Identity element

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.