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I know any f.g. flat module over a PID is projective. I am searching about does any flat module over a PID have the same feature? I consider $\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$ which are not free and flat, am I right ?

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Over $\Bbb Z$, $\Bbb Q$ is flat but not free. For $\Bbb Z$-modules, flatness is the same as torsion-freeness, but $\Bbb Q$ is divisible, and a nonzero projective module is never divisible.

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  • $\begingroup$ I don’t know anything about torsion freeness yet, and I’m not allowed to use it, all I know is Q is not free over Z, so Q is not projective, Q/Z is not flat and in any PID any projective module is free, I have to answer this question with these also I don’t know how to show Projective modules are not divisible, could you please help me with these ? $\endgroup$
    – Math90
    Jan 26 '18 at 7:20
  • $\begingroup$ A projective module is a subset of a free module. A non-zero element of a free Abelian group can only be divided by finitely many positive integers. $\endgroup$ Jan 26 '18 at 7:25

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