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{{Distinguish|Unit ring}}

In [[mathematics]], an '''invertible element''' or a '''unit''' in a [[ring (mathematics)|ring with identity]] {{mvar|R}} is any element {{mvar|u}} that has an [[inverse element]] in the multiplicative [[monoid]] of {{mvar|R}}, i.e. an element {{mvar|v}} such that
 
:{{math|1=''uv'' = ''vu'' = 1<sub>''R''</sub>}}, where {{math|1<sub>''R''</sub>}} is the [[multiplicative identity]].<ref>{{cite book | last1=Dummit | first1=David S. | last2=Foote | first2=Richard M. | title=Abstract Algebra | publisher=[[John Wiley & Sons]] | year=2004 | edition=3rd | isbn=0-471-43334-9}}</ref><ref>{{cite book | last=Lang | first=Serge | authorlink=Serge Lang | title=Algebra | publisher=[[Springer Science+Business Media|Springer]] | series=[[Graduate Texts in Mathematics]] | year=2002 | isbn=0-387-95385-X}}</ref>

The set of units of any ring is closed under [[multiplication]] (the product of two units is again a unit), and forms a [[group (mathematics)|group]] for this operation. It never contains the [[additive identity|element 0]] (except in the case of the [[zero ring]]), and is therefore not closed under addition; its [[complement (set theory)|complement]] however might be a group under addition, which happens if and only if the ring is a [[local ring]].

The term ''unit'' is also used to refer to the identity element {{math|1<sub>''R''</sub>}} of the ring, in expressions like ''ring with a unit'' or ''[[Ring_(mathematics)#Multiplicative_identity:_mandatory_vs._optional|unit ring]]'', and also e.g. ''[[identity matrix|'unit' matrix]]''. For this reason, some authors call {{math|1<sub>''R''</sub>}} "unity" or "identity", and say that {{mvar|R}} is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".

{{anchor|−1}}The multiplicative identity {{math|1<sub>''R''</sub>}} and its additive inverse {{math|−1<sub>''R''</sub>}} are always units. Hence, pairs of [[additive inverse]] elements<ref>In a ring, the additive inverse of a non-zero element can equal the element itself.</ref> {{math|''x''}} and {{math|−''x''}} are always [[Associated element|associated]].

==Examples==
1 is a unit in any ring. More generally, any [[root of unity]] in a ring ''R'' is a unit: if {{math|1=''r''<sup>''n''</sup> = 1}}, then {{math|1=''r''<sup>''n'' − 1</sup>}} is a multiplicative inverse of ''r''.
On the other hand, 0 is never a unit (except in the zero ring). A ring ''R'' is called a [[skew-field]] (or a division ring) if {{math|1=U(''R'') = ''R'' - {0}{{void}}}}, where U(''R'') is the group of units of ''R'' (see below). A commutative skew-field is called a [[field (mathematics)|field]]. For example, the units of the [[real number]]s {{math|'''R'''}} are {{math|'''R''' - {0}}}.

===Integers===
In the ring of [[integers]] {{math|'''Z'''}}, the only units are {{math|[[1 (number)|+1]]}} and {{math|{{num|−1}}}}.

[[ring of integers|Rings of integers]] <math>R= \mathfrak O_F</math> in a [[number field]] ''F'' have, in general, more units. For example,
:{{math|1= ({{sqrt|5}} + 2)({{sqrt|5}} − 2) = 1}}
in the ring {{math|[[quadratic integer{{!}}'''Z'''[{{sfrac|1 + {{sqrt|5}}|2}}]]]}}, and in fact the unit group of this ring is infinite.

In fact, [[Dirichlet's unit theorem]] describes the structure of {{math|U(''R'')}} precisely: it is isomorphic to a group of the form
:<math>\mathbf Z^n \oplus \mu_R</math>
where <math>\mu_R</math> is the (finite, cyclic) group of roots of unity in ''R'' and ''n'', the [[rank of a module|rank]] of the unit group is
:<math>n=r_1 + r_2 -1, </math>
where <math>r_1, r_2</math> are the numbers of real embeddings and the number of pairs of complex embeddings of ''F'', respectively.

This recovers the above example: the unit group of (the ring of integers of) a [[real quadratic field]] is infinite of rank 1, since <math>r_1=2, r_2=0</math>.  

In the ring {{math|'''Z'''/''n'''''Z'''}} of [[Modular arithmetic#Integers modulo n|integers modulo {{mvar|n}}]], the units are the congruence classes {{math|(mod ''n'')}} represented by integers [[coprime]] to {{mvar|n}}. They constitute the [[multiplicative group of integers modulo n|multiplicative group of integers modulo {{math|''n''}}]].

