Editing Unit (ring theory)

{{Short description|In mathematics, element with a multiplicative inverse}}
{{Distinguish|Unit ring}}
In [[algebra]], a '''unit''' or '''invertible element'''{{efn|In the case of rings, the use of "invertible element" is taken as self-evidently referring to multiplication, since all elements of a ring are invertible for addition.}} of a [[ring (mathematics)|ring]] is an [[invertible element]] for the multiplication of the ring. That is, an element {{mvar|u}} of a ring {{mvar|R}} is a unit if there exists {{mvar|v}} in {{mvar|R}} such that
<math display="block">vu = uv = 1,</math>
where {{math|1}} is the [[multiplicative identity]]; the element {{mvar|v}} is unique for this property and is called the [[multiplicative inverse]] of {{mvar|u}}.{{sfn|Dummit|Foote|2004|ps=}}{{sfn|Lang|2002|ps=}} The set of units of {{mvar|R}} forms a [[Group (mathematics)|group]] {{math|''R''{{sup|×}}}} under multiplication, called the '''group of units''' or '''unit group''' of {{mvar|R}}.{{efn|The notation {{math|''R''{{sup|×}}}}, introduced by [[André Weil]], is commonly used in [[number theory]], where unit groups arise frequently.{{sfn|Weil|1974|ps=}}  The symbol {{math|×}} is a reminder that the group operation is multiplication.  Also, a superscript × is not frequently used in other contexts, whereas a superscript {{math|*}} often denotes dual.}}  Other notations for the unit group are {{math|''R''<sup>∗</sup>}}, {{math|U(''R'')}}, and {{math|E(''R'')}} (from the German term {{lang|de|[[wikt:Einheit|Einheit]]}}).

Less commonly, the term ''unit'' is sometimes used to refer to the element {{math|1}} of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also [[unit matrix]]. Because of this ambiguity, {{math|1}} is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a [[rng (algebra)|rng]].

== Examples ==
{{anchor|−1}}The multiplicative identity {{math|1}} and its additive inverse {{math|−1}} are always units.  More generally, any [[root of unity]] in a ring {{mvar|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 {{mvar|r}}.
In a [[zero ring|nonzero ring]], the [[additive identity|element 0]] is not a unit, so {{math|''R''{{sup|×}}}} is not closed under addition.
A nonzero ring {{mvar|R}} in which every nonzero element is a unit (that is, {{math|1=''R''{{sup|×}} = ''R'' &setminus; {{mset|0}}}}) is called a [[division ring]] (or a skew-field). A commutative division ring is called a [[field (mathematics)|field]]. For example, the unit group of the field of [[real number]]s {{math|'''R'''}} is {{math|'''R''' &setminus; {{mset|0}}}}.

=== Integer ring ===
In the ring of [[integers]] {{math|'''Z'''}}, the only units are {{math|1}} and {{math|−1}}.

In the ring {{math|'''Z'''/''n'''''Z'''}} of [[Modular arithmetic#Integers modulo m|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 {{mvar|n}}]].

=== Ring of integers of a number field ===
In the ring {{math|'''Z'''[{{sqrt|3}}]}} obtained by adjoining the [[quadratic integer]] {{math|{{sqrt|3}}}} to {{math|'''Z'''}}, one has {{math|1= (2 + {{sqrt|3}})(2 − {{sqrt|3}}) = 1}}, so {{math|2 + {{sqrt|3}}}} is a unit, and so are its powers, so {{math|'''Z'''[{{sqrt|3}}]}} has infinitely many units.

More generally, for the [[ring of integers]] {{mvar|R}} in a [[number field]] {{mvar|F}}, [[Dirichlet's unit theorem]] states that {{math|''R''{{sup|×}}}} is isomorphic to the group 
<math display="block">\mathbf Z^n \times \mu_R</math>
where <math>\mu_R</math> is the (finite, cyclic) group of roots of unity in {{mvar|R}} and {{mvar|n}}, the [[rank of a module|rank]] of the unit group, is
<math display="block">n = r_1 + r_2 -1, </math>
where <math>r_1, r_2</math> are the number of real embeddings and the number of pairs of complex embeddings of {{mvar|F}}, respectively.

