Are $\mathbb{Q}$ or $\mathbb{Z}$ flat modules?
- $\Bbb Z$ is a free, hence projective, hence flat $\Bbb Z$ module.
- $\Bbb Z$ modules are flat iff they are torsion-free, and $\Bbb Q_\Bbb Z$ is torsion-free. (And this is another reason $\Bbb Z_\Bbb Z$ is flat.)
- $\Bbb Z$ modules are injective iff they are divisible, so to produce a nonflat, noninjective module, it suffices to think of a nontorsion-free module and a nondivisible module and take their product. So, for example, $\Bbb Z/(n)\oplus\Bbb Z.$