Ring of Idempotents of Commutative and Unitary Ring is Boolean Ring

Theorem

Let $\struct {R, +, \circ}$ be a commutative and unitary ring.

Let $\struct {A, \oplus, \circ}$ be its ring of idempotents.

Then $\struct {A, \oplus, \circ}$ is a Boolean ring.

Proof

From Ring of Idempotents is Idempotent Ring, $\struct {A, \oplus, \circ}$ is an idempotent ring.

By Unity is Unity in Ring of Idempotents, $\struct {A, \oplus, \circ}$ is also a unitary ring.

Hence, by definition, $\struct {A, \oplus, \circ}$ is a Boolean ring.

$\blacksquare$

Also see

• Ring of Idempotents is Idempotent Ring

Sources

• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$: Exercise $7$