# Flat not projective, projective not free [duplicate]

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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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## marked as duplicate by rschwieb, Did, Micah, Erick Wong, AmzotiMar 20 '13 at 14:25

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

Note that torsion-free = flat for abelian groups –  t.b. Mar 24 '11 at 11:42
More generally, torsion-free = flat for modules over a Dedekind domain. –  Amitesh Datta Aug 29 '11 at 0:43

## 1 Answer

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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