TITLE: Example of modules that are projective but not free; torsion-free but not free
QUESTION [45 upvotes]: 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~

REPLY [4 votes]: Given any two rings $R_\pm$.
Take the ring $R:=R_+\oplus R_-$.
Consider the free $R$-module $F:=R$.
Then it is projective as it is free.
Consider the summand $R$-module $P:=R_+\oplus 0$.
Then it is projective as a summand of a projective.
Now for any finite rings $\# R_\pm<\infty$:
By counting argument the constructed module is not free!