Ring Element is Zero Divisor iff not Cancellable

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Theorem

Let $\struct {R, +, \circ}$ 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

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.

$\Box$


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:

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


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 $x \ne y$.

$\blacksquare$


Sources

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