identity element

English

Noun

identity element (plural identity elements)

(algebra) An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element.
- 1990, Daniel M. Fendel, Diane Resek, Foundations of Higher Mathematics, Volume 1, Addison-Wesley, page 269,
  Therefore the number ${\\displaystyle 0}$ is not considered an identity element for subtraction, even though ${\\displaystyle x-0=x}$ for all ${\\displaystyle x}$ , since ${\\displaystyle 0-x\eq x}$ .
- 2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17,
  Let ${\\displaystyle (G,\\cdot )}$ be a group. Then the identity element ${\\displaystyle e\\in G}$ is unique. […]
  Proof. If ${\\displaystyle e}$ and ${\\displaystyle e'}$ are both identity elements, then we have ${\\displaystyle ee'=e}$ since ${\\displaystyle e'}$ is an identity element, and ${\\displaystyle ee'=e'}$ since ${\\displaystyle e}$ is an identity element. Thus
  ${\\displaystyle e=ee'=e'}$ .
- 2015, Martyn R. Dixon, Leonid A. Kurdachenko, Igor Ya. Subbotin, An Introduction to Essential Algebraic Structures, Wiley, page 41,
  Sometimes, to avoid ambiguity, we may use the notation ${\\displaystyle e_{M}}$ for the identity element of ${\\displaystyle M}$ .
  If multiplicative notation is used then we use the term identity element, and often use the notation ${\\displaystyle 1}$ , or ${\\displaystyle 1_{M}}$ , for the neutral element ${\\displaystyle e}$ .

Usage notes

For binary operation ${\\displaystyle *}$ defined on a given algebraic structure, an element ${\\displaystyle i}$ is:

a left identity if ${\\displaystyle i*x=x}$ for any ${\\displaystyle x}$ in the structure,
a right identity, ${\\displaystyle x*i=x}$ for any ${\\displaystyle x}$ in the structure,
simply an identity element or (for emphasis) a two-sided identity if both are true.

Where a given structure ${\\displaystyle M}$ is equipped with an operation called addition, the notation ${\\displaystyle 0_{M}}$ may be used for the additive identity. Similarly, the notation ${\\displaystyle 1_{M}}$ denotes a multiplicative identity.

Synonyms

(element that when applied with a binary operation leaves any other element unchanged): identity, neutral element

Hyponyms

(element that when applied with a binary operation leaves any other element unchanged): additive identity, multiplicative identity, zero, zero element

Related terms

left identity
right identity

Translations

member of a structure

Dutch: identiteit (nl) f
Finnish: identiteettialkio
Hebrew: איבר יחידה‎ m
Irish: ball ionannais m
Italian: elemento neutro m

Maori: tūmau
Russian: то́ждество (ru) n (tóždestvo), тожде́ственность (ru) f (toždéstvennostʹ)
Spanish: elemento neutro m
Swedish: identitetselement (sv) n