This recovers the {{math|'''Z'''[{{sqrt|3}}]}} 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>.

=== Polynomials and power series ===
For a commutative ring {{mvar|R}}, the units of the [[polynomial ring]] {{math|''R''[''x'']}} are the polynomials
<math display="block">p(x) = a_0 + a_1 x + \dots + a_n x^n</math>
such that {{math|''a''<sub>0</sub>}} is a unit in {{mvar|R}} and the remaining coefficients <math>a_1, \dots, a_n</math> are [[nilpotent]], i.e., satisfy <math>a_i^N = 0</math> for some {{math|''N''}}.{{sfn|Watkins|2007|loc=Theorem 11.1|ps=}}
In particular, if {{mvar|R}} is a [[domain (ring theory)|domain]] (or more generally [[reduced ring|reduced]]), then the units of {{math|''R''[''x'']}} are the units of {{mvar|R}}.
The units of the [[power series ring]] <math>R[[x]]</math> are the power series
<math display="block">p(x)=\sum_{i=0}^\infty a_i x^i</math>
such that {{math|''a''<sub>0</sub>}} is a unit in {{mvar|R}}.{{sfn|Watkins|2007|loc=Theorem 12.1|ps=}}

=== Matrix rings ===
The unit group of the ring {{math|M<sub>''n''</sub>(''R'')}} of [[square matrix|{{math|''n'' × ''n''}} matrices]] over a ring {{mvar|R}} is the group {{math|[[general linear group|GL<sub>''n''</sub>(''R'')]]}} of [[invertible matrix|invertible matrices]].  For a commutative ring {{mvar|R}}, an element {{mvar|A}} of {{math|M<sub>''n''</sub>(''R'')}} is invertible if and only if the [[determinant]] of {{mvar|A}} is invertible in {{mvar|R}}.  In that case, {{math|''A''{{sup|−1}}}} can be given explicitly in terms of the [[adjugate matrix]].

=== In general ===
For elements {{mvar|x}} and {{mvar|y}} in a ring {{mvar|R}}, if <math>1 - xy</math> is invertible, then <math>1 - yx</math> is invertible with inverse <math>1 + y(1-xy)^{-1}x</math>;{{sfn|Jacobson|2009|loc=§2.2 Exercise 4|ps=}} this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:
<math display="block">(1-yx)^{-1} = \sum_{n \ge 0} (yx)^n = 1 + y \left(\sum_{n \ge 0} (xy)^n \right)x = 1 + y(1-xy)^{-1}x.</math>
See [[Hua's identity]] for similar results.

== Group of units ==
A [[commutative ring]] is a [[local ring]] if {{math|''R'' &setminus; ''R''{{sup|×}}}} is a [[maximal ideal]].

As it turns out, if {{math|''R'' &setminus; ''R''{{sup|×}}}} is an ideal, then it is necessarily a [[maximal ideal]] and {{math|''R''}} is [[local ring|local]] since a [[maximal ideal]] is disjoint from {{math|''R''{{sup|×}}}}.

If {{mvar|R}} is a [[finite field]], then {{math|''R''{{sup|×}}}} is a [[cyclic group]] of order {{math|{{abs|''R''}} − 1}}.

Every [[ring homomorphism]] {{math|''f'' : ''R'' → ''S''}} induces a [[group homomorphism]] {{math|''R''{{sup|×}} → ''S''{{sup|×}}}}, since {{mvar|f}} maps units to units.  In fact, the formation of the unit group defines a [[functor]] from the [[category of rings]] to the [[category of groups]].  This functor has a [[left adjoint]] which is the integral [[group ring]] construction.{{sfn|Cohn|2003|loc=§2.2 Exercise 10|ps=}}

