Unit (ring theory)

In

invertible element

for the multiplication of the ring. That is, an element

u

of a ring

R

is a unit if there exists

v

in

R

such that

vu=uv=1,

where

1

is the

multiplicative identity; the element

v

is unique for this property and is called the multiplicative inverse of

u

.^[1]^[2] The set of units of

R

forms a group

R \times

under multiplication, called the group of units or unit group of

R

.^[b] Other notations for the unit group are

R *

,

U(R)

, and

E(R)

(from the German term Einheit

).

Less commonly, the term unit is sometimes used to refer to the element $1$ of the ring, in expressions like ring with a unit or unit ring, and also

unit matrix. Because of this ambiguity,

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

The multiplicative identity $1$ and its additive inverse $-1$ are always units. More generally, any root of unity in a ring $R$ is a unit: if $r n = 1$ , then $r n -1$ is a multiplicative inverse of $r$ . In a nonzero ring, the element 0 is not a unit, so $R \times$ is not closed under addition. A nonzero ring $R$ in which every nonzero element is a unit (that is, $R \times = R ∖ {0}$ ) 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 $R$ is $R ∖ {0}$ .

Integer ring

In the ring of

integers

Z

, the only units are

1

and

-1

.

In the ring $Z / n Z$ of

coprime to

n

. They constitute the multiplicative group of integers modulo

n

.

Ring of integers of a number field

In the ring $Z [\sqrt 3]$ obtained by adjoining the quadratic integer $\sqrt 3$ to $Z$ , one has $(2 + \sqrt 3)(2 - \sqrt 3) = 1$ , so $2 + \sqrt 3$ is a unit, and so are its powers, so $Z [\sqrt 3]$ has infinitely many units.

More generally, for the

number field

F

, Dirichlet's unit theorem

states that

R \times

is isomorphic to the group

\mathbf {Z} ^{n}\times \mu _{R}

where

\mu _{R}

is the (finite, cyclic) group of roots of unity in

R

and

n

, the

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

F

, respectively.

This recovers the $Z [\sqrt 3]$ 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 $R$ , the units of the polynomial ring $R [x]$ are the polynomials

p(x)=a_{0}+a_{1}x+\dots +a_{n}x^{n}

such that

a 0

is a unit in

R

and the remaining coefficients

a_{1},\dots ,a_{n}

are nilpotent, i.e., satisfy

a_{i}^{N}=0

for some

N

.^[4] In particular, if

R

is a domain (or more generally reduced), then the units of

R [x]

are the units of

R

. The units of the

power series ring

R[[x]]

are the power series

p(x)=\sum _{i=0}^{\infty }a_{i}x^{i}

such that

a 0

is a unit in

R

.^[5]

Matrix rings

The unit group of the ring $M n (R)$ of $n \times n$ matrices over a ring $R$ is the group $GL n (R)$ of invertible matrices. For a commutative ring $R$ , an element $A$ of $M n (R)$ is invertible if and only if the determinant of $A$ is invertible in $R$ . In that case, $A -1$ can be given explicitly in terms of the adjugate matrix.

In general

For elements $x$ and $y$ in a ring $R$ , if $1-xy$ is invertible, then $1-yx$ is invertible with inverse $1+y(1-xy)^{-1}x$ ;^[6] this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:

(1-yx)^{-1}=\sum _{n\geq 0}(yx)^{n}=1+y\left(\sum _{n\geq 0}(xy)^{n}\right)x=1+y(1-xy)^{-1}x.

See Hua's identity for similar results.

Group of units

A commutative ring is a local ring if $R ∖ R \times$ is a maximal ideal.

As it turns out, if $R ∖ R \times$ is an ideal, then it is necessarily a maximal ideal and $R$ is local since a maximal ideal is disjoint from $R \times$ .

If $R$ is a finite field, then $R \times$ is a cyclic group of order $| R | - 1$ .

Every

left adjoint which is the integral group ring construction.^[7]

The group scheme $\operatorname {GL} _{1}$ is isomorphic to the

multiplicative group scheme

\mathbb {G} _{m}

over any base, so for any commutative ring

R

, the groups

\operatorname {GL} _{1}(R)

and

\mathbb {G} _{m}(R)

are canonically isomorphic to

U (R)

. Note that the functor

\mathbb {G} _{m}

(that is,

R \mapsto U (R)

) is representable in the sense:

\mathbb {G} _{m}(R)\simeq \operatorname {Hom} (\mathbb {Z} [t,t^{-1}],R)

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 {Z} [t,t^{-1}]\to R

and the set of unit elements of

R

(in contrast,

\mathbb {Z} [t]

represents the additive group

\mathbb {G} _{a}

, the forgetful functor from the category of commutative rings to the category of abelian groups).

Associatedness

Suppose that $R$ is commutative. Elements $r$ and $s$ of $R$ are called associate if there exists a unit $u$ in $R$ such that $r = us$ ; then write $r ~ s$ . In any ring, pairs of

associate. For example, 6 and −6 are associate in

Z

. In general,

~

is an equivalence relation

on

R

.

Associatedness can also be described in terms of the

orbit

.

In an integral domain, the set of associates of a given nonzero element has the same cardinality as $R \times$ .

The equivalence relation $~$ can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring $R$ .

Notes

^ 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.
^ The notation $R \times$ , introduced by André Weil, is commonly used in number theory, where unit groups arise frequently.^[3] The symbol $\times$ is a reminder that the group operation is multiplication. Also, a superscript × is not frequently used in other contexts, whereas a superscript $*$ often denotes dual.
^ $x$ and $- x$ are not necessarily distinct. For example, in the ring of integers modulo 6, one has $3 = -3$ even though $1 \neq -1$ .

Citations

^ Dummit & Foote 2004
^ Lang 2002
^ Weil 1974
^ Watkins 2007, Theorem 11.1
^ Watkins 2007, Theorem 12.1
^ Jacobson 2009, §2.2 Exercise 4
^ Cohn 2003, §2.2 Exercise 10

Sources

[1] 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.

[5] The notation $R \times$ , introduced by André Weil, is commonly used in number theory, where unit groups arise frequently.^[3] The symbol $\times$ is a reminder that the group operation is multiplication. Also, a superscript × is not frequently used in other contexts, whereas a superscript $*$ often denotes dual.

[10] $x$ and $- x$ are not necessarily distinct. For example, in the ring of integers modulo 6, one has $3 = -3$ even though $1 \neq -1$ .

[FOOTNOTEDummitFoote2004-2] Dummit & Foote 2004

[FOOTNOTELang2002-3] Lang 2002

[FOOTNOTEWeil1974-4] Weil 1974

[FOOTNOTEWatkins2007Theorem_11.1-6] Watkins 2007, Theorem 11.1

[FOOTNOTEWatkins2007Theorem_12.1-7] Watkins 2007, Theorem 12.1

[FOOTNOTEJacobson2009§2.2_Exercise_4-8] Jacobson 2009, §2.2 Exercise 4

[FOOTNOTECohn2003§2.2_Exercise_10-9] Cohn 2003, §2.2 Exercise 10

[1]

[2]

[b]

[4]

[5]

[6]

[7]

[3]

Examples

Integer ring

Ring of integers of a number field

Polynomials and power series

Matrix rings

In general

Group of units

Associatedness

See also

Notes

Citations

Sources