identity element

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English

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Noun

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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 is not considered an identity element for subtraction, even though for all , since .
    • 2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17,
      Let be a group. Then the identity element is unique. []
      Proof. If and are both identity elements, then we have since is an identity element, and since is an identity element. Thus
      .
    • 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 for the identity element of .
      If multiplicative notation is used then we use the term identity element, and often use the notation , or , for the neutral element .

Usage notes

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

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

Where a given structure is equipped with an operation called addition, the notation may be used for the additive identity. Similarly, the notation denotes a multiplicative identity.

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Further reading

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