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 ?
2026-02-22 19:26:18.1771788378
Flat modules over a PID
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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.