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* The [[identity element]] of ''G'' is an element of ''S''; The group itself and the set consisting of the identity element are always subgroups.

{{r|Identity element}}

...al homomorphism''' between [[unital ring]]s (rings with a multiplicative [[identity element]]) must also satisfy

...the [[binary operation]] of concatenation (juxtaposition) of words. The [[identity element]] for this operation is the empty string. (So far we have described the [[

...which multiplication is commutative and every element except the additive identity element (0) has a multiplicative inverse (reciprocal) is called a [[field]]: for ex ...bly [[Nicholas Bourbaki|Bourbaki]], demand that their rings should have an identity element, and call rings without an identity ''pseudorings''.

* Every [[monoid]] is a semigroup, by "forgetting" the identity element. * Every [[group (mathematics)|group]] is a semigroup, by "forgetting" the identity element and inverse operation.

...ommute if and only if the commutator [''x'',''y''] is equal to the group [[identity element|identity]]. The '''commutator subgroup''' or '''derived group''' of ''G''

Note that this means the [[identity element]] of the group is the [[identity map]] on <math>S</math>, which is the map

...er binary operation ''*'' on F such that F\{0} is a commutative group with identity element 1. [[Distributivity]] of ''*'' over ''+'' holds: that is, for any <math>a,

* ''The group has an [[identity element]]:'' There is an element <math>e</math>, such that <math>x \cdot e = x</mat ...\cdot y = e</math> and <math>y \cdot x = e</math>. (<math>e</math> is the identity element)

The neutral element of a group is often called the [[identity element]] if the operation is written in [[multiplicative group|multiplicative]] no * 0 is an integer and for any integer ''a'', 0 + ''a'' = ''a'' + 0 = ''a''. (Identity element)

...homomorphism <math>\phi: G \to K</math> such that the inverse image of the identity element of ''K'' is ''H''. and the coset <math>N = N1</math> as [[identity element]]. It is easy to check that these define a group structure on the set of c

...However, if we take the positive natural numbers and addition, there is no identity element. * An identity element ''e'' exists, such that for every member ''a'' of ''S'', ''e'' * ''a'' and

...the fractional ideal ''O''_''K'' = ''O''_''K''.1 as [[identity element]]. The principal ideals, fractional ideals of the form ''O''<sub>''K''</su

| existence of an [[identity element]]: || ''a''&nbsp;+&nbsp;0&nbsp;&nbsp;=&nbsp;&nbsp;''a'' || ''a''&nbsp;&time

Thus, for instance, ''Gal(C/R)'' consists of two elements: the identity element