TITLE: Flat not projective, projective not free
QUESTION [8 upvotes]: I am looking for examples of a flat but not projective module, and of a projective but not free module.

REPLY [13 votes]: The rational numbers are a flat but not projective $\mathbb Z$-module.
$\mathbb Z\oplus 0$ is a projective but not free 
$\mathbb Z\oplus \mathbb Z$-module.