Identity element

Definition

Let $S$ be a set and let $\ast$ be a binary operation on $S$. An element $e \in S$ is a left identity when $e \ast a = a$ for all $a \in S$. An element $e \in S$ is a right identity when $a \ast e = a$ for all $a \in S$. An element $e \in S$ is an identity element (or two-sided identity) when it is both a left identity and a right identity.

Remarks

The number $0$ is the identity element for addition on $\mathbb{Z}$, and $1$ is the identity element for multiplication. If $e$ is a left identity and $f$ is a right identity, then $e = e \ast f = f$. Thus an operation has at most one two-sided identity element.

Explanations

L1 human

An identity element is an element that does nothing when you combine it with another element. Adding $0$ to a number gives the same number, and multiplying a number by $1$ gives the same number. Some operations have no identity element, for example addition on the positive integers.