In

mathematics
Mathematics (from Greek: ) includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (analysis). It has no generally accepted definition.
Mathematicians seek and use patterns to formulate ...

, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation
In mathematics, a binary operation or dyadic operation is a calculation that combines two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.
More specifically, a binary operatio ...

on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' of finite arity (typically binary operations), and a finite set of identities, known as ...

s such as group
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic identi ...

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 âˆ—. 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
* Left ...

identity if for all in , and a right
Rights are legal, social, or ethical principles of freedom or entitlement; that is, rights are the fundamental normative rules about what is allowed of people or owed to people according to some legal system, social convention, or ethical theory. ...

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, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element ''x'' in the set, yields ''x''. One of the most familiar additive identities is the number 0 from elementary ...

(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 multiplicationâ€”as the underlying operation could be rather arbitrary. In the case of a group
A group is a number of people or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic identi ...

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, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural setti ...

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 together with an associative binary operation.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''Â·''y'', or simply ''xy'', denotes the result ...

. 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, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in three-dimensional space \mathbb^3, and is denoted by the symbol \times. Given ...

of vectors, where the absence of an identity element is related to the fact that the direction of any nonzero cross product is always orthogonal
In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements ''u'' and ''v'' of a vector space with bilinear form ''B'' are orthogonal when . Depending on the bilin ...

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 together with an associative binary operation.
The binary operation of a semigroup is most often denoted multiplicatively: ''x''Â·''y'', or simply ''xy'', denotes the result ...

of positive
Positive is a property of 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 polarity of electrical charge
* ...

natural number
In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). In common mathematical terminology, words colloquially used ...

s.
See also

*Absorbing elementIn mathematics, an absorbing element (or annihilating element) is a special type of element of a set with respect to a binary operation on that set. The result of combining an absorbing element with any element of the set is the absorbing element it ...

* Additive inverse
In mathematics, the additive inverse of a number is the number that, when added to , yields zero. This number is also known as the opposite (number), sign change, and negation. For a real number, it reverses its sign: the additive inverse (opposi ...

* Generalized inverse
In mathematics, and in particular, algebra, a generalized inverse (or, g-inverse) of an element ''x'' is an element ''y'' that has some properties of an inverse element but not necessarily all of them. Generalized inverses can be defined in any ma ...

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

* Inverse element
In abstract algebra, the idea of an inverse element generalises the concepts of negation (sign reversal) (in relation to addition) and reciprocation (in relation to multiplication). The intuition is of an element that can 'undo' the effect of combi ...

* Monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element.
Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics.
...

* Pseudo-ring
* Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible. Quasigroups differ from groups mainly in that they are not necessarily associative.
A quasigr ...

* Unital (disambiguation)Unital may refer to:
* A unital algebra â€“ an algebra that contains a multiplicative identity element.
* A geometric unital â€“ a block design for integer .
* A unital algebraic structure, such as a unital magma.
* A unital map on C*-algebras â ...

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. 14â€“15Algebraic properties of elementsAbstract algebra ...

*Identity element
1 (number)In mathematics, 1 is the number of things in a singleton, and it is the identity element for multiplication. This category is for concepts that are related to the number one.
{{Commons cat
Integers
Mathematical constants ...