Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

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

share|improve this question
Multiplication with 2 is injective $\mathbb Z \rightarrow \mathbb Z$. What happens if you $\otimes_{\mathbb Z} \mathbb F_2$ –  Blah Mar 26 '12 at 20:03

2 Answers 2

up vote 10 down vote accepted

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.

share|improve this answer

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

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.