Is the following proof that $\mathbb{Q}$ is not finitely generated correct?
If $\mathbb{Q}$ were finitely generated so would the quotient module $\mathbb{Q}/\mathbb{Z}$ (generated by the cosets of the generators of $\mathbb{Q}$). But the latter is a torsion module, so by the classification theorem it would have the form $\bigoplus_{i=1}^n \mathbb{Z} / p_i^{r_i}$ (torsion module -> no free part). But this is finite, whereas $\mathbb{Q} / \mathbb{Z}$ is clearly infinite.