# identity element

## English

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### Noun

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 ${\displaystyle 0}$ is not considered an identity element for subtraction, even though ${\displaystyle x-0=x}$ for all ${\displaystyle x}$, since ${\displaystyle 0-x\neq x}$.
• 2003, Houshang H. Sohrab, Basic Real Analysis, Birkhäuser, page 17,
Let ${\displaystyle (G,\cdot )}$ be a group. Then the identity element ${\displaystyle e\in G}$ is unique. []
Proof. If ${\displaystyle e}$ and ${\displaystyle e'}$ are both identity elements, then we have ${\displaystyle ee'=e}$ since ${\displaystyle e'}$ is an identity element, and ${\displaystyle ee'=e'}$ since ${\displaystyle e}$ is an identity element. Thus
${\displaystyle e=ee'=e'}$.
• 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 ${\displaystyle e_{M}}$ for the identity element of ${\displaystyle M}$.
If multiplicative notation is used then we use the term identity element, and often use the notation ${\displaystyle 1}$, or ${\displaystyle 1_{M}}$, for the neutral element ${\displaystyle e}$.

#### Usage notes

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

1. a left identity if ${\displaystyle i*x=x}$ for any ${\displaystyle x}$ in the structure,
2. a right identity, ${\displaystyle x*i=x}$ for any ${\displaystyle x}$ in the structure,
3. simply an identity element or (for emphasis) a two-sided identity if both are true.

Where a given structure ${\displaystyle M}$ is equipped with an operation called addition, the notation ${\displaystyle 0_{M}}$ may be used for the additive identity. Similarly, the notation ${\displaystyle 1_{M}}$ denotes a multiplicative identity.