Identity element
Identity element
This concept is used in algebraic structures such as groups and rings. The term identity element is often shortened to identity (as will be done in this article) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
Let (S, ∗) be a set S with a binary operation ∗ on it. Then an element e of S is called a left identity if e ∗ a = a for all a in S, and a right identity if a ∗ e = a for all a in S. If e is both a left identity and a right identity, then it is called a two-sided identity, or simply an identity.
An identity with respect to addition is called an additive identity (often denoted as 0) and an identity with respect to multiplication is called a multiplicative identity (often denoted as 1). These need not be ordinary addition and multiplication, but rather arbitrary operations. The distinction is used most often for sets that support both binary operations, such as rings and fields. The multiplicative identity is often called unity in the latter context (a ring with unity). This should not be confused with a unit in ring theory, which is any element having a multiplicative inverse. Unity itself is necessarily a unit.
Video: Identity Element
Examples
| Set | Operation | Identity |
|---|---|---|
|
Real numbers
|
+ (addition)
|
0
|
|
Real numbers
|
· (multiplication)
|
1
|
|
Positive integers
|
Least common multiple
|
1
|
|
Non-negative integers
|
Greatest common divisor
|
0 (under most definitions of GCD)
|
| ...... | ...... | ....... |
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