Unit (ring theory)
Inalgebra,aunitorinvertible element[a]of aringis aninvertible elementfor the multiplication of the ring. That is, an elementuof a ringRis a unit if there existsvinRsuch that where1is themultiplicative identity;the elementvis unique for this property and is called themultiplicative inverseofu.[1][2]The set of units ofRforms agroupR×under multiplication, called thegroup of unitsorunit groupofR.[b]Other notations for the unit group areR∗,U(R),andE(R)(from the German termEinheit).
Less commonly, the termunitis sometimes used to refer to the element1of the ring, in expressions likering with a unitorunit ring,and alsounit matrix.Because of this ambiguity,1is 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 arng.
Examples
[edit]The multiplicative identity1and its additive inverse−1are always units. More generally, anyroot of unityin a ringRis a unit: ifrn= 1,thenrn−1is a multiplicative inverse ofr. In anonzero ring,theelement 0is not a unit, soR×is not closed under addition. A nonzero ringRin which every nonzero element is a unit (that is,R×=R∖ {0}) is called adivision ring(or a skew-field). A commutative division ring is called afield.For example, the unit group of the field ofreal numbersRisR∖ {0}.
Integer ring
[edit]In the ring ofintegersZ,the only units are1and−1.
In the ringZ/nZofintegers modulon,the units are the congruence classes(modn)represented by integerscoprimeton.They constitute themultiplicative group of integers modulon.
Ring of integers of a number field
[edit]In the ringZ[√3]obtained by adjoining thequadratic integer√3toZ,one has(2 +√3)(2 −√3) = 1,so2 +√3is a unit, and so are its powers, soZ[√3]has infinitely many units.
More generally, for thering of integersRin anumber fieldF,Dirichlet's unit theoremstates thatR×is isomorphic to the group whereis the (finite, cyclic) group of roots of unity inRandn,therankof the unit group, is whereare the number of real embeddings and the number of pairs of complex embeddings ofF,respectively.
This recovers theZ[√3]example: The unit group of (the ring of integers of) areal quadratic fieldis infinite of rank 1, since.
Polynomials and power series
[edit]For a commutative ringR,the units of thepolynomial ringR[x]are the polynomials such thata0is a unit inRand the remaining coefficientsarenilpotent,i.e., satisfyfor someN.[4] In particular, ifRis adomain(or more generallyreduced), then the units ofR[x]are the units ofR. The units of thepower series ringare the power series such thata0is a unit inR.[5]
Matrix rings
[edit]The unit group of the ringMn(R)ofn × nmatricesover a ringRis the groupGLn(R)ofinvertible matrices.For a commutative ringR,an elementAofMn(R)is invertible if and only if thedeterminantofAis invertible inR.In that case,A−1can be given explicitly in terms of theadjugate matrix.
In general
[edit]For elementsxandyin a ringR,ifis invertible, thenis invertible with inverse;[6]this formula can be guessed, but not proved, by the following calculation in a ring of noncommutative power series: SeeHua's identityfor similar results.
Group of units
[edit]Acommutative ringis alocal ringifR∖R×is amaximal ideal.
As it turns out, ifR∖R×is an ideal, then it is necessarily amaximal idealandRislocalsince amaximal idealis disjoint fromR×.
IfRis afinite field,thenR×is acyclic groupof order|R| − 1.
Everyring homomorphismf:R→Sinduces agroup homomorphismR×→S×,sincefmaps units to units. In fact, the formation of the unit group defines afunctorfrom thecategory of ringsto thecategory of groups.This functor has aleft adjointwhich is the integralgroup ringconstruction.[7]
Thegroup schemeis isomorphic to themultiplicative group schemeover any base, so for any commutative ringR,the groupsandare canonically isomorphic toU(R).Note that the functor(that is,R↦U(R)) is representable in the sense:for commutative ringsR(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 homomorphismsand the set of unit elements ofR(in contrast,represents the additive group,the forgetful functor from the category of commutative rings to the category of abelian groups).
Associatedness
[edit]Suppose thatRis commutative. ElementsrandsofRare calledassociateif there exists a unituinRsuch thatr=us;then writer~s.In any ring, pairs ofadditive inverseelements[c]xand−xareassociate,since any ring includes the unit−1.For example, 6 and −6 are associate inZ.In general,~is anequivalence relationonR.
Associatedness can also be described in terms of theactionofR×onRvia multiplication: Two elements ofRare associate if they are in the sameR×-orbit.
In anintegral domain,the set of associates of a given nonzero element has the samecardinalityasR×.
The equivalence relation~can be viewed as any one ofGreen's semigroup relationsspecialized to the multiplicativesemigroupof a commutative ringR.
See also
[edit]Notes
[edit]- ^In the case of rings, the use of "invertible element" is taken as self-evidently referring to multiplication, since all elements of a ring are invertible for addition.
- ^The notationR×,introduced byAndré Weil,is commonly used innumber 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.
- ^xand−xare not necessarily distinct. For example, in the ring of integers modulo 6, one has3 = −3even though1 ≠ −1.
Citations
[edit]- ^Dummit & Foote 2004
- ^Lang 2002
- ^Weil 1974
- ^Watkins 2007,Theorem 11.1
- ^Watkins 2007,Theorem 12.1
- ^Jacobson 2009,§2.2 Exercise 4
- ^Cohn 2003,§2.2 Exercise 10
Sources
[edit]- Cohn, Paul M.(2003).Further algebra and applications(Revised ed. of Algebra, 2nd ed.). London:Springer-Verlag.ISBN1-85233-667-6.Zbl1006.00001.
- Dummit, David S.; Foote, Richard M. (2004).Abstract Algebra(3rd ed.).John Wiley & Sons.ISBN0-471-43334-9.
- Jacobson, Nathan(2009).Basic Algebra 1(2nd ed.). Dover.ISBN978-0-486-47189-1.
- Lang, Serge(2002).Algebra.Graduate Texts in Mathematics.Springer.ISBN0-387-95385-X.
- Watkins, John J. (2007),Topics in commutative ring theory,Princeton University Press,ISBN978-0-691-12748-4,MR2330411
- Weil, André(1974).Basic number theory.Grundlehren der mathematischen Wissenschaften. Vol. 144 (3rd ed.).Springer-Verlag.ISBN978-3-540-58655-5.