Definition:Ring of Idempotents

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 $7$