# Idempotent Ring has Characteristic Two/Corollary

## Theorem

Let $\struct {R, +, \circ}$ be an idempotent non-null ring.

Then:

$\forall x \in R: -x = x$

## Proof

Let $0_R$ denote the zero of $R$.

Let $x \in R$.

Then:

 $\ds x + x$ $=$ $\ds 0_R$ Idempotent Ring has Characteristic Two $\ds \leadsto \ \$ $\ds -x + x + x$ $=$ $\ds -x + 0_R$ $\ds \leadsto \ \$ $\ds x$ $=$ $\ds -x$

Hence the result.

$\blacksquare$