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Summary

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. The set of units of R forms a group R× under multiplication, called the group of units or unit group of R.[a] Other notations for the unit group are R, U(R), and E(R) (from the German term Einheit).

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 rn = 1, then rn − 1 is a multiplicative inverse of r. In a nonzero ring, the element 0 is not a unit, so R× is not closed under addition. A nonzero ring R in which every nonzero element is a unit (that is, 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}.

Integer ring

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

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

More generally, for the ring of integers R in a number field F, Dirichlet's unit theorem states that R× 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 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. In particular, if R is a domain, 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.

Matrix rings

The unit group of the ring Mn(R) of n × n matrices over a ring R is the group GLn(R) of invertible matrices. For a commutative ring R, an element A of Mn(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 $1+y(1-xy)^{-1}x$ ; 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 RR× is a maximal ideal.

As it turns out, if RR× is an ideal, then it is necessarily a maximal ideal and R is local since a maximal ideal is disjoint from R×.

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

Every ring homomorphism f : RS induces a group homomorphism R×S×, 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.

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 rs. In any ring, pairs of additive inverse elements[b] 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 R× on R via multiplication: Two elements of R are associate if they are in the same R×-orbit.

In an integral domain, the set of associates of a given nonzero element has the same cardinality as 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

1. ^ The notation R×, introduced by André Weil, is commonly used in number theory, where unit groups arise frequently. The symbol × 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.
2. ^ x and x are not necessarily distinct. For example, in the ring of integers modulo 6, one has 3 = −3 even though 1 ≠ −1.

Citations

1. ^
2. ^
3. ^
4. ^ Watkins (2007, Theorem 11.1)
5. ^ Watkins (2007, Theorem 12.1)
6. ^ Jacobson 2009, § 2.2. Exercise 4.
7. ^ 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.