identity or neutral element, n. (for a set endowed with a binary operation) a member of the set such that the result of applying the operation to that element and any member of the set is equal to the latter member. A multiplicative identity is a unity; an additive identity is often called a zero. For example, the identity for multiplication in ordinary arithmetic is 1, since
x × 1 = x = 1 × x.
In some mathematical theories it is necessary to distinguish between a left identity, l, such that l × x = x for all x, and a right identity, r, such that x × r = x for all x; when an identity element does exist, it is both the unique left and the unique right identity for the theory. See also inverse.