Flat modules and non-vanishing of simple tensors

Question

Let $R$ be a noncommutative ring, and $M$ and $N$ be a right and a left $R$-module respectively. If $M$ or $N$ is projective, then $$ 0 \neq m \otimes n \in M \otimes_R N, ~~~ \textrm{for all } n \in N, m \in M. $$ Is this true is $M$ or $N$ are flat, or torsion free.