$0_M\otimes n=0$ for tensor product over non-commutative algebra.

Question

Let $R$ be a non-commutative ring, $M$ a right module and $N$ a left module and $M\otimes_R N$ their tensor product. I want to show that $0_M\otimes n=0$. If we would have the tensor product over a commutative ring, $M\otimes_R N$ would be a module and I would say $0_M\otimes n=0_R\cdot 0_M\otimes n = 0_R \cdot (0_M\otimes n) = 0_{M\otimes_R N}$ (I am not sure this is the way to do it). But here $R$ is actually non-commutative, so $M\otimes_R N$ is an abelian group and I am not too sure how to proceed.

Answer 1

Let $\alpha=0_M\otimes n$. Then $$\alpha+\alpha=0_M\otimes n+0_M\otimes n=(0_M+0_M)\otimes n=0_M\otimes n=\alpha$$ where I have used that $\otimes$ is additive in the first variable. As we are in a group, we may subtract $\alpha$ and deduce that $\alpha$ is the zero element of $M\otimes_R N$.