Intersection and tensor product of flat modules

Question

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?

Answer 1

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.