Identity element

English

Noun

1. Template:Lb An element of an algebraic structure which when applied, in either order, to any other element via a binary operation yields the other element.
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   • 2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17,

     Let $(G,\cdot )$ be a group. Then the identity element $e\in G$ is unique.Template:...

     Proof. If $e$ and $e^{\prime }$ are both identity elements, then we have $e{e}^{\prime }=e$ since $e^{\prime }$ is an identity element, and $e{e}^{\prime }=e^{\prime }$ since $e$ is an identity element. Thus

     $e=e{e}^{\prime }=e^{\prime }$.

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Usage notes

For binary operation $*$ defined on a given algebraic structure, an element $i$ is:

1. a Template:M if $i*x=x$ for any $x$ in the structure,
2. a Template:M, $x*i=x$ for any $x$ in the structure,
3. simply an identity element or (for emphasis) a two-sided identity if both are true.

Where a given structure $M$ is equipped with an operation called addition, the notation $0_M$ may be used for the Template:M. Similarly, the notation $1_M$ denotes a Template:M.

Synonyms

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Hyponyms

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Related terms

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Translations

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See also

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Further reading

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