How many units can a commutative ring have? SunilK. Chebolu and Keir Lockridge 7 1 0 Abstract.La´szlo´ Fuchsposedthefollowingproblemin1960,whichremainsopen:classify 2 the abelian groups occurring as the group of all units in a commutative ring. In this note, n we provide an elementary solution to a simpler, related problem: find all cardinal numbers a occurringasthecardinalityofthegroupofallunitsinacommutativering.Asaby-product, J weobtainasolutiontoFuchs’problemfortheclassoffiniteabelianp-groupswhenpisan 9 oddprime. ] C 1. INTRODUCTION It is well known that a positive integer k is the order of the A multiplicative group of a finite field if and only if k is one less than a prime power. . The correspondingfact for finite commutative rings, however,is not as well known. h Our motivation for studying this—and the question raised in the title—stems from a t a problem posedby La´szlo´ Fuchs in 1960:characterizethe abelian groups that are the m groupofunitsinacommutativering(see[8]).ThoughFuchs’problemremainsopen,it [ hasbeensolvedforvariousspecializedclassesofgroups,wheretheringisnotassumed 1 tobecommutative.Examplesincludecyclicgroups([10]),alternating,symmetricand v finitesimplegroups([4,5]),indecomposableabeliangroups([1]),anddihedralgroups 1 ([2]).InthisnoteweconsideramuchweakerversionofFuchs’problem,determining 4 only the possiblecardinalnumbers|R×|, whereR is a commutativering with group 3 ofunitsR×. 2 0 InapreviousnoteintheMONTHLY([6]),DitorshowedthatafinitegroupGofodd . orderis thegroupofunitsofaringifandonlyifGisisomorphictoadirectproduct 1 ofcyclicgroupsG ,where|G | = 2ni −1forsomepositiveintegern .Thisimplies 0 i i i 7 that an odd positive integer is the number of units in a ring if and only if it is of the 1 form t (2ni −1)forsomepositiveintegersn ,...,n .AsDitormentionedinhis Qi=1 1 t v: paper, this theorem may be derived from the work of Eldridge in [7] in conjunction i with the Feit-Thompson theorem which says that every finite group of odd order is X solvable. However, the purpose of his note was to give an elementary proof of this r resultusingclassicalstructuretheory.Specifically,Ditor’sproofusesthefollowingkey a ingredients:Mashke’stheorem,whichclassifies(forfinitegroups)thegroupalgebras overafieldthataresemisimplerings;theArtin-Wedderburntheorem,whichdescribes the structure of semisimple rings; and Wedderburn’s little theorem, which states that everyfinitedivisionringisafield. Inthis note wegiveanotherelementaryproofofDitor’s theoremforcommutative rings.Wealsoextendthetheoremtoevennumbersandinfinitecardinals,providinga completeanswertothequestionposedinthetitle;seeTheorem8.Ourapproachalso givesanelementarysolutionto Fuchs’problemforfinite abelianp-groupswhenp is anoddprime;seeCorollary4. 2. FINITECARDINALS Webeginwithtwolemmas.Foraprimep,letF denote p thefieldofpelements.Recallthatanyringhomomorphismφ: A −→ B mapsunits tounits,soφinducesagrouphomomorphismφ×: A× −→ B×. 1 Lemma1. Letφ: A −→ Bbeahomomorphismofcommutativerings.Iftheinduced grouphomomorphismφ×: A× −→ B× issurjective,thenthereisaquotientA′ ofA suchthat(A′)× ∼= B×. Proof. By the first isomorphism theorem for rings, Imφ is isomorphic to a quotient of A. It therefore suffices to prove that (Imφ)× = B×. Since ring homomorphisms mapunitstounitsandφ×issurjective,wehaveB× ⊆(Imφ)×.Thereverseinclusion (Imφ)× ⊆ B× holdssince everyunit in the subringImφmustalso be a unitin the ambientringB. Lemma2. LetV andW befinitefieldsofcharacteristic2. 