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.

Let $M \otimes_R N$ denote their tensor product.

Then:


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

is the zero in $M \otimes_R N$.

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

Then

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