1. Tensorial Product

Hi,
I have to solve this question, I appreciate your help.
The ring $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C}$ isn't an Integral Domain
The multiplication operation in this ring is the usual complex multiplication by components.

Thanks

2. let $x = 1 \otimes i - i \otimes 1$ and $y = 1 \otimes i + i \otimes 1$. what is $xy$? note that both $x$ and $y$ are non-zero because $\{1,i\}$ is linearly independent over $\mathbb{R}$.

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