Ring Element is Zero Divisor iff not Cancellable

Theorem
Let $$\left({R, +, \circ}\right)$$ be a non-null ring.

Let $$z \in R^*$$.

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

Proof

 * If $$z \circ x = 0_R$$ or $$x \circ z = 0_R$$ for some $$x \in R^*$$, then $$z$$ can not be cancellable for $$\circ$$ since $$z \circ 0_R = 0_R = 0_R \circ z$$.


 * Let $$z \circ x = z \circ y$$ where $$x \ne y$$. Then:

$$ $$ $$

But $$x \ne y$$, so $$x + \left({- y}\right) \ne 0$$.

Thus $$z$$ is a zero divisor.

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