# Unit (ring theory)

In algebra, a unit or invertible element 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. The set of units of R forms a group R$^×$ under multiplication, called the group of units or unit group of R. 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$^×$ 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[√3] 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[√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$^×$ 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[√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. 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.

### Matrix rings

The unit group of the ring M$_n$(R) of n × 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 $1+y(1-xy)^{-1}x$; 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$^×$ is a maximal ideal.
As it turns out, if R ∖ R$^×$ 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 : R → S 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 ↦ 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 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$^×$ 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.