Idempotent Element/Examples/Zero is Idempotent for Addition

Example of Idempotent Element
$0$ is idempotent under the operation of addition in the set of integers $\Z$, but no other element of $\Z$ is so.

Proof
From Integers under Addition form Abelian Group, $\struct {\Z, +}$ forms an abelian group whose identity element is $0$.

The result follows from Identity is only Idempotent Element in Group.