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I am looking for examples of a flat but not projective module, and of a projective but not free module.

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    More generally, torsion-free = flat for modules over a Dedekind domain.2011-08-29

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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.