Definition:Identity (Abstract Algebra)

Definition
Let $\left({S, \circ}\right)$ be an algebraic structure.

Left Identity
An element $e_L \in S$ is called a left identity iff:
 * $\forall x \in S: e_L \circ x = x$

Right Identity
An element $e_R \in S$ is called a right identity iff:
 * $\forall x \in S: x \circ e_R = x$

Identity
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:
 * $\forall x \in S: x \circ e = x = e \circ x$

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

 * Identity is Unique