Unit of Ring is not Zero Divisor

Theorem
A unit of a ring is not a zero divisor.

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

Let $x$ be a unit of $\struct {R, +, \circ}$.

$x$ is such that:
 * $x \circ y = 0_R, y \ne 0_R$

Then:

From this contradiction it follows that $x$ cannot have such a property.

Thus by Proof by Contradiction $x$ is not a zero divisor.