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

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Definition

Let $\left({S, \circ}\right)$ be an algebraic structure.

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.


Also see


Sources

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