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

Definition
Let $\struct {S, \circ}$ be an algebraic structure. An element $e \in S$ is called an identity (element) it is both a left identity and a right identity:


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

In Identity is Unique it is established that an identity element, if it exists, is unique within $\struct {S, \circ}$.

Thus it is justified to refer to it as the identity (of a given algebraic structure).

This identity is often denoted $e_S$, or $e$ if it is clearly understood what structure is being discussed.

Also see

 * Definition:Left Identity
 * Definition:Right Identity


 * Definition:Identity Mapping


 * Identity is Unique