# Unit 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 $\struct {R, +, \circ}$.

Aiming for a contradiction, suppose $x$ is such that:

- $x \circ y = 0_R, y \ne 0_R$

Then:

\(\displaystyle \paren {x^{-1} \circ x} \circ y\) | \(=\) | \(\displaystyle 0_R\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 1_R \circ y\) | \(=\) | \(\displaystyle 0_R\) | $\quad$ Definition of Inverse Element | $\quad$ | ||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 1_R\) | \(=\) | \(\displaystyle 0_R\) | $\quad$ as $y \ne 0_R$ | $\quad$ |

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

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

$\blacksquare$