Show that $\mathbb{C} \otimes_\mathbb{R} \mathbb{C}$ is not an integral domain

Question

Why is $\mathbb{C} \otimes_\mathbb{R} \mathbb{C}$ not an integral domain?

My professor laid out a proof in class today. Some clarification would be very helpful.

Consider: $$(i \otimes i)^2 = -1 \otimes -1 = 1 \otimes 1$$

This implies that: $$(i \otimes i - 1 \otimes 1)(i \otimes i + 1 \otimes 1) = 0$$ But neither factors are 0, therefore $\mathbb{C} \otimes_\mathbb{R} \mathbb{C}$ is not an integral domain. $\blacksquare$

Can anyone show me why $(i \otimes i)^2 = -1 \otimes -1 = 1 \otimes 1$ and how this implies that the product is equal to 0?

Answer 1

Observe that

$$\begin{align} (i\otimes i)^2 &=i^2\otimes i^2\\ &=(-1)\otimes (-1)\\ &=(-1)(-1)\otimes 1\\ &=1\otimes 1 \end{align}$$

and

$$\begin{align} (i\otimes i-1\otimes 1)(i\otimes i+1\otimes 1)&=(i\otimes i)(i\otimes i)+(i\otimes i)(1\otimes 1)\\ &-(1\otimes 1)(i\otimes i)-(1\otimes 1)^2\\ &=(i\otimes i)^2-(1\otimes 1)=0. \end{align}$$