^{a_i+1}$. Let now $N$ be a submodule
of $M$. How can i prove that $N$ can be written a direct sum of cyclic modules with
annihilators of the form $p^{b_1} | \cdots | p^{b_t}, t\leq s$ and $\ p^{b_i}| p^
{a_(s-t+i)}$?
I've already shown that $t\leq s$ considering the epimorphism from a free module to $M$ and from its submodule to $N$.

just look at the size of the subgroups annihilated by p^k for various k.
