Unit (ring theory)

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In algebra, a unit or invertible element^[a] of a ring is an 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 integers modulo $n$ , the units are the congruence classes $(mod n)$ represented by integers 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 ring of integers $R$ in a 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 ring homomorphism $f : R \to S$ induces a group homomorphism $R \times \to S \times$ , since $f$ 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.^[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 additive inverse elements^[c] $x$ and $- x$ are associate, since any ring includes the unit $-1$ . 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 action of $R \times$ on $R$ via multiplication: Two elements of $R$ are associate if they are in the same $R \times$ -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

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 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

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