Faithfully Flat Abelian Groups

Question

I need some help to find faithfully flat abelian groups.

Flat abelian groups are torsion free $\mathbb{Z}$-modules. But what about faithfully flat abelian groups. $\mathbb{Q}$ is an example that is torsionfree (flat) but not faithfully flat $\mathbb{Z}$-module.

How do we describe all faithfully flat abelian groups?

Thanks for your help...

Answer 1

A $\mathbb Z$-module $M$ is faithfully flat iff it is torsion free and $pM\ne M$ for any prime number $p$.