so I am learning about flat modules and I found this criterion for flatness. An $R$-module $M$ is flat iff for every finitely generated ideal $I$, we get that the multiplication map $I\otimes_R M\rightarrow M$ is an injection (Eisenbud Proposition 6.1)
I am confused because say we have any $M$ (an $R$ module), and say that $i\otimes m\mapsto 0$. This means that $im=0$, but wouldn't this imply that $i\otimes m=1\otimes im=1\otimes 0=0$, so the kernel is trivial, and hence the map is injective always?
I am obviously doing something wrong, so I'd appreciate anyone that could clarify things a bit!