Idempotent Ring has Characteristic Two/Corollary

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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:

\(\displaystyle x + x\) \(=\) \(\displaystyle 0_R\) Idempotent Ring has Characteristic Two
\(\displaystyle \leadsto \ \ \) \(\displaystyle -x + x + x\) \(=\) \(\displaystyle -x + 0_R\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle x\) \(=\) \(\displaystyle -x\)

Hence the result.

$\blacksquare$


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