identity element is unique

Theorem.  The identity elementMathworldPlanetmath of a monoid is unique.

Proof.  Let e and e be identity elements of a monoid  (G,).  Since e is an identity element, one has  ee=e.  Since e is an identity element, one has also  ee=e.  Thus


i.e. both identity elements are the same (in inferring this result from the two first equations, one has used the symmetryPlanetmathPlanetmath ( and transitivity of the equality relation).

Note.  The theorem also proves the uniqueness of e.g. the identity element of a group, the additive identity ( 0 of a ring or a field, and the multiplicative identityPlanetmathPlanetmath ( 1 of a field.

Title identity element is unique
Canonical name IdentityElementIsUnique
Date of creation 2013-03-22 18:01:20
Last modified on 2013-03-22 18:01:20
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 11
Author pahio (2872)
Entry type Theorem
Classification msc 20M99
Synonym neutral element is unique
Synonym uniqueness of identity element
Related topic Group
Related topic UniquenessOfInverseForGroups
Related topic ZeroVectorInAVectorSpaceIsUnique
Related topic AbsorbingElement