identity element

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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[edit]

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[edit]