The [[group scheme]] <!-- shouldn't we avoid scheme? --><math>\operatorname{GL}_1</math> is isomorphic to the [[multiplicative group scheme]] <math>\mathbb{G}_m</math> over any base, so for any commutative ring {{mvar|R}}, the groups <math>\operatorname{GL}_1(R)</math> and <math>\mathbb{G}_m(R)</math> are canonically isomorphic to {{math|''U''(''R'')}}. Note that the functor <math>\mathbb{G}_m</math> (that is, {{math|''R'' ↦ ''U''(''R'')}}) is [[Representable functor|representable]] in the sense: <math>\mathbb{G}_m(R) \simeq \operatorname{Hom}(\mathbb{Z}[t, t^{-1}], R)</math> for commutative rings {{mvar|R}} (this for instance follows from the aforementioned adjoint relation with the group ring construction). Explicitly this means that there is a natural bijection between the set of the ring homomorphisms <math>\mathbb{Z}[t, t^{-1}] \to R</math> and the set of unit elements of {{mvar|R}} (in contrast, <math>\mathbb{Z}[t]</math> represents the additive group <math>\mathbb{G}_a</math>, the [[forgetful functor]] from the category of commutative rings to the [[category of abelian groups]]).

== Associatedness ==
Suppose that {{mvar|R}} is commutative.  Elements {{mvar|r}} and {{mvar|s}} of {{mvar|R}} are called ''{{visible anchor|associate}}'' if there exists a unit {{mvar|u}} in {{mvar|R}} such that {{math|1=''r'' = ''us''}}; then write {{math|''r'' ~ ''s''}}.  In any ring, pairs of [[additive inverse]] elements{{efn|{{mvar|x}} and {{math|−''x''}} are not necessarily distinct.  For example, in the ring of integers modulo 6, one has {{math|1=3 = −3}} even though {{math|1 ≠ −1}}.}} {{math|''x''}} and {{math|−''x''}} are [[Associated element|associate]], since any ring includes the unit {{math|−1}}.  For example, 6 and −6 are associate in {{math|'''Z'''}}.  In general, {{math|~}} is an [[equivalence relation]] on {{mvar|R}}.

Associatedness can also be described in terms of the [[Group action (mathematics)|action]] of {{math|''R''{{sup|×}}}} on {{mvar|R}} via multiplication: Two elements of {{mvar|R}} are associate if they are in the same {{math|''R''{{sup|×}}}}-[[orbit (group theory)|orbit]].

In an [[integral domain]], the set of associates of a given nonzero element has the same [[cardinality]] as {{math|''R''{{sup|×}}}}.

The equivalence relation {{math|~}} can be viewed as any one of [[Green's relations|Green's semigroup relations]] specialized to the multiplicative [[semigroup]] of a commutative ring {{mvar|R}}.

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

== Notes ==
{{notelist}}

== Citations ==
{{reflist|3}}

== Sources ==
{{refbegin}}
* {{cite book
 | last=Cohn | first=Paul M. | author-link=Paul Cohn
 | year=2003
 | title=Further algebra and applications
 | edition=Revised ed. of Algebra, 2nd
 | location=London | publisher=[[Springer-Verlag]]
 | isbn=1-85233-667-6
 | zbl=1006.00001
}}
* {{cite book
 | last1 = Dummit | first1 = David S.
 | last2 = Foote | first2 = Richard M.
 | year = 2004
 | title = Abstract Algebra
 | edition = 3rd
 | publisher = [[John Wiley & Sons]]
 | isbn = 0-471-43334-9
}}
* {{cite book
 | last = Jacobson | first = Nathan | author-link = Nathan Jacobson
 | year = 2009
 | title = Basic Algebra 1
 | edition = 2nd
 | publisher = Dover
 | isbn = 978-0-486-47189-1
}}
* {{cite book
 | title = Algebra
 | last = Lang | first = Serge | author-link = Serge Lang
 | year = 2002
 | series = [[Graduate Texts in Mathematics]]
 | publisher = [[Springer Science+Business Media|Springer]]
 | isbn = 0-387-95385-X
}}
* {{citation
 | last = Watkins | first = John J.
 | year = 2007
 | title = Topics in commutative ring theory
 | publisher = Princeton University Press
 | isbn = 978-0-691-12748-4
 | mr = 2330411
}}
* {{cite book
 | title = Basic number theory
 | last = Weil | first = André | author-link = André Weil
 | year = 1974
 | series = Grundlehren der mathematischen Wissenschaften
 | volume = 144
 | edition = 3rd
 | publisher = [[Springer-Verlag]]
 | isbn = 978-3-540-58655-5
}}
{{refend}}

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