An identity element in a set 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.