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This is probably one of those questions with a super obvious counterexample, but here goes.

Is a field necessarily a flat $\mathbb Z$-module?

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    $\begingroup$ Multiplication with 2 is injective $\mathbb Z \rightarrow \mathbb Z$. What happens if you $\otimes_{\mathbb Z} \mathbb F_2$ $\endgroup$ – Blah Mar 26 '12 at 20:03
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Over a PID (such as $\mathbb{Z}$) being flat is equivalent to being torsion-free. Therefore, if your field is torsion-free, it is flat, and if it has torsion, it is not flat.

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A $\mathbb{Z}$-module is flat if and only if it is torsion free, so it might depend on the characteristic of the field.

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