Identity element

An identity element in a set $S$ with a binary operation $\ast$ 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:

1. For all $a\in S$, $i\ast a=a$. If only this statement is true then $i$ is said to be a left identity.
2. For all $a\in S$, $a\ast i=a$. 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.