Let $R$ be a commutative ring, $M$ be an $R$-module, and $N$ be a submodule of $M$. Assume that both $M$ and $N$ are flat, so we can identify $N\otimes\_RN$, $M\otimes\_RN$, and $N \otimes\_RN$ as submodules of $M\otimes\_RM$.


Is it true that $N\otimes\_R N = (M\otimes\_RN)\cap (N\otimes\_R M)$? If not in general, under which conditions is this true?



Intersection and tensor product of flat modules
This is not true in general: take $M=R$, $N=Rx$ for some $x\in R$. Then $N\otimes \_RN=Rx^2$ while $M\otimes \_RN=N\otimes \_RM=Rx$.


It is true if $M/N$ is flat: this follows from Proposition 7 of §2.6 in Bourbaki's *Commutative algebra*, ch. I.


