identity element is unique

Theorem.  The identity element of a monoid is unique.

Proof.  Let $e$ and $e^{\prime }$ be identity elements of a monoid  $(G,\cdot )$.  Since $e$ is an identity element, one has  $e\cdot e^{\prime }=e^{\prime }$.  Since $e^{\prime }$ is an identity element, one has also  $e\cdot e^{\prime }=e$.  Thus

$$e^{\prime }=e\cdot e^{\prime }=e,$$

i.e. both identity elements are the same (in inferring this result from the two first equations, one has used the symmetry (http://planetmath.org/Symmetric) 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 (http://planetmath.org/Ring) 0 of a ring or a field, and the multiplicative identity (http://planetmath.org/Ring) 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