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NounEdit

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 notesEdit

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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