Unity of Ring is Idempotent

Theorem

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

Then $1_R$ is an idempotent element of $R$ under the ring product $\circ$:

$1_R \circ 1_R = 1_R$

Proof

By definition of ring with unity, $\left({R, \circ}\right)$ is a monoid whose identity element is $1_R$.

From Identity Element is Idempotent (applied to $1_R$):

$1_R \circ 1_R = 1_R$

which was to be proven.

$\blacksquare$