What is "identity element"

n. (context algebra English) A member of a structure which, when applied to any other element via a binary operation, induces an identity mapping; more specifically, given an operation ''*'', an element ''I'' is

n. an operator that leaves unchanged the element on which it operates; "the identity under numerical multiplication is 1" [syn: identity, identity operator]

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, which leaves other elements unchanged when combined with them. This concept is used in algebraic structures such as groups. The term identity element is often shortened to identity (as will be done in this article) when there is no possibility of confusion.

Let be a set with a binary operation ∗ on it. Then an element of is called a left identity if for all in , and a right 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 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.