# Unit (ring theory)

__: In mathematics, element with a multiplicative inverse__

**Short description**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
[math]\displaystyle{ vu = uv = 1, }[/math]
where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u.^{[1]}^{[2]} 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 *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**/*n***Z** 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
[math]\displaystyle{ \mathbf Z^n \times \mu_R }[/math]
where [math]\displaystyle{ \mu_R }[/math] is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group, is
[math]\displaystyle{ n = r_1 + r_2 -1, }[/math]
where [math]\displaystyle{ r_1, r_2 }[/math] 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 [math]\displaystyle{ r_1=2, r_2=0 }[/math].

### Polynomials and power series

For a commutative ring R, the units of the polynomial ring *R*[*x*] are the polynomials
[math]\displaystyle{ p(x) = a_0 + a_1 x + \dots + a_n x^n }[/math]
such that [math]\displaystyle{ a_0 }[/math] is a unit in R and the remaining coefficients [math]\displaystyle{ a_1, \dots, a_n }[/math] are nilpotent, i.e., satisfy [math]\displaystyle{ a_i^N = 0 }[/math] for some *N*.^{[4]}
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 [math]\displaystyle{ Rx }[/math] are the power series
[math]\displaystyle{ p(x)=\sum_{i=0}^\infty a_i x^i }[/math]
such that [math]\displaystyle{ a_0 }[/math] is a unit in R.^{[5]}

### 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 [math]\displaystyle{ 1 - xy }[/math] is invertible, then [math]\displaystyle{ 1 - yx }[/math] is invertible with inverse [math]\displaystyle{ 1 + y(1-xy)^{-1}x }[/math];^{[6]} this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:
[math]\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. }[/math]
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 [math]\displaystyle{ |R| - 1 }[/math].

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.^{[7]}

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

## See also

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

## Notes

- ↑ The use of "invertible element" without specifying the operation is not ambiguous in the case of rings, since all elements of a ring are invertible for addition.
- ↑ The notation
*R*^{×}, introduced by André Weil, is commonly used in number theory, where unit groups arise frequently.^{[3]}The symbol × is a reminder that the group operation is multiplication. Also, a superscript × is not frequently used in other contexts, whereas a superscript * often denotes dual. - ↑ x and −
*x*are not necessarily distinct. For example, in the ring of integers modulo 6, one has 3 = −3 even though 1 ≠ −1.

### Citations

- ↑ Dummit & Foote 2004.
- ↑ Lang 2002.
- ↑ Weil 1974.
- ↑ (Watkins 2007)
- ↑ (Watkins 2007)
- ↑ Jacobson 2009, § 2.2. Exercise 4.
- ↑ Exercise 10 in § 2.2. of Cohn, Paul M. (2003).
*Further algebra and applications*(Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. ISBN 1-85233-667-6.

## Sources

- Dummit, David S.; Foote, Richard M. (2004).
*Abstract Algebra*(3rd ed.).*John Wiley & Sons*. ISBN 0-471-43334-9. - Jacobson, Nathan (2009).
*Basic Algebra 1*(2nd ed.). Dover. ISBN 978-0-486-47189-1. - Lang, Serge (2002).
*Algebra*. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X. - Watkins, John J. (2007),
*Topics in commutative ring theory*, Princeton University Press, ISBN 978-0-691-12748-4 - Weil, André (1974).
*Basic number theory*. Grundlehren der mathematischen Wissenschaften.**144**(3rd ed.). Springer-Verlag. ISBN 978-3-540-58655-5.

Original source: https://en.wikipedia.org/wiki/Unit (ring theory).
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