# Unit (ring theory)

Short description: In mathematics, element with a multiplicative inverse

In algebra, a unit or invertible element[lower-alpha 1] 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 $\displaystyle{ vu = uv = 1, }$ 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.[lower-alpha 2] 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 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 $\displaystyle{ \mathbf Z^n \times \mu_R }$ where $\displaystyle{ \mu_R }$ is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group, is $\displaystyle{ n = r_1 + r_2 -1, }$ where $\displaystyle{ r_1, r_2 }$ 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 $\displaystyle{ r_1=2, r_2=0 }$.

### Polynomials and power series

For a commutative ring R, the units of the polynomial ring R[x] are the polynomials $\displaystyle{ p(x) = a_0 + a_1 x + \dots + a_n x^n }$ such that $\displaystyle{ a_0 }$ is a unit in R and the remaining coefficients $\displaystyle{ a_1, \dots, a_n }$ are nilpotent, i.e., satisfy $\displaystyle{ a_i^N = 0 }$ 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 $\displaystyle{ Rx }$ are the power series $\displaystyle{ p(x)=\sum_{i=0}^\infty a_i x^i }$ such that $\displaystyle{ a_0 }$ 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 $\displaystyle{ 1 - xy }$ is invertible, then $\displaystyle{ 1 - yx }$ is invertible with inverse $\displaystyle{ 1 + y(1-xy)^{-1}x }$; this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series: $\displaystyle{ (1-yx)^{-1} = \sum_{n \ge 0} (yx)^n = 1 + y \left(\sum_{n \ge 0} (xy)^n \right)x = 1 + y(1-xy)^{-1}x. }$ 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 $\displaystyle{ |R| - 1 }$.

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 $\displaystyle{ \operatorname{GL}_1 }$ is isomorphic to the multiplicative group scheme $\displaystyle{ \mathbb{G}_m }$ over any base, so for any commutative ring R, the groups $\displaystyle{ \operatorname{GL}_1(R) }$ and $\displaystyle{ \mathbb{G}_m(R) }$ are canonically isomorphic to $\displaystyle{ U(R) }$. Note that the functor $\displaystyle{ \mathbb{G}_m }$ (that is, $\displaystyle{ R \mapsto U(R) }$) is representable in the sense: $\displaystyle{ \mathbb{G}_m(R) \simeq \operatorname{Hom}(\mathbb{Z}[t, t^{-1}], R) }$ 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 $\displaystyle{ \mathbb{Z}[t, t^{-1}] \to R }$ and the set of unit elements of R (in contrast, $\displaystyle{ \mathbb{Z}[t] }$ represents the additive group $\displaystyle{ \mathbb{G}_a }$, 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[lower-alpha 3] 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.