Let $\displaystyle \mathbb{H}$ be the quaternion algebra.

Show that $\displaystyle \mathbb{H}$ is a division algebra, i.e.

if xy = 0, then x = 0 or y = 0.

I understand this but I am not sure how to prove it. Is it as simple as just multiplying by x and y inverse?

$\displaystyle x^{-1}xy=y=0$ and $\displaystyle xyy^{-1}=x=0$

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