Invertible element

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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 Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
$R$ 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 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 th ...
. The set of units of forms a
group A group is a number of people 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 ident ...
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 ...
'', 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 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 ...
(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 ...

, 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 n, multiplicative group of integers modulo .

## Ring of integers of a number field

In the ring obtained by adjoining the quadratic integer 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 , Dirichlet's unit theorem 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 of a module, 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 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 :$p\left(x\right)=a_0 + a_1 x + \dots a_n x^n$ such that $a_0$ is a unit in and the remaining coefficients $a_1, \dots, a_n$ are nilpotent, i.e., satisfy $a_i^N =0$ for some ''N''. In particular, if is a domain (ring theory), domain, then the units of are the units of . The units of the power series ring $Rx$ 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 square matrix, matrices over a ring is the group of invertible matrix, invertible matrices. 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 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 is a local ring if is a maximal ideal. As it turns out, if is an ideal, then it is necessarily a maximal ideal and ''R'' is local ring, local since a maximal ideal is disjoint from . If is a finite field, then is a cyclic group of order $, R, - 1$. Every ring homomorphism induces a group homomorphism , since 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.Exercise 10 in § 2.2. of The group scheme $\operatorname_1$ 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: $\mathbb_m\left(R\right) \simeq \operatorname\left(\mathbb\left[t, t^\right], R\right)$ 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 $\mathbb\left[t, t^\right] \to R$ and the set of unit elements of ''R'' (in contrast, $\mathbb\left[t\right]$ 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 Associated element, 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 Group action (mathematics), action of on via multiplication: Two elements of are associate if they are in the same -orbit (group theory), orbit. In an integral domain, 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 relations, Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring .