Unity is Unity in Ring of Idempotents

Theorem

Let $\left({R, +, \circ}\right)$ be a commutative and unitary ring whose unity is $1_R$.

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

Then $1_R$ is also a unity for $\left({A, \oplus, \circ}\right)$.

Proof

From Unity of Ring is Idempotent, $1_R$ is an idempotent element of $R$.

Hence $1_R \in A$.

Recall that the ring product of $A$ is a restriction from that of $R$.

Hence, for each $x \in A$:

$x \circ 1_R = x = 1_R \circ x$

so that $1_R$ is a unity for $A$, as desired.

$\blacksquare$