Module tensor product of an element with zero Is it true that for any two $R$-modules, $N$, $M$, that the tensor product $n \otimes 0 = 0 \otimes m = 0 \space\space \forall \space n \in N, m \in M$?
The reason I am thinking this is true is because since $r(n\otimes m) = nr \otimes m = n \otimes rm$ then if we take $r = 0$: $0(n \otimes m) = n0 \otimes m = n \otimes 0m = 0 \otimes m = n \otimes 0 = 0$.
Is there anything wrong with my reasoning?

No this is right, in every tensor product the elements $x \otimes 0$ and $0 \otimes y$ represents the trivial element. This follows from your argument.
