Unit Not Zero Divisor

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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:

\(\displaystyle \paren {x^{-1} \circ x} \circ y\) \(=\) \(\displaystyle 0_R\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle 1_R \circ y\) \(=\) \(\displaystyle 0_R\) Definition of Inverse Element
\(\displaystyle \leadsto \ \ \) \(\displaystyle 1_R\) \(=\) \(\displaystyle 0_R\) as $y \ne 0_R$

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

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

$\blacksquare$