[Math] Torsion module over PID

Question

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 $N$.

Answer 1

Here's an ugly argument that I know of (which works for abelian groups, and so I assume it works over arbitrary PID). You need to make three observations:

1. Maximal cyclic $p$-submodules of a $p$-module $M$ are the direct summands of $M$.
2. Every cyclic $p$-submodule of $M$ is contained in a maximal cyclic $p$-submodule.
3. If $N_1\subset M_1$ and $N_2\subset M_2$ are cyclic $p$-submodules of $M$, then $M_1$ and $M_2$ are disjoint if and only if $N_1$ and $N_2$ (this follows from observing that the submodules of a cyclic $p$-module are totally ordered under inclusion, and considering where $M_1\cap M_2$ is relative to $N_1$ and $N_2$ as a submodule of $M_1$ or $M_2$; you may have to use the fact that cyclic $p$-modules are indecomposable).

Once you have proven those, take a direct sum decomposition of $N$ into cylic $p$-submodules $N_1\oplus N_2\oplus\dots\oplus N_i$ (which exists by structure theorem of finitely generated modules over a PID), and use 1., 2. and 3. to obtain a decomposition $M=M_1\oplus\dots M_n\oplus M'$, where $M_i$ is the maximal cyclic $p$-submodule that contains the cyclic $p$-submodule $N_i$.

*by $p$-module I mean a module annihilated by some power of the prime $p$ in the PID $R$ (it's the primality of $p$ that matters, not irreducibility, for the total ordering of submodules, though the two coincide over PIDs, which is where you need to be anyway for the structure theorem to work)