Division Ring has No Proper Zero Divisors

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Theorem

Let $\left({R, +, \circ}\right)$ be a division ring.


Then $\left({R, +, \circ}\right)$ has no proper zero divisors.


Proof

Let $\left({R, +, \circ}\right)$ be a division ring whose zero is $\left\{{0_R}\right\}$ and whose unity is $1_R$.

By definition of division ring, every element $x$ of $R^* = R \setminus \left\{{0_R}\right\}$ has an element $y$ such that:

$y \circ x = x \circ y = 1_R$

That is, by definition, every element of $R^*$ is a unit of $R$.

The result follows from Unit Not Zero Divisor.

$\blacksquare$


Sources