Division Ring has No Proper Zero Divisors

Theorem
Let $\struct {R, +, \circ}$ be a division ring.

Then $\struct {R, +, \circ}$ has no proper zero divisors.

Proof
Let $\struct {R, +, \circ}$ be a division ring whose zero is $0_R$ and whose unity is $1_R$.

By definition of division ring, every element $x$ of $R^* = R \setminus \set {0_R}}$ 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 of Ring is not Zero Divisor.