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Identity Element
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:
''For other uses, see identity (disambiguation).''
In
mathematics, an '''identity element''' (or '''neutral element''') is a special type of element of a
set with respect to a
binary operation on that set. It leaves other elements unchanged when combined with them. This is used for
group (mathematics) groups and
magma (algebra) related concepts.
The term ''identity element'' is often shortened to ''identity'' when there is no possibility of confusion; we do so in this article.
Let (''S'',*) be a set ''S'' with a binary operation * on it (known as a
magma (algebra) magma). 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''' and an identity with respect to multiplication is called a '''multiplicative identity'''. The distinction is used most often for sets that support both binary operations (such as with
ring (mathematics) rings).
Examples
{| border=1, align=top
!set!!operation!!identity
|-
|
real numbers.html">0 (number)
0
|-
|
real numbers.html">1 (number)
1
|-
|''n''-by-''n'' square
matrix (mathematics) matrices|| + (addition)||
zero matrix
|-
|''n''-by-''n'' square
matrix (mathematics) matrices|| • (multiplication)||
identity matrix
|-
|all
function (mathematics) functions from a set ''M'' to itself|| function composition||
identity map
|-
|character strings|| concatenation || empty string
|-
|only two elements {''e'', ''f''}||* defined by
''e'' * ''e'' = ''f'' * ''e'' = ''e'' and
''f'' * ''f'' = ''e'' * ''f'' = ''f''||both ''e'' and ''f'' are left identities, but there is no right or two-sided identity
|}
As the last example shows, it is possible for (''S'',*) to have several left identities. In fact, every element can be a left identity. Similarly, there can be several right identities. But if there is both a right identity and a left identity, then they are equal and there is just a single two-sided identity. To see this, note that if ''l'' is a left identity and ''r'' is a right identity then ''l'' = ''l'' * ''r'' = ''r''. In particular, there can never be more than one two-sided identity.
See also
*
Inverse element
*
Additive inverse
*
Group (mathematics) Group
*
Monoid
*
Unital
*
Quasigroup
Category:Abstract algebra
Category:Algebra
Category:Binary operations *Identity element
ar:عنصر Øيادي
cs:Neutrálnà prvek
de:Neutrales Element
et:Ãœhikelement
es:Elemento neutro
fr:Élément neutre
ko:í•ë“±ì›?
it:Elemento neutro
he:×?יבר יחידה
hu:Neutrális elem
nl:Neutraal element
ja:��元
pl:Element neutralny
pt:Elemento neutro
sk:Neutrálny prvok
sl:Enak element
sv:Neutralt element
vi:Phần tỠđơn vị
zh:單�元
see
Identity element
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