# Definition:Identity (Abstract Algebra)

## Contents

## Definition

Let $\left({S, \circ}\right)$ be an algebraic structure.

### Left Identity

An element $e_L \in S$ is called a **left identity** if and only if:

- $\forall x \in S: e_L \circ x = x$

### Right Identity

An element $e_R \in S$ is called a **right identity** if and only if:

- $\forall x \in S: x \circ e_R = x$

### Two-Sided Identity

An element $e \in S$ is called an **identity (element)** if and only if it is both a left identity and a right identity:

- $\forall x \in S: x \circ e = x = e \circ x$

In Identity is Unique it is established that an identity element, if it exists, is unique within $\struct {S, \circ}$.

Thus it is justified to refer to it as **the** identity (of a given algebraic structure).

This identity is often denoted $e_S$, or $e$ if it is clearly understood what structure is being discussed.

## Also known as

Other terms which are seen that mean the same as **identity** are:

**Two-sided identity**, to reflect the fact that it is both a left identity and a right identity.**Neutral element**, which is perfectly okay, but considered slightly old-fashioned.**Unit element**, but this is not recommended as it is too easy to confuse it with a unit of a ring.**Unity**, but this is generally reserved for a ring unity.**Zero**, but it is best to reserve that term for a zero element.- The
**trivial element**, in the context of a group.

The symbols used for an **identity element** are often found to include $0$ and $1$. In the context of the general algebraic structure, these are not recommended, as this can cause confusion.

Some sources use $I$ for the **identity**.