Recently I came across the following statement about the criterion of flat modules which looks somewhat like Baer's Criterion for injective modules:
Statement. Let $R$ be a commutative ring. An $R$-module $M$ is flat if and only if for every ideal $I\subset R$ the canonical map $I\otimes_{R} M \rightarrow M, \; r\otimes m \mapsto rm$ is an injective $R$-homomorphism.
We use the definition of flatness: an $R$-module $M$ is called a flat module if the functor $- \otimes_R M$ is an exact functor.
I think this is true, and one direction is obvious by the definition. But I have no idea how to prove (or disprove) the other direction.
Any help is appreciated.