Unit (ring theory)

Template:Short description In algebra, a unit or invertible elementTemplate:Efn of a ring is an invertible element for the multiplication of the ring. That is, an element Template:Mvar of a ring Template:Mvar is a unit if there exists Template:Mvar in Template:Mvar such that $$vu=uv=1,$$ where 1 is the multiplicative identity; the element Template:Mvar is unique for this property and is called the multiplicative inverse of Template:Mvar.Template:SfnTemplate:Sfn The set of units of Template:Mvar forms a group R^× under multiplication, called the group of units or unit group of Template:Mvar.Template:Efn Other notations for the unit group are R^∗, U(R), and E(R) (from the German term ).

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 Template:Mvar is a unit: if rⁿ = 1, then rⁿ⁻¹ is a multiplicative inverse of Template:Mvar. In a nonzero ring, the element 0 is not a unit, so R^× is not closed under addition. A nonzero ring Template:Mvar in which every nonzero element is a unit (that is, R^× = R ∖ Template:Mset) 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 ∖ Template:Mset.

Integer ring

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

In the ring Z/nZ of [[Modular arithmetic#Integers modulo m|integers modulo Template:Mvar]], the units are the congruence classes (mod n) represented by integers coprime to Template:Mvar. They constitute the [[multiplicative group of integers modulo n|multiplicative group of integers modulo Template:Mvar]].

Ring of integers of a number field

In the ring Z[

1. REDIRECT Template:Radic

Template:Rcat shell] obtained by adjoining the quadratic integer

1. REDIRECT Template:Radic

Template:Rcat shell to Z, one has (2 +

1. REDIRECT Template:Radic

Template:Rcat shell)(2 −

1. REDIRECT Template:Radic

Template:Rcat shell) = 1, so 2 +

1. REDIRECT Template:Radic

Template:Rcat shell is a unit, and so are its powers, so Z[

1. REDIRECT Template:Radic

Template:Rcat shell] has infinitely many units.

More generally, for the ring of integers Template:Mvar in a number field Template:Mvar, 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 Template:Mvar and Template:Mvar, 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 Template:Mvar, respectively.

This recovers the Z[

1. REDIRECT Template:Radic

Template:Rcat shell] 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 Template:Mvar, 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 a₀ is a unit in Template:Mvar and the remaining coefficients $a_1,\dots ,a_n$ are nilpotent, i.e., satisfy $a_i^N=0$ for some N.Template:Sfn In particular, if Template:Mvar is a domain (or more generally reduced), then the units of R[x] are the units of Template:Mvar. The units of the power series ring $R[[x]]$ are the power series $$p(x)=\sum _{i=0}^{\mathrm{\infty }}{a}_i{x}^i$$ such that a₀ is a unit in Template:Mvar.Template:Sfn

Matrix rings

The unit group of the ring M_n(R) of n × n matrices over a ring Template:Mvar is the group GL_n(R) of invertible matrices. For a commutative ring Template:Mvar, an element Template:Mvar of M_n(R) is invertible if and only if the determinant of Template:Mvar is invertible in Template:Mvar. In that case, A⁻¹ can be given explicitly in terms of the adjugate matrix.

In general

For elements Template:Mvar and Template:Mvar in a ring Template:Mvar, if $1-xy$ is invertible, then $1-yx$ is invertible with inverse $1+y(1-xy{)}^{-1}x$;Template:Sfn this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series: $$(1-yx{)}^{-1}=\sum _{n\ge 0}(yx{)}^n=1+y(\sum _{n\ge 0}(xy{)}^n)x=1+y(1-xy{)}^{-1}x.$$ 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 Template:Mvar is a finite field, then R^× is a cyclic group of order Template:Abs − 1.

Every ring homomorphism f : R → S induces a group homomorphism R^× → S^×, since Template:Mvar 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.Template:Sfn

The group scheme ${\mathrm{GL}}_1$ is isomorphic to the multiplicative group scheme ${\mathbb{𝔾}}_m$ over any base, so for any commutative ring Template:Mvar, the groups ${\mathrm{GL}}_1(R)$ 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: ${\mathbb{𝔾}}_m(R)\simeq \mathrm{Hom}(\mathbb{\mathbb{Z}}[t,t^{-1}],R)$ for commutative rings Template:Mvar (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 $\mathbb{\mathbb{Z}}[t,t^{-1}]\to R$ and the set of unit elements of Template:Mvar (in contrast, $\mathbb{\mathbb{Z}}[t]$ 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 Template:Mvar is commutative. Elements Template:Mvar and Template:Mvar of Template:Mvar are called Template:Visible anchor if there exists a unit Template:Mvar in Template:Mvar such that r = us; then write r ~ s. In any ring, pairs of additive inverse elementsTemplate:Efn 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 Template:Mvar.

Associatedness can also be described in terms of the action of R^× on Template:Mvar via multiplication: Two elements of Template:Mvar 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 Template:Mvar.

See also

• S-units
• Localization of a ring and a module