Unit (ring theory)

Inalgebra,aunitorinvertible element^[a]of aringis aninvertible elementfor 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 themultiplicative identity;the element $v$ is unique for this property and is called themultiplicative inverseof $u$ .^[1]^[2]The set of units of $R$ forms agroup $R \times$ under multiplication, called thegroup of unitsorunit groupof $R$ .^[b]Other notations for the unit group are $R *$ , $U(R)$ ,and $E(R)$ (from the German termEinheit).

Less commonly, the termunitis sometimes used to refer to the element $1$ of the ring, in expressions likering with a unitorunit ring,and alsounit 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 arng.

Examples

The multiplicative identity $1$ and its additive inverse $-1$ are always units. More generally, anyroot of unityin a ring $R$ is a unit: if $r n = 1$ ,then $r n -1$ is a multiplicative inverse of $r$ . In anonzero ring,theelement 0is 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 adivision ring(or a skew-field). A commutative division ring is called afield.For example, the unit group of the field ofreal numbers $R$ is $R ∖ {0}$ .

Integer ring

In the ring ofintegers $Z$ ,the only units are $1$ and $-1$ .

In the ring $Z / n Z$ ofintegers modulo $n$ ,the units are the congruence classes $(mod n)$ represented by integerscoprimeto $n$ .They constitute themultiplicative group of integers modulo $n$ .

Ring of integers of a number field

In the ring $Z [\sqrt 3]$ obtained by adjoining thequadratic 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 thering of integers $R$ in anumber field $F$ ,Dirichlet's unit theoremstates 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 $n$ ,therankof 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) areal quadratic fieldis infinite of rank 1, since $r_{1}=2,r_{2}=0$ .

Polynomials and power series

For a commutative ring $R$ ,the units of thepolynomial 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 $a_{1},\dots,a_{n}$ arenilpotent,i.e., satisfy $a_{i}^{N}=0$ for some $N$ .^[4] In particular, if $R$ is adomain(or more generallyreduced), then the units of $R [x]$ are the units of $R$ . The units of thepower 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 $n \times n$ matricesover a ring $R$ is the group $GL n (R)$ ofinvertible matrices.For a commutative ring $R$ ,an element $A$ of $M n (R)$ is invertible if and only if thedeterminantof $A$ is invertible in $R$ .In that case, $A -1$ can be given explicitly in terms of theadjugate 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: $(1-yx)^{-1}=\sum _{n\geq 0}(yx)^{n}=1+y\left(\sum _{n\geq 0}(xy)^{n}\right)x=1+y(1-xy)^{-1}x.$ SeeHua's identityfor similar results.

Group of units

Acommutative ringis alocal ringif $R ∖ R \times$ is amaximal ideal.

As it turns out, if $R ∖ R \times$ is an ideal, then it is necessarily amaximal idealand $R$ islocalsince amaximal idealis disjoint from $R \times$ .

If $R$ is afinite field,then $R \times$ is acyclic groupof order $| R | - 1$ .

Everyring homomorphism $f : R \to S$ induces agroup homomorphism $R \times \to S \times$ ,since $f$ maps units to units. In fact, the formation of the unit group defines afunctorfrom thecategory of ringsto thecategory of groups.This functor has aleft adjointwhich is the integralgroup ringconstruction.^[7]

Thegroup scheme $\operatorname {GL} _{1}$ is isomorphic to themultiplicative 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 calledassociateif there exists a unit $u$ in $R$ such that $r = us$ ;then write $r ~ s$ .In any ring, pairs ofadditive inverseelements^[c] $x$ and $- x$ areassociate,since any ring includes the unit $-1$ .For example, 6 and −6 are associate in $Z$ .In general, $~$ is anequivalence relationon $R$ .

Associatedness can also be described in terms of theactionof $R \times$ on $R$ via multiplication: Two elements of $R$ are associate if they are in the same $R \times$ -orbit.

In anintegral domain,the set of associates of a given nonzero element has the samecardinalityas $R \times$ .

The equivalence relation $~$ can be viewed as any one ofGreen's semigroup relationsspecialized to the multiplicativesemigroupof 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 byAndré Weil,is commonly used innumber 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

Cohn, Paul M.(2003).Further algebra and applications(Revised ed. of Algebra, 2nd ed.). London:Springer-Verlag.ISBN 1-85233-667-6.Zbl 1006.00001.
Dummit, David S.; Foote, Richard M. (2004).Abstract Algebra(3rd ed.).John Wiley & Sons.ISBN 0-471-43334-9.
Jacobson, Nathan(2009).Basic Algebra 1(2nd ed.). Dover.ISBN 978-0-486-47189-1.
Lang, Serge(2002).Algebra.Graduate Texts in Mathematics.Springer.ISBN 0-387-95385-X.
Watkins, John J. (2007),Topics in commutative ring theory,Princeton University Press,ISBN 978-0-691-12748-4,MR 2330411
Weil, André(1974).Basic number theory.Grundlehren der mathematischen Wissenschaften. Vol. 144 (3rd ed.).Springer-Verlag.ISBN 978-3-540-58655-5.

[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 byAndré Weil,is commonly used innumber 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

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