If $M$ is a flat $R$-module and $rm=0$ for some $r$ and $m$, show $m=0$.

Question

Let $R$ be a commutative ring and let $M$ be a flat $R$-module. If $r\in R$ is not a zero-divisor and $m\in M$ is such that $rm=0$, prove that $m=0$.

I can't seem to figure out how to use flatness here. I've tried tensoring the exact sequence $$ 0\rightarrow (r)\rightarrow R\rightarrow R/(r)\rightarrow 0 $$ with $M$ but the induced exact sequence doesn't seem to help much. Is there something else I should try? Hints or an outright solution would be welcome.

Answer 1

Hint:

$r$ is a non-zero divisor on $R$ if and only if the sequence $$0\longrightarrow R\xrightarrow{\:\times r\:} R$$ is exact. This sequence remains exact on tensoring by $M$.