\begin{definition}[Definition:Ring of Idempotents]
Let $\struct {R, +, \circ}$ be a commutative ring.
Let $A$ be the set of all idempotent elements of $R$ under $\circ$:
:$A = \set {x \in R: x \circ x = x}$
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 $\struct {A, \oplus, \circ}$ is called the '''ring of idempotents''' of $R$.
It is an idempotent ring, as shown on Ring of Idempotents is Idempotent Ring.
\end{definition}