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

Definition
Let $\left({S, \circ}\right)$ 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 $\left({S, \circ}\right)$.

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 known as
Other terms which are seen that mean the same as identity are:
 * Two-sided identity, to reflect the fact that it is both a left identity and a right identity.
 * Neutral element, which is perfectly okay, but considered slightly old-fashioned.
 * Unit element, but this is not recommended as it is too easy to confuse it with a unit of a ring.
 * Unity, but this is generally reserved for a ring unity.
 * Zero, but it is best to reserve that term for a zero element.

Similarly, the symbols used for an identity element are often found to include $0$ and $1$. Again, in the context of the general algebraic structure, these are not recommended for the same reason.

Also see

 * Definition:Left Identity
 * Definition:Right Identity


 * Definition:Identity Mapping


 * Identity is Unique