let be the torsion submodule of we have for some it's very easy to see that and thus:

the exact sequence tells us that is finitely generated and thus for some so the claim is that right now i don't know a quick way to prove

b) We have an exact sequence of modules over a PID . Show that is a free -module of rank n-m ( denoting the submodule of torsionelements).

the claim but i'm sure it's not hard. one idea is to use the given exact sequence to find an exact sequence which obviously implies that