When do tensor products of elements coincide

Question

Let $M,N$ be $R$-modules and $m \otimes n, m' \otimes n' \in M \otimes_R N$ non-zero (EDIT) elements. When does $m \otimes n = m' \otimes n'$ hold? Obviously, this is true if either $(m',n)=(rm,rn')$ or $(m,n')=(sm',sn)$ for some $r,s \in R$. Is this condition also necessary? I have the strong feeling that the abelian group structure can do no harm here, but I'm uncapable of proving it...

Answer 1

$1\otimes1=0\otimes0$ in $\mathbb{Z}/2\mathbb{Z}\otimes_\mathbb{Z}\mathbb{Z}/3\mathbb{Z}$, so your condition is not necessary.