Idempotent Ring is Commutative

Theorem
Let $\struct {R, +, \circ}$ be an idempotent ring.

Denote with $0_R$ the zero of $R$.

Then $\struct {R, +, \circ}$ is a commutative ring.

Proof
Let $x, y \in R$.

Then:

Subtracting $x + y$ from both sides yields:

Finally, subtracting $y \circ x$ from both sides, we obtain:


 * $x \circ y = y \circ x$

and conclude $R$ is a commutative ring.