Unit (ring theory)

{{anchor|−1}}The multiplicative identity {{math|1}} and its additive inverse {{math|−1}} are always units. More generally, any root of unity in a ring {{mvar|R}} is a unit: if {{math|1=rⁿ = 1}}, then {{math|1=rⁿ⁻¹}} is a multiplicative inverse of {{mvar|r}}.

In a nonzero ring, the element 0 is not a unit, so {{math|R{{sup|×}}}} is not closed under addition.

A nonzero ring {{mvar|R}} in which every nonzero element is a unit (that is, {{math|1=R{{sup|×}} = R ∖ {{mset|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 {{math|R}} is {{math|R ∖ {{mset|0}}}}.

In the ring of integers {{math|Z}}, the only units are {{math|1}} and {{math|−1}}.

In the ring {{math|Z/nZ}} of Modular arithmetic#Integers modulo m, the units are the congruence classes {{math|(mod n)}} represented by integers coprime to {{mvar|n}}. They constitute the multiplicative group of integers modulo n.

In the ring {{math|Z[{{sqrt|3}}]}} obtained by adjoining the quadratic integer {{math|{{sqrt|3}}}} to {{math|Z}}, one has {{math|1= (2 + {{sqrt|3}})(2 − {{sqrt|3}}) = 1}}, so {{math|2 + {{sqrt|3}}}} is a unit, and so are its powers, so {{math|Z[{{sqrt|3}}]}} has infinitely many units.

More generally, for the ring of integers {{mvar|R}} in a number field {{mvar|F}}, Dirichlet's unit theorem states that {{math|R{{sup|×}}}} is isomorphic to the group

$$\backslash mathbf\; Z^n\; \backslash times\; \backslash mu\_R$$

where $\backslash mu\_R$ is the (finite, cyclic) group of roots of unity in {{mvar|R}} and {{mvar|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 {{mvar|F}}, respectively.

This recovers the {{math|Z[{{sqrt|3}}]}} example: The unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since $r\_1=2,\; r\_2=0$.

For a commutative ring {{mvar|R}}, the units of the polynomial ring {{math|R[x]}} are the polynomials

$$p(x)\; =\; a\_0\; +\; a\_1\; x\; +\; \backslash dots\; +\; a\_n\; x^n$$

such that {{math|a₀}} is a unit in {{mvar|R}} and the remaining coefficients $a\_1,\; \backslash dots,\; a\_n$ are nilpotent, i.e., satisfy $a\_i^N\; =\; 0$ for some {{math|N}}.{{sfn|Watkins|2007|loc=Theorem 11.1|ps=}}

In particular, if {{mvar|R}} is a domain (or more generally reduced), then the units of {{math|R[x]}} are the units of {{mvar|R}}.

The units of the power series ring $Rx$ are the power series

$$p(x)=\backslash sum\_\{i=0\}^\backslash infty\; a\_i\; x^i$$

such that {{math|a₀}} is a unit in {{mvar|R}}.{{sfn|Watkins|2007|loc=Theorem 12.1|ps=}}

The unit group of the ring {{math|M_n(R)}} of square matrix over a ring {{mvar|R}} is the group {{math|GL_n(R)}} of invertible matrices. For a commutative ring {{mvar|R}}, an element {{mvar|A}} of {{math|M_n(R)}} is invertible if and only if the determinant of {{mvar|A}} is invertible in {{mvar|R}}. In that case, {{math|A{{sup|−1}}}} can be given explicitly in terms of the adjugate matrix.

For elements {{mvar|x}} and {{mvar|y}} in a ring {{mvar|R}}, if $1\; -\; xy$ is invertible, then $1\; -\; yx$ is invertible with inverse $1\; +\; y(1-xy)^\{-1\}x$;{{sfn|Jacobson|2009|loc=§2.2 Exercise 4|ps=}} this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:

$$(1-yx)^\{-1\}\; =\; \backslash sum\_\{n\; \backslash ge\; 0\}\; (yx)^n\; =\; 1\; +\; y\; \backslash left(\backslash sum\_\{n\; \backslash ge\; 0\}\; (xy)^n\; \backslash right)x\; =\; 1\; +\; y(1-xy)^\{-1\}x.$$

See Hua's identity for similar results.

A commutative ring is a local ring if {{math|R ∖ R{{sup|×}}}} is a maximal ideal.

As it turns out, if {{math|R ∖ R{{sup|×}}}} is an ideal, then it is necessarily a maximal ideal and {{math|R}} is local since a maximal ideal is disjoint from {{math|R{{sup|×}}}}.

If {{mvar|R}} is a finite field, then {{math|R{{sup|×}}}} is a cyclic group of order {{math|{{abs|R}} − 1}}.

Every ring homomorphism {{math|f : R → S}} induces a group homomorphism {{math|R{{sup|×}} → S{{sup|×}}}}, since {{mvar|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.{{sfn|Cohn|2003|loc=§2.2 Exercise 10|ps=}}

The group scheme $\backslash operatorname\{GL\}\_1$ is isomorphic to the multiplicative group scheme $\backslash mathbb\{G\}\_m$ over any base, so for any commutative ring {{mvar|R}}, the groups $\backslash operatorname\{GL\}\_1(R)$ and $\backslash mathbb\{G\}\_m(R)$ are canonically isomorphic to {{math|U(R)}}. Note that the functor $\backslash mathbb\{G\}\_m$ (that is, {{math|R ↦ U(R)}}) is representable in the sense: $\backslash mathbb\{G\}\_m(R)\; \backslash simeq\; \backslash operatorname\{Hom\}(\backslash mathbb\{Z\}[t,\; t^\{-1\}],\; R)$ for commutative rings {{mvar|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 $\backslash mathbb\{Z\}[t,\; t^\{-1\}]\; \backslash to\; R$ and the set of unit elements of {{mvar|R}} (in contrast, $\backslash mathbb\{Z\}[t]$ represents the additive group $\backslash mathbb\{G\}\_a$, the forgetful functor from the category of commutative rings to the category of abelian groups).

Suppose that {{mvar|R}} is commutative. Elements {{mvar|r}} and {{mvar|s}} of {{mvar|R}} are called {{visible anchor|associate}} if there exists a unit {{mvar|u}} in {{mvar|R}} such that {{math|1=r = us}}; then write {{math|r ~ s}}. In any ring, pairs of additive inverse elements{{efn|{{mvar|x}} and {{math|−x}} are not necessarily distinct. For example, in the ring of integers modulo 6, one has {{math|1=3 = −3}} even though {{math|1 ≠ −1}}.}} {{math|x}} and {{math|−x}} are associate, since any ring includes the unit {{math|−1}}. For example, 6 and −6 are associate in {{math|Z}}. In general, {{math|~}} is an equivalence relation on {{mvar|R}}.

Associatedness can also be described in terms of the action of {{math|R{{sup|×}}}} on {{mvar|R}} via multiplication: Two elements of {{mvar|R}} are associate if they are in the same {{math|R{{sup|×}}}}-orbit.

In an integral domain, the set of associates of a given nonzero element has the same cardinality as {{math|R{{sup|×}}}}.

The equivalence relation {{math|~}} can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring {{mvar|R}}.

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

{{notelist}}

{{reflist|3}}

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{{refend}}

{{DEFAULTSORT:Unit (Ring Theory)}}

Category:1 (number)

Category:Algebraic number theory

Category:Group theory

Category:Ring theory

Category:Algebraic properties of elements