What are the free and torsion parts of $\mathbb{Z}\times\mathbb{Z}$, $\mathbb{Z}\times\mathbb{Z}_6$, and $\mathbb{Q}/\mathbb{Z}$?

Question

So I'm given the $\mathbb{Z}$-modules: $\mathbb{Z} \times \mathbb{Z}$, $\mathbb{Z} \times \mathbb{Z}_{6}$, and $\mathbb{Q}/\mathbb{Z}$.

I am told to find state whether it is a free or torsion $\mathbb{Z}$-module and if neither give the torsion and free part. Looking at the first one I know it would be free as the basis could be $\{(1, 0), (0, 1)\}$ however I am a little lost on the others. Please let me know. Any help appreciated.

Answer 1

For the final $\mathbb{Z}$-module, let $\left[\frac{p}{q}\right] \in \mathbb{Q}/\mathbb{Z}$. Note that $k\left[\frac{p}{q}\right] = \left[\frac{kp}{q}\right]$ and if $a$ is an integer, $[a] = 0$. Can you find an integer $k \neq 0$ such that $\frac{kp}{q}$ is an integer? If so $\left[\frac{p}{q}\right]$ is an element of the torsion submodule.