Unit (ring theory)

In the branch of abstract algebra known as ring theory, 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 $1$ is the multiplicative identity.[1][2] The set of units $U(R)$ of a ring forms a group under multiplication.

Less commonly, the term unit is also used to refer to the element $1$ of the ring, in expressions like ring with a unit or unit ring, and also e.g. 'unit' matrix. For this reason, some authors call $1$ "unity" or "identity", and say that $R$ is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".

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 $U(R)$ is not closed under addition. A ring $R$ in which every nonzero element is a unit (that is, $U(R) = R -{0}$ ) 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 $R - {0$ }.

Integers

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

The ring of integers in a number field may have more units in general. For example, in the ring $Z [.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px;white-space:nowrap} 1 + \sqrt 5 / 2]$ that arises by adjoining the quadratic integer $1 + \sqrt 5 / 2$ to $Z$ , one has

(\sqrt 5 + 2)(\sqrt 5 - 2) = 1

in the ring, so $\sqrt 5 + 2$ is a unit. (In fact, the unit group of this ring is infinite.)

In fact, Dirichlet's unit theorem describes the structure of $U(R)$ precisely: it is isomorphic to a group of the form

\mathbf {Z} ^{n}\oplus \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 numbers of real embeddings and the number of pairs of complex embeddings of $F$ , respectively.

This recovers the above example: the unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since $r_{1}=2,r_{2}=0$ .

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 $n$ .

Polynomials and power series

For a commutative ring $R$ , the units of the polynomial ring $R [x]$ are precisely those 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 elements, i.e., satisfy $a_{i}^{N}=0$ for some N.[3] In particular, if R is a domain (has no zero divisors), then the units of $R [x]$ agree with the ones of $R$ . The units of the power series ring $R[[x]]$ are precisely those power series

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

such that $a_{0}$ is a unit in $R$ .[4]

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, $A -1$ is explicitly given by Cramer's rule.

In general

For elements $x$ and $y$ in a ring $R$ , if $1-xy$ is invertible, then $1-yx$ is invertible with the inverse $1+y(1-xy)^{-1}x$ .[5] The formula for the inverse 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

The units of a ring $R$ form a group $U(R)$ under multiplication, the group of units of $R$ .

Other common notations for $U(R)$ are $R *$ , $R \times$ , and $E(R)$ (from the German term Einheit).

A commutative ring is a local ring if $R - U(R)$ is a maximal ideal.

As it turns out, if $R - U(R)$ is an ideal, then it is necessarily a maximal ideal and R is local since a maximal ideal is disjoint from $U(R)$ .

If $R$ is a finite field, then $U(R)$ is a cyclic group of order $|R|-1$ .

The formulation of the group of units defines a functor $U$ from the category of rings to the category of groups:

every ring homomorphism $f : R \to S$ induces a group homomorphism $U(f) : U(R) \to U(S)$ , since $f$ maps units to units.

This functor has a left adjoint which is the integral group ring construction.[6]

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 $r \sim s$ . In any ring, pairs of additive inverse elements[lower-alpha 1] $x$ and $- x$ are 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 action of $U(R)$ on $R$ via multiplication: Two elements of $R$ are associate if they are in the same $U(R)$ -orbit.

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

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

$x$ 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.
Watkins (2007, Theorem 11.1)
Watkins (2007, Theorem 12.1)
Jacobson 2009, § 2.2. Exercise 4.
Exercise 10 in § 2.2. of 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.

Sources

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

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

[FOOTNOTEDummitFoote2004-1] Dummit & Foote 2004.

[FOOTNOTELang2002-2] Lang 2002.

[3] Watkins (2007, Theorem 11.1)

[4] Watkins (2007, Theorem 12.1)

[FOOTNOTEJacobson2009§_2.2._Exercise_4-5] Jacobson 2009, § 2.2. Exercise 4.

[6] Exercise 10 in § 2.2. of 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.