[summary: An identity element in a set $~$S$~$ with a binary operation $~$*$~$ is an element $~$i$~$ that leaves any element $~$a \in S$~$ unchanged when combined with it in that operation.]

An identity element in a set $~$S$~$ with a binary operation $~$*$~$ is an element $~$i$~$ that leaves any element $~$a \in S$~$ unchanged when combined with it in that operation.

Formally, we can define an element $~$i$~$ to be an identity element if the following two statements are true:

- For all $~$a \in S$~$, $~$i * a = a$~$. If only this statement is true then $~$i$~$ is said to be a
*left identity*. - For all $~$a \in S$~$, $~$a * i = a$~$. If only this statement is true then $~$i$~$ is said to be a
*right identity*.

The existence of an identity element is a property of many algebraic structures, such as groups, rings, and fields.