Right Identity Element is Idempotent

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

Let $e_R \in S$ be a right identity with respect to $\circ$.

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

Proof
By the definition of a right identity:
 * $\forall x \in S: x \circ e_R = x$

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

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