This article will be permanently flagged as inappropriate and made unaccessible to everyone. Are you certain this article is inappropriate? Excessive Violence Sexual Content Political / Social
Email Address:
Article Id: WHEBN0000014962 Reproduction Date:
In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. This is used for groups and related concepts.
The term identity element is often shortened to identity (as will be done in this article) when there is no possibility of confusion.
Let (S, ∗) be a set with a binary operation ∗ on it. Then an element of S is called a left and right (algebra) identity if e ∗ a = a for all a in S, and a left and right (algebra) identity if a ∗ e = a for all a in S. If e 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 identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). The distinction is used most often for sets that support both binary operations, such as rings. The multiplicative identity is often called the unit in the latter context, where, though, a unit is often used in a broader sense, to mean an element with a multiplicative inverse.
As the last example (a semigroup) shows, it is possible for (S, ∗) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if l is a left identity and r is a right identity then l = l ∗ r = r. In particular, there can never be more than one two-sided identity. If there were two, e and f, then e ∗ f would have to be equal to both e and f.
It is also quite possible for (S, ∗) to have no identity element. A common example of this is the cross product of vectors. The absence of an identity element is related to the fact that the direction of any nonzero cross product is always orthogonal to any element multiplied – so that it is not possible to obtain a non-zero vector in the same direction as the original. Another example would be the additive semigroup of positive natural numbers.
Logic, Set theory, Statistics, Number theory, Mathematical logic
Set theory, Pi, Venn diagram, Empty set, Axiomatic set theory
Number theory, Abstract algebra, Simple group, Hydrogen, Abelian group
O, 1 (number), Computer science, 2 (number), Hinduism
Monoid, Group (mathematics), Identity element, Magma (algebra), Binary operation
Linear algebra, Mathematics, Column space, Row space, Vector space
Linear algebra, Number theory, Arithmetic, Mathematics, Geometry
Computer science, Semigroup, Universal algebra, MathWorld, Groupoid
0 (number), 2 (number), Latin, 3 (number), 4 (number)