As a counter example for a projective module which is not free my instructor gave this one:
$\Bbb Z_6=\Bbb Z_2\oplus\Bbb Z_3$
$\Bbb Z_6$ projective, so $\Bbb Z_2$ and $\Bbb Z_3$ are projective but $\Bbb Z_2$ is not free.
But he also gave
Any ring with unity as an example of a free module. I know $\Bbb Z_2$ is a ring with unity, So shouldn't it be free.
$\mathbf Z_6$ is a free $\mathbf Z_6$-module, and the Chinese remainder theorem says it is isomorphic to $\;\mathbf Z_2\oplus\mathbf Z_3$ (both as $\mathbf Z$-modules and as $\mathbf Z_6$-modules). Projective modules are direct summands of free modules, hence $\;\mathbf Z_2\;$ and $\;\mathbf Z_3\;$ are projective $\mathbf Z_6$-modules.
Note:
To remove any ambiguity, none of these $$\mathbf Z$-modules is projective, as they have torsion.