Ring Element is Zero Divisor iff not Cancellable

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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 $x \ne y$.

$\blacksquare$


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