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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 $N$.

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