Division Ring has No Proper Zero Divisors

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.