How describe the structure of $M/\mathrm{Tor}(M)$?

Question

The following problem is from Dummit and Foote, 12.1.13:

If $M$ is a finitely generated module over the P.I.D. $R$, describe the structure of $M/\mathrm{Tor}(M)$.

What do they want? I know I can write $\mathrm{Tor}(M)$ as a direct product of Quotients and similarly for $M$ but including and extra $R^r$. Should I show that $M/\mathrm{Tor}(M)$ is torsion free? I just don't know.

Answer 1

Using the fundamental theorem of finitely generated modules over a PID, and a bit of elbow grease, you should be able to prove that $$M/\mathrm{Tor}(M) \simeq R^t,$$ for an appropriate choice of $t$.

Answer 2

The fundamental theorem for finitely generated modules over a PID can be proven in the following manner: first one shows that

$$ M \simeq t(M) \oplus M/t(M) \tag{1}. $$

Then, $t(M)$ can be further studied via observing $Ann_R(t(M))$ primary decomposition, etc. The other component, however, is much simpler to describe (which is what you are being asked to do): $M/t(M)$ is always a free module when $M$ is a finitely generated module over a PID.