identity element

English

Noun

identity element (plural identity elements)

1. (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 $0$ is not considered an identity element for subtraction, even though $x-0=x$ for all $x$, since $0-x\neq x$.

   • 2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17,

     Let $(G,\cdot )$ be a group. Then the identity element $e\in G$ is unique. […]

     Proof. If $e$ and $e'$ are both identity elements, then we have $ee'=e$ since $e'$ is an identity element, and $ee'=e'$ since $e$ is an identity element. Thus

     $e=ee'=e'$.

   • 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 $e_{M}$ for the identity element of $M$.
     If multiplicative notation is used then we use the term identity element, and often use the notation $1$, or $1_{M}$, for the neutral element $e$.

Usage notes

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

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

Where a given structure $M$ is equipped with an operation called addition, the notation $0_{M}$ may be used for the additive identity. Similarly, the notation $1_{M}$ 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
• Hungarian: egységelem (hu)
• 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

See also

• unity

Further reading

• Identity matrix on Wikipedia.Wikipedia
• Inverse element on Wikipedia.Wikipedia