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

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

Without loss of generality, 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:

\(\ds y\)

\(=\)

\(\ds 1_R \circ y\)

Definition of Unity of Ring

\(\ds \)

\(=\)

\(\ds x^{-1} \circ x \circ y\)

Definition of Unit of Ring

\(\ds \)

\(=\)

\(\ds x^{-1} \circ 0_R\)

$x \circ y = 0_R$

\(\ds \)

\(=\)

\(\ds 0_R\)

Ring Product with Zero

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

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

$\blacksquare$