mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an identity element, or neutral element, of a

binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, an internal binary op ...

operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. 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'' (typically binary operations such as addition and multiplication), and a finite set of ...

s such as groups 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

∗. Then an element of is called a if for all in , and a if for all in . If is both a left identity and a right identity, then it is called a , or simply an . An identity with respect to addition is called an (often denoted as 0) and an identity with respect to multiplication is called a (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 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 domains, and fields. The multiplicative identity is often called 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'' (a ...

in ring theory, which is any element having a multiplicative inverse. By its own definition, unity itself is necessarily a unit.

Examples

Properties

In the example ''S'' = with the equalities given, ''S'' is a semigroup. 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 a three-dimensional oriented Euclidean vector space (named here E), and is ...

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 geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

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 structure without identity element involves the additive semigroup of positive natural numbers.

Notes and references

Bibliography

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Definitions

Examples

Properties

See also

Notes and references

Bibliography

Further reading