Identity element

https://arbital.com/p/identity_element

by Joe Zeng Jul 5 2016 updated Aug 20 2016

An element in a set with a binary operation that leaves every element unchanged when used as the other operand.


[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.