Unit (ring theory)

Short description: In mathematics, element with a multiplicative inverse

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 [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. 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 [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 a₀ 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. 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 a₀ is a unit in R.

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]; 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 |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 [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 U(R). Note that the functor [math]\displaystyle{ \mathbb{G}_m }[/math] (that is, R ↦ U(R)) 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 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

Sources

• Further algebra and applications (Revised ed. of Algebra, 2nd ed.). London: Springer-Verlag. 2003. ISBN 1-85233-667-6. 
• Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9. 
• Basic Algebra 1 (2nd ed.). Dover. 2009. ISBN 978-0-486-47189-1. 
• Algebra. Graduate Texts in Mathematics. Springer. 2002. ISBN 0-387-95385-X. 
• Watkins, John J. (2007), Topics in commutative ring theory, Princeton University Press, ISBN 978-0-691-12748-4 
• Basic number theory. Grundlehren der mathematischen Wissenschaften. 144 (3rd ed.). Springer-Verlag. 1974. ISBN 978-3-540-58655-5.