Definition:Identity (Abstract Algebra)
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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:
- $\forall x \in S: x \circ e_R = x$
Two-Sided 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:
- $\forall x \in S: x \circ e = x = e \circ x$
Such an element is usually referred to as just an identity (element).
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:
- 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.
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.
