# identity element

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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 $0$ is not considered an identity element for subtraction, even though $x-0=x$ for all $x$ , since $0-x\neq x$ .
• 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. []
Proof. If $e$ and $e'$ are both identity elements, then we have $ee'=e$ since $e'$ is an identity element, and $ee'=e'$ since $e$ is an identity element. Thus
$e=ee'=e'$ .
• 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 $e_{M}$ for the identity element of $M$ .
If multiplicative notation is used then we use the term identity element, and often use the notation $1$ , or $1_{M}$ , for the neutral element $e$ .

#### Usage notes

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

1. a left identity if $i*x=x$ for any $x$ in the structure,
2. a right identity, $x*i=x$ 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 additive identity. Similarly, the notation $1_{M}$ denotes a multiplicative identity.