# Ring Element is Zero Divisor iff not Cancellable

## Theorem

Let $\left({R, +, \circ}\right)$ be a ring which is not null.

Let $z \in R^*$.

Then $z$ is a zero divisor if and only if $z$ 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:

 $\displaystyle z \circ \left({x + \left({- y}\right)}\right)$ $=$ $\displaystyle z \circ x + z \circ \left({- y}\right)$ $\circ$ distributes over $+$ $\displaystyle$ $=$ $\displaystyle z \circ x + \left({- z \circ y}\right)$ Product with Ring Negative $\displaystyle$ $=$ $\displaystyle 0_R$ as $z \circ x = z \circ y$

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 $x \ne y$.

$\blacksquare$