===Polynomials and power series===
For a commutative ring ''R'', the units of the [[polynomial ring]] ''R''[''x''] are precisely those polynomials
:<math>p(x)=a_0 + a_1 x + \dots a_n x^n</math>
such that <math>a_0</math> is a unit in ''R'', and the remaining coefficients <math>a_1, \dots, a_n</math> are [[nilpotent]] elements, i.e., satisfy <math>a_i^N =0</math> for some ''N''.<ref>{{harvtxt|Watkins|2007|loc=Theorem 11.1}}</ref> In particular, if ''R'' is a [[domain (ring theory)|domain]] (has no [[zero divisors]]), then the units of ''R''[''x''] agree with the ones of ''R''.
The units of the [[power series ring]] <math>R[[x]]</math> are precisely those power series
:<math>p(x)=\sum_{i=0}^\infty a_i x^i</math>
such that <math>a_0</math> is a unit in ''R''.<ref>{{harvtxt|Watkins|2007|loc=Theorem 12.1}}</ref>

===Matrix rings===
The unit group of the ring {{math|M<sub>''n''</sub>(''R'')}} of [[square matrix|{{math|''n'' × ''n''}} matrices]] over a [[commutative ring]] {{mvar|R}} (for example, a [[field (mathematics)|field]]) is the group [[general linear group|{{math|GL<sub>''n''</sub>(''R'')}}]] of [[invertible matrix|invertible matrices]].

An element of the matrix ring <math>\operatorname{M}_n(R)</math> is invertible if and only if the [[determinant]] of the element is invertible in ''R'', with the inverse explicitly given by [[Cramer's rule]].

===In general===
Let <math>R</math> be a ring. For any <math>x, y</math> in <math>R</math>, if <math>1 - xy</math> is invertible, then <math>1 - yx</math> is invertible with the inverse <math>1 + y(1-xy)^{-1}x</math>.<ref>{{harvnb|Jacobson|loc=§ 2.2. Exercise 4.}}</ref> The formula for the inverse can be found as follows: thinking formally, suppose <math>1-yx</math> is invertible and that the inverse is given by a geometric series: <math>(1-yx)^{-1} = \sum_0^{\infty} (yx)^n</math>. Then, manipulating it formally,
:<math>(1-yx)^{-1} = 1 + y \left(\sum_0^{\infty} (xy)^n \right)x = 1 + y(1-xy)^{-1}x.</math>

See also [[Hua's identity]] for a similar type of results.

==Group of units==

The units of a ring {{mvar|R}} form a [[group (mathematics)|group]] {{math|U(''R'')}} under multiplication, the ''group of units'' of {{mvar|R}}. Other common notations for {{math|U(''R'')}} are {{math|''R''<sup>∗</sup>}}, {{math|''R''<sup>×</sup>}}, and {{math|E(''R'')}} (from the German term [[wikt:Einheit|''Einheit'']]).

A commutative ring is a [[local ring]] if {{math|''R'' &minus; U(''R'')}} is a [[maximal ideal]]. As it turns out, if {{math|''R'' &minus; U(''R'')}} is an ideal, then it is necessarily a maximal ideal and ''R'' is local since a maximal ideal is disjoint from {{math|U(''R'')}}.

If {{mvar|R}} is a [[finite field]], then {{math|U(''R'')}} is a [[cyclic group]] of order <math>|R| - 1</math>.

The formulation of the group of units defines a [[functor]] {{math|U}} from the [[category of rings]] to the [[category of groups]]: every [[ring homomorphism]] {{math|1=''f'' : ''R'' → ''S''}} induces a [[group homomorphism]] {{math|1=U(''f'') : U(''R'') → U(''S'')}}, since {{mvar|f}} maps units to units. This functor has a [[left adjoint]] which is the integral [[group ring]] construction.

==Associatedness==
{{Main|Green's relations#The H and D relations}}
In a commutative unital ring {{mvar|R}}, the group of units {{math|U(''R'')}} [[Group action (mathematics)|acts]] on {{mvar|R}} via multiplication. The [[orbit (group theory)|orbits]] of this action are called sets of ''{{visible anchor|associates}}''; in other words, there is an [[equivalence relation]] ∼ on {{mvar|R}} called ''associatedness'' such that

:{{math|''r'' ∼ ''s''}}

means that there is a unit {{mvar|u}} with {{math|1=''r'' = ''us''}}.

In an [[integral domain]] the [[cardinality]] of an equivalence class of associates is the same as that of {{math|U(''R'')}}.

==See also==
* [[S-units]]
* [[Localization of a ring and a module]]

==References==
{{reflist}}
* Jacobson, Nathan (2009), Basic Algebra 1 (2nd ed.), Dover, {{ISBN|978-0-486-47189-1}}
* {{citation|author=Watkins|first=John J.|title=Topics in commutative ring theory|publisher=Princeton University Press|year=2007|isbn=978-0-691-12748-4|mr=2330411}}

{{DEFAULTSORT:Unit (Ring Theory)}}
[[Category:Ring theory]]
[[Category:Group theory]]
[[Category:Algebraic number theory]]
[[Category:1 (number)]]