Definition:Ring of Idempotents
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Definition
Let $\left({R, +, \circ}\right)$ be a commutative ring.
Let $A$ be the set of all idempotent elements of $R$ under $\circ$:
- $A = \left\{{x \in R: x \circ x = x}\right\}$
Define a binary operation $\circ$ on $A$ by:
- $\forall x, y \in A: x \oplus y := x + y - 2 x \circ y$
Denote also with $\circ$ the restriction of $\circ$ to $A$.
The algebraic structure $\left({A, \oplus, \circ}\right)$ is called the ring of idempotents of $R$.
It is an idempotent ring, as shown on Ring of Idempotents is Idempotent Ring.
Sources
- 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$: Exercise $7$