Tensor with Zero Element is Zero in Tensor

Theorem
Let $R$ be a ring

Let $M$ be a right $R$-module

Let $N$ be a left $R$-module

and $M\otimes_R N$ their tensor product

Then


 * $0\otimes_R n = m\otimes_R 0 = 0\otimes_R 0$

is the $0$ in $M\otimes_R N$.

Proof
Let $m \in M$ and $n \in N$

Then

which implies that $0\otimes_R n$, $m \otimes_R 0$ and $0 \otimes_R 0$ must all be identity elements for $M \otimes_R N$ as a left module