Unit (ring theory)

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 is the multiplicative identity. The set of units of forms a group under multiplication, called the group of units or unit group of . Other notations for the unit group are , , and (from the German term ).

Less commonly, the term ''unit'' is also used to refer to the element 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 "unity" or "identity", and say that is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".

Examples

The multiplicative identity and its additive inverse are always units. More generally, any root of unity in a ring is a unit: if , then is a multiplicative inverse of .
In a nonzero ring, the element 0 is not a unit, so is not closed under addition.
A nonzero ring in which every nonzero element is a unit (that is, ) 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 is .

Integer ring

In the ring of

integers

, the only units are and .

In the ring of integers modulo , the units are the congruence classes represented by integers coprime to . They constitute the multiplicative group of integers modulo .

Ring of integers of a number field

In the ring obtained by adjoining the quadratic integer to , one has , so is a unit, and so are its powers, so has infinitely many units.

More generally, for the ring of integers in a number field , Dirichlet's unit theorem states that is isomorphic to the group
:$\mathbf Z^n \times \mu_R$
where $\mu_R$ is the (finite, cyclic) group of roots of unity in and , 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 , respectively.

This recovers the 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 , the units of the polynomial ring are the polynomials
:$p(x)=a_0 + a_1 x + \dots a_n x^n$
such that $a_0$ is a unit in and the remaining coefficients $a_1, \dots, a_n$ are nilpotent, i.e., satisfy $a_i^N =0$ for some ''N''.
In particular, if is a domain, then the units of are the units of .
The units of the power series ring $R$

x

are the power series
:$p(x)=\sum_^\infty a_i x^i$
such that $a_0$ is a unit in .

Matrix rings

The unit group of the ring of matrices over a ring is the group of invertible matrices. For a commutative ring , an element of is invertible if and only if the

determinant

of is invertible in . In that case, can be given explicitly in terms of the adjugate matrix.

In general

For elements and in a ring , if $1 - xy$ is invertible, then $1 - yx$ is invertible with inverse $1 + y(1-xy)^x$; this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:
:$(1-yx)^ = \sum_ (yx)^n = 1 + y \left(\sum_ (xy)^n \right)x = 1 + y(1-xy)^x.$
See Hua's identity for similar results.

Group of units

A commutative ring is a local ring if is a maximal ideal.

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

If is a finite field, then is a

cyclic group

of order $, R, - 1$.

Every ring homomorphism induces a

group homomorphism

, since 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.Exercise 10 in § 2.2. of

The group scheme $\operatorname_1$ is isomorphic to the multiplicative group scheme $\mathbb_m$ over any base, so for any commutative ring , the groups $\operatorname_1(R)$ and $\mathbb_m(R)$ are canonically isomorphic to $U(R)$. Note that the functor $\mathbb_m$ (that is, $R \mapsto U(R)$) is representable in the sense: \mathbb_m(R) \simeq \operatorname(\mathbb , t^ 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 , t^\to R and the set of unit elements of ''R'' (in contrast, \mathbb /math> represents the additive group $\mathbb_a$, the forgetful functor from the category of commutative rings to the category of abelian groups).

Associatedness

Suppose that is commutative. Elements and of are called ' if there exists a unit in such that ; then write . In any ring, pairs of additive inverse elements and are associate. For example, 6 and −6 are associate in . In general, is an equivalence relation on .

Associatedness can also be described in terms of the action of on via multiplication: Two elements of are associate if they are in the same -orbit.

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

The equivalence relation can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring .

See also

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

Notes

Citations

Sources

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{{DEFAULTSORT:Unit (Ring Theory)
1 (number)
Algebraic number theory
Group theory
Ring theory
Algebraic properties of elements