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, $$\left({R, +, \circ}\right)$$ and suppose $$x \circ y = 0_R, y \ne 0_R$$. Then:

... thus deriving a contradiction.