I am working on the following exercise.
If $M$ is a finitely generated module over a PID $A$ and $M′$ is a submodule, can we write $M = F ⊕ T$ and $M′ = F′ ⊕ T′$ such that $F$ and $F′$ are free, $T$ and $T′$ are torsion, and $F′ \subset F$ and $T′ \subset T$?
Following a hint, I've assumed $A$ contains an irreducible $p$. I then showed $M′ = A(p, \bar{1})$ is a free submodule of $M = A ⊕ A/(p)$.
Now I should show that $M'$ cannot be contained in the free part of $M$, but it seems to me that $M'$ is indeed contained in the free part of $M$. I'm not seeing how to use that $p$ is irreducible.
I would appreciate any help.