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:
\(\ds x + x\) | \(=\) | \(\ds 0_R\) | Idempotent Ring has Characteristic Two | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds -x + x + x\) | \(=\) | \(\ds -x + 0_R\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds -x\) |
Hence the result.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $9$: Rings: Exercise $1$