Flat modules are torsion-free

Question

I need to prove the following assertion:

Let $A$ be an integral domain. If $M$ is a flat $A$-module, then $M$ is torsion-free.

My definition of a flat module is: an $R$-module $F$ is flat if the functor $F \otimes_R \star \colon M \mapsto F \otimes_R M$ transforms exact sequences in exact sequences.

I am confused since I cannot find the relation between the definition and to be torsion-free.

Does anyone have some recommendation for me?

Thank you.

Answer 1

Hint : let $r\in A$ and let $f: A\to A$ be the map "multiplication by $r$".

What happens if you tensor it with $M$ ?