Free modules are projective, and projective modules are direct summands of free modules. 


  Are there examples of projective modules that are not free? 


(I know this is not possible for modules of fields.)

Free modules are torsion-free. But is the inverse true? 


  Are there examples of torsion-free modules that are not free?


Thank you~

Let $R=\mathbb{Z}/6\mathbb{Z}$. Obviously, $R$ is a free module over itself. Because $\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/3\mathbb{Z}\cong R$, we have that $\mathbb{Z}/2\mathbb{Z}$, considered as an $R$-module, is projective, but it cannot be free - any non-trivial direct sum of $R$'s would have at least 6 elements. 

The Baer-Specker group is an example of a torsion-free $\mathbb{Z}$-module which is not free.
