# 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 $\circ$.

$\blacksquare$