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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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Note that torsion-free = flat for abelian groups –  t.b. Mar 24 '11 at 11:42
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More generally, torsion-free = flat for modules over a Dedekind domain. –  Amitesh Datta Aug 29 '11 at 0:43
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marked as duplicate by rschwieb, Did, Micah, Erick Wong, Amzoti Mar 20 '13 at 14:25

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up vote 9 down vote accepted

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.

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