Idempotent Ring has Characteristic Two
Jump to navigation
Jump to search
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.
$\blacksquare$
Sources
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$