Unit (ring theory)

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 is a unit in and the remaining coefficients a_1, \dots, a_n are nilpotent, i.e., satisfy a_i^N = 0 for some . In particular, if is a domain (or more generally reduced), then the units of are the units of . The units of the power series ring Rx are the power series p(x)=\sum_{i=0}^\infty a_i x^i such that 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)^{-1}x; this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series: (1-yx)^{-1} = \sum_{n \ge 0} (yx)^n = 1 + y \biggl(\sum_{n \ge 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 is a maximal ideal. As it turns out, if is an ideal, then it is necessarily a maximal ideal and is local since a maximal ideal is disjoint from . If is a finite field, then is a cyclic group of order . 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. The group scheme \operatorname{GL}_1 is isomorphic to the multiplicative group scheme \mathbb{G}_m over any base, so for any commutative ring , the groups \operatorname{GL}_1(R) and \mathbb{G}_m(R) are canonically isomorphic to . Note that the functor \mathbb{G}_m (that is, ) is representable in the sense: \mathbb{G}_m(R) \simeq \operatorname{Hom}(\mathbb{Z}[t, t^{-1}], R) for commutative rings (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{Z}[t, t^{-1}] \to R and the set of unit elements of (in contrast, \mathbb{Z}[t] represents the additive group \mathbb{G}_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, since any ring includes the unit . 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 ==