Definition:Identity (Abstract Algebra)/Two-Sided Identity

Definition
Let $\left({S, \circ}\right)$ be an algebraic structure. An element $e \in S$ is called a two-sided identity or simply identity) iff it is both a left identity and a right identity:
 * $\forall x \in S: x \circ e = x = e \circ x$

Such an element is usually referred to as just an identity (element).

Alternative names
Other terms which are seen that mean the same as identity are:
 * Neutral element, which is perfectly okay, but considered slightly old-fashioned.
 * Unit element or Unity, but these are not recommended as it is too easy to confuse them with ring unity and unit of a ring.
 * Zero, but it is best to reserve that term for a zero element.

Also see

 * Right Identity
 * Two-Sided Identity