# Idempotent Ring has Characteristic Two

## Theorem

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

Denote with $0_R$ the zero of $R$.

Then $\struct {R, +, \circ}$ has characteristic $2$.

### Corollary

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

## Proof

Let $x \in R$.

Then:

 $\ds x + x$ $=$ $\ds \paren {x + x}^2$ Definition of Idempotent Ring $\ds$ $=$ $\ds \paren {x + x} \paren {x + x}$ $\ds$ $=$ $\ds x \paren {x + x} + x \paren {x + x}$ Ring Axiom $\text D$: Distributivity of Product over Addition $\ds$ $=$ $\ds x^2 + x^2 + x^2 + x^2$ Ring Axiom $\text D$: Distributivity of Product over Addition again $\ds$ $=$ $\ds x + x + x + x$ Definition of Idempotent Ring $\ds \leadsto \ \$ $\ds 0_R$ $=$ $\ds x + x$ $\struct {\R, +}$ is a group, so Cancellation Laws apply

Hence the result.

$\blacksquare$