Boolean Ring has Proper Zero Divisor

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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.


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.