Identity element

In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set which leaves unchanged every element of the set when the operation is applied.[1][2] This concept is used in algebraic structures such as groups and rings. The term identity element is often shortened to identity (as in the case of additive identity and multiplicative identity)[3] when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.

Let (S, ∗) be a set S equipped with a binary operation ∗. Then an element e of S is called a left identity if ea = a for all a in S, and a right identity if ae = a for all a in S.[4] If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.[5][6][7][8][9]

To see this, note that if l is a left identity and r is a right identity, then l = lr = r. In particular, there can never be more than one two-sided identity: if there were two, say e and f, then ef would have to be equal to both e and f.

It is also quite possible for (S, ∗) to have no identity element,[15] such as the case of even integers under the multiplication operation.[3] Another common example is the cross product 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 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 of positive natural numbers.