This is a follow up to the post Showing $ m\otimes n$ is free given $m,n$ are free.
I am curious about conditions that would eliminate the possibility for a counterexample in the case that was outlined above.
Let $R$ be a commutative ring with identity. Let $M, N$ be $R$-modules with $m \in M$ and $0\neq n\in N$.
If $Rm$ is free, how do you show $m \otimes n \neq 0$?
I was thinking of showing the contrapositive... Does the problem reduce to showing that if $m \otimes n = 0 $ then $Rm$ is not free?