Let $R$ be a commutative ring. If $M$ is an $R$-module such that for every finitely generated prime ideal $P$ of $R$, the map $i\otimes Id: P\otimes_R M \to R \otimes_R M\cong M$ is injective, then is $M$ a flat $R$-module ? If this is not true in general, then what if we assume the injectivity for all prime ideals ? Or assume $R$ to be Noetherian ?
I know that to show flat-ness, it would be enough to show injectiveness of $I\otimes_R M \to R \otimes_R M\cong M$ for every finitely generated ideal $I$ of $R$, but I'm unable to see whether that is true here or not.