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[parent] identity element is unique (Theorem)

Theorem. The identity element of a monoid is unique.

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

$\displaystyle e' = e \cdot e' = e,$
i.e. both identity elements are same (in inferring this result from the two first equations, one has used the symmetry 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 identity 1 of a field.



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"identity element is unique" is owned by pahio. [ full author list (2) ]
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See Also: group, uniqueness of inverse (for groups), zero vector in a vector space is unique, absorbing element

Other names:  neutral element is unique, uniqueness of identity element
Keywords:  monoid

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Cross-references: field, ring, group, equality relation, transitivity, equations, proof, monoid, identity element

This is version 7 of identity element is unique, born on 2008-04-24, modified 2008-07-11.
Object id is 10539, canonical name is IdentityElementIsUnique.
Accessed 435 times total.

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AMS MSC: 20M99 (Group theory and generalizations :: Semigroups :: Miscellaneous)
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