I have to show that if $R$ is an Euclidean domain and $M$ is a finitely generated torsion-free $R$-module, then $M$ is free using the Structure Theorem. I can prove it using basic facts but can't see how to apply the Structure Theorem. Does it suffice to show that Tor(M) is isomorphic to the direct sum of $\frac{R}{<d_i>}$ where the $d_i$'s are the invariant factors from the Structure Theorem and if yes, why?
Cheers