Unit of Ring is not Zero Divisor

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

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

Then $x$ is neither a left zero divisor nor a right zero divisor of $\struct {R, +, \circ}$.

Proof
$x$ is either a left zero divisor or a right zero divisor of $\struct {R, +, \circ}$.

, suppose $x$ is a left zero divisor of $\struct {R, +, \circ}$.

That is:


 * $x \circ y = 0_R$

for some $y \in R \setminus \set {0_R}$.

Then:

This contradicts the assertion that $y \ne 0_R$.

Thus by Proof by Contradiction $x$ is neither a left zero divisor nor a right zero divisor of $\struct {R, +, \circ}$.