{{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 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 {{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|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.

== 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 nonzero ring, the 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. For example, the unit group of the field of real numbers {{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 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 {{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 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 (or more generally reduced), then the units of {{math|''R''[''x'']}} are the units of {{mvar|R}}. The units of the power series ring <math>Rx</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 {{math|''n'' × ''n''}} matrices over a ring {{mvar|R}} is the group {{math|GL<sub>''n''</sub>(''R'')}} of 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 \biggl(\sum_{n \ge 0} (xy)^n \biggr)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 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 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 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 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.

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 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 | 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