# Definition:Identity (Abstract Algebra)/Two-Sided Identity

## Definition

Let $\struct {S, \circ}$ be an algebraic structure.

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 or unity of field, and can also mean the number $1$.**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**.

## Examples

### Symmetry Group of Square

Consider the **symmetry group of the square**:

Let $\SS = ABCD$ be a square.

The various symmetry mappings of $\SS$ are:

- the identity mapping $e$
- the rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ around the center of $\SS$ anticlockwise respectively
- the reflections $t_x$ and $t_y$ are reflections in the $x$ and $y$ axis respectively
- the reflection $t_{AC}$ in the diagonal through vertices $A$ and $C$
- the reflection $t_{BD}$ in the diagonal through vertices $B$ and $D$.

This group is known as the **symmetry group of the square**, and can be denoted $D_4$.

The mapping $e$ which leaves $\SS$ unchanged is the **identity element**.

## Also see

- Results about
**identity elements**can be found**here**.

## Sources

- 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 1$. Rings and Fields - 1964: Walter Ledermann:
*Introduction to the Theory of Finite Groups*(5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 2$: The Axioms of Group Theory - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 4.3$. Units and zeros: Definition $1$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses - 1967: John D. Dixon:
*Problems in Group Theory*... (previous) ... (next): Introduction: Notation - 1967: George McCarty:
*Topology: An Introduction with Application to Topological Groups*... (previous) ... (next): Chapter $\text{II}$: Groups: The Group Property - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $1$: Integral Domains: $\S 3$. Definition of an Integral Domain - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $1$: The definition of a ring: Definitions $1.1 \ \text{(b)}$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 27$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Operations - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 5$: Groups $\text{I}$ - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $2$: Examples of Groups and Homomorphisms: $2.2$ Definitions $\text{(i)}$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 31$. Identity element and inverses - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.1$: Monoids - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**group** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**identity** - 1999: J.C. Rosales and P.A. García-Sánchez:
*Finitely Generated Commutative Monoids*... (previous) ... (next): Chapter $1$: Basic Definitions and Results - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**group** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**identity** - 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): Chapter $\S 2.4$: Composition of continuous linear transformations