# Boolean Ring has Proper Zero Divisor

## Theorem

Let $\left({R, +, \circ}\right)$ be a Boolean ring whose zero is $0_R$.

Suppose that $R$ has more than two elements.

Then $R$ has a proper zero divisor.

## Proof

Since $R$ has more than two elements, there exist distinct non-zero elements $x, y \in R$.

Note that $x + y \ne 0_R$ since $x$ and $y$ are distinct (by Idempotent Ring has Characteristic Two).

If $x \circ y = 0$, $x$ is a proper zero divisor.

If $x \circ y \ne 0$, then:

 $\ds \left({x \circ y}\right) \circ \left({x + y}\right)$ $=$ $\ds x \circ y \circ x + x \circ y \circ y$ $\ds$ $=$ $\ds x \circ y + x \circ y$ $R$ is an idempotent ring, Idempotent Ring is Commutative $\ds$ $=$ $\ds 0_R$ Idempotent Ring has Characteristic Two

Hence $x \circ y$ is a proper zero divisor.

The result follows, from Proof by Cases.

$\blacksquare$