In algebra, a unit or invertible element 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 v u = u v = 1 , {\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. The set of units of R forms a group R× under multiplication, called the group of units or unit group of R. 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 Z n × μ R {\displaystyle \mathbf {Z} ^{n}\times \mu _{R}} where μ R {\displaystyle \mu _{R}} is the (finite, cyclic) group of roots of unity in R and n, the rank of the unit group, is n = r 1 + r 2 − 1 , {\displaystyle n=r_{1}+r_{2}-1,} where r 1 , r 2 {\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 r 1 = 2 , r 2 = 0 {\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 p ( x ) = a 0 + a 1 x + ⋯ + a n x n {\displaystyle p(x)=a_{0}+a_{1}x+\dots +a_{n}x^{n}} such that a0 is a unit in R and the remaining coefficients a 1 , … , a n {\displaystyle a_{1},\dots ,a_{n}} are nilpotent, i.e., satisfy a i N = 0 {\displaystyle a_{i}^{N}=0} for some N. 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 R [ [ x ] ] {\displaystyle R[[x]]} are the power series p ( x ) = ∑ i = 0 ∞ a i x i {\displaystyle p(x)=\sum _{i=0}^{\infty }a_{i}x^{i}} such that a0 is a unit in R.

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 1 − x y {\displaystyle 1-xy} is invertible, then 1 − y x {\displaystyle 1-yx} is invertible with inverse 1 + y ( 1 − x y ) − 1 x {\displaystyle 1+y(1-xy)^{-1}x}; this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series: ( 1 − y x ) − 1 = ∑ n ≥ 0 ( y x ) n = 1 + y ( ∑ n ≥ 0 ( x y ) n ) x = 1 + y ( 1 − x y ) − 1 x . {\displaystyle (1-yx)^{-1}=\sum _{n\geq 0}(yx)^{n}=1+y{\biggl (}\sum _{n\geq 0}(xy)^{n}{\biggr )}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 R is a finite field, then R× is a cyclic group of order |R| − 1.

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.

The group scheme GL 1 {\displaystyle \operatorname {GL} _{1}} is isomorphic to the multiplicative group scheme G m {\displaystyle \mathbb {G} _{m}} over any base, so for any commutative ring R, the groups GL 1 ⁡ ( R ) {\displaystyle \operatorname {GL} _{1}(R)} and G m ( R ) {\displaystyle \mathbb {G} _{m}(R)} are canonically isomorphic to U(R). Note that the functor G m {\displaystyle \mathbb {G} _{m}} (that is, R ↦ U(R)) is representable in the sense: G m ( R ) ≃ Hom ⁡ ( Z [ t , t − 1 ] , R ) {\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 Z [ t , t − 1 ] → R {\displaystyle \mathbb {Z} [t,t^{-1}]\to R} and the set of unit elements of R (in contrast, Z [ t ] {\displaystyle \mathbb {Z} [t]} represents the additive group G a {\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 r ~ s. In any ring, pairs of additive inverse elements 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 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

Notes

Citations

Sources