# Unit (ring theory)

In algebra, a unit or invertible element[a] 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.[1][2] The set of units of R forms a group R× under multiplication, called the group of units or unit group of R.[b] 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[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

${\displaystyle \mathbf {Z} ^{n}\times \mu _{R}}$
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[3] 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.[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 ${\displaystyle R[[x]]}$ 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.[5]

### 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 ${\displaystyle 1+y(1-xy)^{-1}x}$;[6] this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series:

${\displaystyle (1-yx)^{-1}=\sum _{n\geq 0}(yx)^{n}=1+y\left(\sum _{n\geq 0}(xy)^{n}\right)x=1+y(1-xy)^{-1}x.}$
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.[7]

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[c] 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.

## Notes

1. ^ 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.
2. ^ 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.
3. ^ 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

1. ^
2. ^
3. ^
4. ^ Watkins (2007, Theorem 11.1)
5. ^ Watkins (2007, Theorem 12.1)
6. ^ Jacobson 2009, § 2.2. Exercise 4.
7. ^ 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. Zbl 1006.00001.