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Suppose I have $R$ a ring, and let M,N be two $R$-bimodules.

Suppose I have $$ (m_1 \otimes_Rn_1)=(m_2 \otimes_R n_2). $$

Then what can I say about $m_1$ and $m_2$ and, respectively, $n_1$ and $n_2$?

Thanks!

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    I think you can say that there are $r,s\in R$ such that $sm_1=rm_2$ and $rn_1=sn_2$, and $m\in M, n\in N$ such that these two are $rsm$ abs $rsn$ respectively. At the very least, assuming that such elements exist – 2017-02-01
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    Yeah I would think of a relation like this, I would have hoped to find an $r$ such that $rm_1=m_2$ honestly though. – 2017-02-01
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    There is no guarantee that $s$ is invertible, so it's not that simple. – 2017-02-01

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In general you can not say anything, since you can think for two modules when you tensor them get zero. Take any two elements both should be equal but there are no relation between them. For example take $M=Z_2$ , $N=Z_3$ it is clearly that $$(Z_2\otimes_R Z_3)=0$$.