Identity Element is Idempotent

Theorem
Let $\left({S, \circ}\right)$ be an algebraic structure.

Let $e \in S$ be an identity with respect to $\circ$.

Then $e$ is idempotent under $\circ$.

Proof
By the definition of an identity element, $e \circ x = x$ for each $x \in S$.

Thus in particular $e \circ e = e$.

Therefore $e$ is idempotent under $\circ$.