Identity Element is Idempotent

Theorem
Let $\struct {S, \circ}$ 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:
 * $\forall x \in S: e \circ x = x$

Thus in particular:
 * $e \circ e = e$

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

Also see

 * Left Identity Element is Idempotent
 * Right Identity Element is Idempotent