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In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, a unit of a ring is an
invertible element In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this property and is called the multiplicative inverse of . The set of units of forms a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ). Less commonly, the term ''unit'' is sometimes used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''unit ring'', and also unit matrix. Because of this ambiguity, 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

The multiplicative identity and its additive inverse are always units. More generally, any root of unity in a ring is a unit: if , then is a multiplicative inverse of . In a nonzero ring, the element 0 is not a unit, so is not closed under addition. A nonzero ring in which every nonzero element is a unit (that is, ) is called a
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
(or a skew-field). A commutative division ring is called a field. For example, the unit group of the field of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s is .


Integer ring

In the ring of
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, the only units are and . In the ring of integers modulo , the units are the congruence classes represented by integers
coprime In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to . They constitute the multiplicative group of integers modulo .


Ring of integers of a number field

In the ring obtained by adjoining the
quadratic integer In number theory, quadratic integers are a generalization of the usual integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and (usual) integers. When algebra ...
to , one has , so is a unit, and so are its powers, so has infinitely many units. More generally, for the ring of integers in a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
,
Dirichlet's unit theorem In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring of algebraic integers of a number field . The regulator is a pos ...
states that is isomorphic to the group \mathbf Z^n \times \mu_R where \mu_R is the (finite, cyclic) group of roots of unity in and , the
rank Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * ...
of the unit group, is n = r_1 + r_2 -1, where r_1, r_2 are the number of real embeddings and the number of pairs of complex embeddings of , respectively. This recovers the example: The unit group of (the ring of integers of) a
real quadratic field In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...
is infinite of rank 1, since r_1=2, r_2=0.


Polynomials and power series

For a commutative ring , the units of the polynomial ring are the polynomials p(x) = a_0 + a_1 x + \dots + a_n x^n such that a_0 is a unit in and the remaining coefficients a_1, \dots, a_n are nilpotent, i.e., satisfy a_i^N = 0 for some ''N''. In particular, if is a domain, then the units of are the units of . The units of the
power series ring In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
R x are the power series p(x)=\sum_^\infty a_i x^i such that a_0 is a unit in .


Matrix rings

The unit group of the ring of matrices over a ring is the group of
invertible matrices In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplicati ...
. For a commutative ring , an element of is invertible if and only if the determinant of is invertible in . In that case, can be given explicitly in terms of the
adjugate matrix In linear algebra, the adjugate or classical adjoint of a square matrix is the transpose of its cofactor matrix and is denoted by . It is also occasionally known as adjunct matrix, or "adjoint", though the latter today normally refers to a differe ...
.


In general

For elements and in a ring , if 1 - xy is invertible, then 1 - yx is invertible with inverse 1 + y(1-xy)^x; this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series: (1-yx)^ = \sum_ (yx)^n = 1 + y \left(\sum_ (xy)^n \right)x = 1 + y(1-xy)^x. See Hua's identity for similar results.


Group of units

A commutative ring is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
if is a
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
. As it turns out, if is an ideal, then it is necessarily a
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
and ''R'' is
local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...
since a
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
is disjoint from . If is a finite field, then is a
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order , R, - 1. Every
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...
induces a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
, since maps units to units. In fact, the formation of the unit group defines a
functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
from the
category of rings In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of ring ...
to the
category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories There a ...
. This functor has a
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
which is the integral
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the giv ...
construction.Exercise 10 in § 2.2. of The
group scheme In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have ...
\operatorname_1 is isomorphic to the multiplicative group scheme \mathbb_m over any base, so for any commutative ring , the groups \operatorname_1(R) and \mathbb_m(R) are canonically isomorphic to U(R). Note that the functor \mathbb_m (that is, R \mapsto U(R)) is representable in the sense: \mathbb_m(R) \simeq \operatorname(\mathbb , t^ R) for commutative rings (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 \mathbb , t^\to R and the set of unit elements of (in contrast, \mathbb /math> represents the additive group \mathbb_a, the forgetful functor from the category of commutative rings to the category of abelian groups).


Associatedness

Suppose that is commutative. Elements and of are called ' if there exists a unit in such that ; then write . In any ring, pairs of additive inverse elements and are associate. For example, 6 and −6 are associate in . In general, is an equivalence relation on . Associatedness can also be described in terms of the
action Action may refer to: * Action (narrative), a literary mode * Action fiction, a type of genre fiction * Action game, a genre of video game Film * Action film, a genre of film * ''Action'' (1921 film), a film by John Ford * ''Action'' (1980 fil ...
of on via multiplication: Two elements of are associate if they are in the same -
orbit In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as ...
. In an
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
, the set of associates of a given nonzero element has the same cardinality as . The equivalence relation can be viewed as any one of Green's semigroup relations specialized to the multiplicative
semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ...
of a commutative ring .


See also

* S-units * Localization of a ring and a module


Notes


Citations


Sources

* * * * * {{DEFAULTSORT:Unit (Ring Theory) 1 (number) Algebraic number theory Group theory Ring theory Algebraic properties of elements