The flat module, module is not flat

Question

Why $\mathbb{Z}$-module $\mathbb{Q}$ is flat and $\mathbb{Z}$-module $\mathbb{Z}_n$ is not flat? P/s: How can I prove them by definition and without functor. Thankyou.

Answer 1

$\Bbb Z$-modules are flat iff they are torsion-free. Hence $\Bbb Q$ is flat, but $\Bbb Z/n$ is not.

References:

Show that a Z-module A is flat if and only if it is torsion free?