---
tag: 0RQPZZ44T4
kind: definition
name: Identity element
msc: [08A02]
defines: 07WYQPB8E3
uses: [0580ZJSA3Q]
sources:
  - url: https://en.wikipedia.org/wiki/Identity_element
    locator: Definitions
---

## Definition

Let $S$ be a set and let $\ast$ be a [[binary operation|0M418FE7BB]] 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.