Are $\mathbb{Q}$ or $\mathbb{Z}$ flat modules?

Question

I have the following three question about flat modules.

1. Why is not $\mathbb{Z}$ a flat $\mathbb{Z}$-module.

2. Why is $\mathbb{Q}$ a flat module $\mathbb{Z}$-module.

3. I need an example of a module which is not flat, nor injective, nor projective.

Thanks a lot!

Note: I need only a hint to attack those problems not the complete answer.

Answer 1

1. $\Bbb Z$ is a free, hence projective, hence flat $\Bbb Z$ module.
2. $\Bbb Z$ 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.)
3. $\Bbb Z$ 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, $\Bbb Z/(n)\oplus\Bbb Z.$