Template:Short description Template:Distinguish In algebra, a unit or invertible elementTemplate:Efn of a ring is an invertible element for the multiplication of the ring. That is, an element Template:Mvar of a ring Template:Mvar is a unit if there exists Template:Mvar in Template:Mvar such that <math display="block">vu = uv = 1,</math> where Template:Math is the multiplicative identity; the element Template:Mvar is unique for this property and is called the multiplicative inverse of Template:Mvar.Template:SfnTemplate:Sfn The set of units of Template:Mvar forms a group Template:Math under multiplication, called the group of units or unit group of Template:Mvar.Template:Efn Other notations for the unit group are Template:Math, Template:Math, and Template:Math (from the German term {{#invoke:Lang|lang}}).
Less commonly, the term unit is sometimes used to refer to the element Template:Math of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, Template:Math 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.
ExamplesEdit
Template:AnchorThe multiplicative identity Template:Math and its additive inverse Template:Math are always units. More generally, any root of unity in a ring Template:Mvar is a unit: if Template:Math, then Template:Math is a multiplicative inverse of Template:Mvar. In a nonzero ring, the element 0 is not a unit, so Template:Math is not closed under addition. A nonzero ring Template:Mvar in which every nonzero element is a unit (that is, Template:Math) 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 Template:Math is Template:Math.
Integer ringEdit
In the ring of integers Template:Math, the only units are Template:Math and Template:Math.
In the ring Template:Math of [[Modular arithmetic#Integers modulo m|integers modulo Template:Mvar]], the units are the congruence classes Template:Math represented by integers coprime to Template:Mvar. They constitute the [[multiplicative group of integers modulo n|multiplicative group of integers modulo Template:Mvar]].
Ring of integers of a number fieldEdit
In the ring Template:Math obtained by adjoining the quadratic integer Template:Math to Template:Math, one has Template:Math, so Template:Math is a unit, and so are its powers, so Template:Math has infinitely many units.
More generally, for the ring of integers Template:Mvar in a number field Template:Mvar, Dirichlet's unit theorem states that Template:Math is isomorphic to the group <math display="block">\mathbf Z^n \times \mu_R</math> where <math>\mu_R</math> is the (finite, cyclic) group of roots of unity in Template:Mvar and Template:Mvar, the rank of the unit group, is <math display="block">n = r_1 + r_2 -1, </math> where <math>r_1, r_2</math> are the number of real embeddings and the number of pairs of complex embeddings of Template:Mvar, respectively.
This recovers the Template:Math example: The unit group of (the ring of integers of) a real quadratic field is infinite of rank 1, since <math>r_1=2, r_2=0</math>.
Polynomials and power seriesEdit
For a commutative ring Template:Mvar, the units of the polynomial ring Template:Math are the polynomials <math display="block">p(x) = a_0 + a_1 x + \dots + a_n x^n</math> such that Template:Math is a unit in Template:Mvar and the remaining coefficients <math>a_1, \dots, a_n</math> are nilpotent, i.e., satisfy <math>a_i^N = 0</math> for some Template:Math.Template:Sfn In particular, if Template:Mvar is a domain (or more generally reduced), then the units of Template:Math are the units of Template:Mvar. The units of the power series ring <math>Rx</math> are the power series <math display="block">p(x)=\sum_{i=0}^\infty a_i x^i</math> such that Template:Math is a unit in Template:Mvar.Template:Sfn
Matrix ringsEdit
The unit group of the ring Template:Math of [[square matrix|Template:Math matrices]] over a ring Template:Mvar is the group Template:Math of invertible matrices. For a commutative ring Template:Mvar, an element Template:Mvar of Template:Math is invertible if and only if the determinant of Template:Mvar is invertible in Template:Mvar. In that case, Template:Math can be given explicitly in terms of the adjugate matrix.
In generalEdit
For elements Template:Mvar and Template:Mvar in a ring Template:Mvar, if <math>1 - xy</math> is invertible, then <math>1 - yx</math> is invertible with inverse <math>1 + y(1-xy)^{-1}x</math>;Template:Sfn this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series: <math display="block">(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 unitsEdit
A commutative ring is a local ring if Template:Math is a maximal ideal.
As it turns out, if Template:Math is an ideal, then it is necessarily a maximal ideal and Template:Math is local since a maximal ideal is disjoint from Template:Math.
If Template:Mvar is a finite field, then Template:Math is a cyclic group of order Template:Math.
Every ring homomorphism Template:Math induces a group homomorphism Template:Math, since Template:Mvar 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.Template:Sfn
The group scheme <math>\operatorname{GL}_1</math> is isomorphic to the multiplicative group scheme <math>\mathbb{G}_m</math> over any base, so for any commutative ring Template:Mvar, the groups <math>\operatorname{GL}_1(R)</math> and <math>\mathbb{G}_m(R)</math> are canonically isomorphic to Template:Math. Note that the functor <math>\mathbb{G}_m</math> (that is, Template:Math) is representable in the sense: <math>\mathbb{G}_m(R) \simeq \operatorname{Hom}(\mathbb{Z}[t, t^{-1}], R)</math> for commutative rings Template:Mvar (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>\mathbb{Z}[t, t^{-1}] \to R</math> and the set of unit elements of Template:Mvar (in contrast, <math>\mathbb{Z}[t]</math> represents the additive group <math>\mathbb{G}_a</math>, the forgetful functor from the category of commutative rings to the category of abelian groups).
AssociatednessEdit
Suppose that Template:Mvar is commutative. Elements Template:Mvar and Template:Mvar of Template:Mvar are called Template:Visible anchor if there exists a unit Template:Mvar in Template:Mvar such that Template:Math; then write Template:Math. In any ring, pairs of additive inverse elementsTemplate:Efn Template:Math and Template:Math are associate, since any ring includes the unit Template:Math. For example, 6 and −6 are associate in Template:Math. In general, Template:Math is an equivalence relation on Template:Mvar.
Associatedness can also be described in terms of the action of Template:Math on Template:Mvar via multiplication: Two elements of Template:Mvar are associate if they are in the same Template:Math-orbit.
In an integral domain, the set of associates of a given nonzero element has the same cardinality as Template:Math.
The equivalence relation Template:Math can be viewed as any one of Green's semigroup relations specialized to the multiplicative semigroup of a commutative ring Template:Mvar.