Unit of Ring is not Zero Divisor

Theorem
A unit of a ring can not be a zero divisor.

Proof
Let $\left({R, +, \circ}\right)$ be a ring.

Let $x$ be a unit of $\left({R, +, \circ}\right)$, and suppose $x \circ y = 0_R, y \ne 0_R$.

Then:

... thus deriving a contradiction.