Ring of Idempotents of Commutative and Unitary Ring is Boolean Ring

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Theorem

Let $\left({R, +, \circ}\right)$ be a commutative and unitary ring.

Let $\left({A, \oplus, \circ}\right)$ be its ring of idempotents.


Then $\left({A, \oplus, \circ}\right)$ is a Boolean ring.


Proof

From Ring of Idempotents is Idempotent Ring, $\left({A, \oplus, \circ}\right)$ is an idempotent ring.

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


Hence, by definition, $\left({A, \oplus, \circ}\right)$ is a Boolean ring.

$\blacksquare$


Also see


Sources