Definition:Identity (Abstract Algebra)
Definition
Let $\struct {S, \circ}$ be an algebraic structure.
Left Identity
An element $e_L \in S$ is called a left identity if and only if:
- $\forall x \in S: e_L \circ x = x$
Right Identity
An element $e_R \in S$ is called a right identity if and only if:
- $\forall x \in S: x \circ e_R = x$
Two-Sided Identity
An element $e \in S$ is called an identity (element) if and only if 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 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 or unity of field.
- Zero, but it is best to reserve that term for a zero element.
- The trivial element, in the context of a group.
The symbols used for an identity element are often found to include $0$ and $1$. In the context of the general algebraic structure, these are not recommended, as this can cause confusion.
Some sources use $I$ for the identity.
Also see
- Results about identity elements can be found here.