In

mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...

, an identity element, or neutral element, of a binary operation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

operating on a set is an element of the set which leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s such as group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

s and rings. The term ''identity element'' is often shortened to ''identity'' (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
Definitions

Let be a set equipped with a binary operation ∗. Then an element of is called aleft
Left may refer to:
Music
* ''Left'' (Hope of the States album), 2006
* ''Left'' (Monkey House album), 2016
* ''Left'' (Sharlok Poems album)
Direction
* Left (direction), the relative direction opposite of right
* Left-handedness
Politics
* ...

identity if for all in , and a right
Rights are legal
Law is a system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
A system, surrounded and influenced by its environment, is describe ...

identity if for all in . If is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.
An identity with respect to addition is called an additive identityIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

(often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). These need not be ordinary addition and multiplication—as the underlying operation could be rather arbitrary. In the case of a group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

for example, the identity element is sometimes simply denoted by the symbol $e$. The distinction between additive and multiplicative identity is used most often for sets that support both binary operations, such as rings, integral domain
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, and field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

s. The multiplicative identity is often called unity in the latter context (a ring with unity). This should not be confused with a unit
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* Unit (album), ' ...

in ring theory, which is any element having a multiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

. By its own definition, unity itself is necessarily a unit.
Examples

Properties

In the example ''S'' = with the equalities given, ''S'' is asemigroup
In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation.
The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', o ...

. It demonstrates the possibility for to have several left identities. In fact, every element can be a left identity. In a similar manner, there can be several right identities. But if there is both a right identity and a left identity, then they must be equal, resulting in a single two-sided identity.
To see this, note that if is a left identity and is a right identity, then . In particular, there can never be more than one two-sided identity: if there were two, say and , then would have to be equal to both and .
It is also quite possible for to have ''no'' identity element, such as the case of even integers under the multiplication operation. Another common example is the cross product
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

of vectors
Vector may refer to:
Biology
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector
*Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...

, where the absence of an identity element is related to the fact that the direction
Direction may refer to:
*Relative direction, for instance left, right, forward, backwards, up, and down
** Anatomical terms of location for those used in anatomy
*Cardinal direction
Mathematics and science
*Direction vector, a unit vector that ...

of any nonzero cross product is always orthogonal
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

to any element multiplied. That is, it is not possible to obtain a non-zero vector in the same direction as the original. Yet another example of group without identity element involves the additive semigroup
In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation.
The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', o ...

of positive
Positive is a property of Positivity (disambiguation), positivity and may refer to:
Mathematics and science
* Converging lens or positive lens, in optics
* Plus sign, the sign "+" used to indicate a positive number
* Positive (electricity), a po ...

natural number
File:Three Baskets.svg, Natural numbers can be used for counting (one apple, two apples, three apples, ...)
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and total order, o ...

s.
See also

*Absorbing elementIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

* Additive inverse
In mathematics, the additive inverse of a number
A number is a mathematical object
A mathematical object is an abstract concept arising in mathematics.
In the usual language of mathematics, an ''object'' is anything that has been (or could be) ...

* Generalized inverse
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

* Identity (equation)
*Identity function
Graph of the identity function on the real numbers
In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function (mathematics), function that always returns the same value that was ...

* Inverse element
In abstract algebra, the idea of an inverse element generalises the concepts of negation (sign reversal) (in relation to addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic
Arithmetic ...

* Monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...

* Pseudo-ring
* Quasigroup
* Unital (disambiguation)
Notes and references

Bibliography

* * * *Further reading

* M. Kilp, U. Knauer, A.V. Mikhalev, ''Monoids, Acts and Categories with Applications to Wreath Products and Graphs'', De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, {{ISBN, 3-11-015248-7, p. 14–15 Algebraic properties of elements Binary operations, *Identity element Properties of binary operations 1 (number)