Identity element

[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:

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