unit (ring theory)

TheInfoList

In the branch of abstract algebra known as
ring theory In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. ...
, a unit of a ring $R$ is any element $u \in R$ that has a multiplicative inverse in $R$: an element $v \in R$ such that :$vu = uv = 1$, where is the
multiplicative identity In mathematics, an identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it. This concept is used in algebraic s ...
. The set of units of forms a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
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 also used to refer to the element of the ring, in expressions like ''ring with a unit'' or ''
unit ring In mathematics, rings are algebraic structures that generalize field (mathematics), fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a Set (mathematics), set equipped with t ...
'', and also e.g. ''''. For this reason, some authors call "unity" or "identity", and say that is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".

# Examples

The multiplicative identity and its additive inverse are always units. More generally, any
root of unity The 5th roots of unity (blue points) in the complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are containe ...
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 ringIn algebra, a division ring, also called a skew field, is a ring (mathematics), ring in which division (mathematics), division is possible. Specifically, it is a zero ring, nonzero ring in which every nonzero element has a multiplicative inverse, th ...
(or a skew-field). A commutative division ring is called a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
. For example, the unit group of the field of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s is .

## Integer ring

In the ring of
integers An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...

, the only units are and . In the ring of integers modulo , the units are the congruence classes represented by integers
coprime In number theory, two integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...
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 integers to quadratic fields. Quadratic integers are algebraic integers of degree two, that is, solutions of equations of the form : with and integers. When algebraic integers are ...
to , one has , so is a unit, and so are its powers, so has infinitely many units. More generally, for the
ring of integersIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
in a
number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
,
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 an abelian group, rank of the group of units in the ring (mathematics), ring of algebraic integer ...
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 A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...
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 ''K'' of Degree of a field extension, degree two over Q, the rational numbers. The map ''d'' ↦ Q() is a bijection from the Set (mathematics), set of all square-f ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
are the polynomials :$p\left(x\right)=a_0 + a_1 x + \dots a_n x^n$ such that $a_0$ is a unit in and the remaining coefficients $a_1, \dots, a_n$ are
nilpotent In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, i.e., satisfy $a_i^N =0$ for some ''N''. In particular, if is a
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
, then the units of are the units of . The units of the
power series ring Power most often refers to: * Power (physics) In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, ...
$R$ are the power series :$p\left(x\right)=\sum_^\infty a_i x^i$ 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 Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and t ...
. For a commutative ring , an element of is invertible if and only if the
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of is invertible in . In that case, can be given explicitly in terms of the
adjugate matrixIn linear algebra, the adjugate or classical adjoint of a square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space ...
.

## In general

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

# Group of units

A
commutative ring In , a branch of , a commutative ring is a in which the multiplication operation is . The study of commutative rings is called . Complementarily, is the study of s where multiplication is not required to be commutative. Definition and first e ...
is a
local ring In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), r ...
if is a
maximal idealIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
. As it turns out, if is an ideal, then it is necessarily a
maximal idealIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
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 administrati ...
since a
maximal idealIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
is disjoint from . If is a
finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, then is a
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

of order $, R, - 1$. Every
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...
induces a
group homomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

, since maps units to units. In fact, the formation of the unit group defines a
functor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

from the
category of rings In mathematics, the category of rings, denoted by Ring, is the category (mathematics), category whose objects are ring (mathematics), rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categor ...
to the
category of groups In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. This functor has a
left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may have. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of ...
which is the integral
group ring In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
construction.Exercise 10 in § 2.2. of The
group scheme In mathematics, a group scheme is a type of Algebraic geometry, algebro-geometric object equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in the se ...
$\operatorname_1$ is isomorphic to the multiplicative group scheme $\mathbb_m$ over any base, so for any commutative ring , the groups $\operatorname_1\left(R\right)$ and $\mathbb_m\left(R\right)$ are canonically isomorphic to $U\left(R\right)$. Note that the functor $\mathbb_m$ (that is, $R \mapsto U\left(R\right)$) is representable in the sense: for commutative rings ''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 and the set of unit elements of ''R'' (in contrast,

# 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 In mathematics, the additive inverse of a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be ...
elements and are associate. For example, 6 and −6 are associate in . In general, is an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
on . Associatedness can also be described in terms of the
action ACTION is a bus operator in , Australia owned by the . History On 19 July 1926, the commenced operating public bus services between Eastlake (now ) in the south and in the north. The service was first known as Canberra City Omnibus Se ...
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 physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificial satellite around an object or po ...
. In an
integral domain In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
, the set of associates of a given nonzero element has the same
cardinality In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), an ...
of a commutative ring .