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Let $R$ be an integral domain, and let $M$ be a faithful $R$-module. Consider the $R$-module $M\otimes_R M$. If $x_1\otimes y_1=x_2\otimes y_2$ for $x_1,x_2,y_1,y_2\in M$, does this imply that there is some $0\neq\alpha\in R$ such that $x_1=\alpha x_2$ and $\alpha y_1=y_2$?

It seems that if we take $M=R^n$ it works, but I don't have an idea for the general case.

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