Flat not projective, projective not free [duplicate]

commutative-algebra modules

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


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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