\begin{definition}[Definition:Identity (Abstract Algebra)/Two-Sided Identity]
Let $\struct {S, \circ}$ be an algebraic structure.
An element $e \in S$ is called an '''identity (element)''' {{iff}} 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.
\end{definition}