Ring Element is Zero Divisor iff not Cancellable

Theorem
Let $\struct {R, +, \circ}$ be a ring which is not null.

Let $z \in R^*$.

Then $z$ is a zero divisor $z$ is not cancellable for $\circ$.

Sufficient Condition
Let $z$ be a zero divisor.

Then either $z \circ x = 0_R$ or $x \circ z = 0_R$ for some $x \in R^*$.

Then:
 * $z \circ 0_R = 0_R = 0_R \circ z$

and so $z$ is not cancellable.

Necessary Condition
Let $z$ not be cancellable in $R$.

Then there exists $x, y \in R$ such that $x \ne y$ and:
 * $z \circ x = z \circ y$

Then:

But $x \ne y$, so $x + \paren {-y} \ne 0$.

Thus $z$ is a zero divisor.

Similarly if $x \circ z = y \circ z$ where $x \ne y$.