Identity element

Definition

Let $S$ be a set and let $\ast$ be a binary operation on $S$ . An element $e \in S$ is a left identity when $e \ast a = a$ for all $a \in S$ . An element $e \in S$ is a right identity when $a \ast e = a$ for all $a \in S$ . An element $e \in S$ is an identity element (or two-sided identity) when it is both a left identity and a right identity.

Remarks

The number $0$ is the identity element for addition on $\mathbb{Z}$ , and $1$ is the identity element for multiplication. If $e$ is a left identity and $f$ is a right identity, then $e = e \ast f = f$ . Thus an operation has at most one two-sided identity element.

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Binary operation definition
Identity element definition

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Tag	`0RQPZZ44T4`
URI	`m2://0RQPZZ44T4`
Kind	definition
Writing	W1 stated
MSC	08A02
Defines	Identity element
Uses	Binary operation

Identity element

Definition

Remarks

Reading path

Where this leads

Further reading

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