1. ThetensorproductV ⊗F2 W is isomorphicasa ringto afinite directproduct offinitefieldsofcharacteristic2. 2. AsF2-vectorspaces,dim(V ⊗F2 W)= (dimV)(dimW). Proof. Toprove(1),letKandLbefinitefieldsofcharacteristic2.Bytheprimitiveel- ementtheorem,wehaveL ∼=F [x]/(f(x)),wheref(x)isanirreduciblepolynomial 2 inF [x].Thisimpliesthat 2 K[x] K ⊗F2 L ∼= (f(x)). Theirreduciblefactorsoff(x)inK[x]aredistinctsincetheextensionL/F is sep- 2 arable.Nowletf(x) = t f (x)bethefactorizationoff(x)intoits distinctirre- Qi=1 i duciblefactorsinK[x].Wethenhavethefollowingseriesofringisomorphisms: t K[x] K[x] K[x] K ⊗F2 L ∼= (f(x)) ∼= ( t f (x)) ∼= Y (f (x)), Qi=1 i i=1 i wherethelastisomorphismfollowsfromtheChineseremaindertheoremforthering K[x]. Since eachfactorK[x]/(f (x)) is afinite field ofcharacteristic2, we seethat i K ⊗F2 Lisisomorphicasaringtoadirectproductoffinitefieldsofcharacteristic2, asdesired. For (2), simply note that if {v ,...,v } is a basis for V and {w ,...,w } is a 1 k 1 l basisforW,then{vi⊗wj|1 ≤ i ≤ k,1 ≤ j ≤ l}isabasisforV ⊗F2 W. Wemaynowclassifythefiniteabeliangroupsofoddorderthatappearasthegroup ofunitsinacommutativering. Proposition3. LetGbeafiniteabeliangroupofoddorder.ThegroupGisisomorphic tothegroupofunitsinacommutativeringifandonlyifGisisomorphictothegroup of units in a finite direct product of finite fields of characteristic 2. In particular, an oddpositiveintegerk is thenumberofunitsinacommutativeringifandonlyifk is oftheform t (2ni −1)forsomepositiveintegersn ,...,n . Qi=1 1 t Proof. The‘if’directionofthesecondstatementfollowsfrom thefactthat,forrings AandB,(A×B)× ∼= A××B×. For the converse, since the trivial group is the group of units of F , let G be a 2 nontrivial finite abelian group of odd order and let R be a commutative ring with group of units G. Since G has odd order, the unit −1 in R must have order 1. This impliesthatRhascharacteristic2. 2 Nowlet G∼= Cpα11 ×···×Cpαkk denoteadecompositionofGasadirectproductofcyclicgroupsofprimepowerorder (theprimesp arenotnecessarilydistinct).Letg denoteageneratoroftheithfactor. i i DefinearingS by F [x ,...,x ] 2 1 k S = . (xpα11 −1,...,xpαkk −1) 1 k Since R is a commutative ring of characteristic 2, there is a natural ring homomor- phism S −→ R sending x to g for all i. Since the g ’s together generate G, this i i i mapinducesasurjectionS× −→ R×,andhencebyLemma1thereisaquotientofS whosegroupofunitsisisomorphictoG. Sinceanyquotientofafinitedirectproductoffieldsisagainafinitedirectproduct of fields (of possibly fewer factors), the proof will be complete if we can show that S isisomorphicasaringtoafinitedirectproductoffieldsofcharacteristic2. Tosee this,observethatthemap F [x ]/(xpα11 −1)×···×F [x ]/(xpαkk −1) −→ S 2 1 1 2 1 k sendingak-tupletotheproductofitsentriesissurjectiveandF -linearineachfactor; 2 bytheuniversalpropertyofthetensorproduct,itinducesasurjectiveringhomomor- phism F2[x1]/(xp1α11 −1)⊗F2 ···⊗F2 F2[x1]/(xpkαkk −1) −→ S. (†) Thedimensionofthesourceof(†)asanF -vectorspaceis pα1···pαk byLemma2 2 1 k (2);thisisalsothedimensionofthetarget(countmonomialsinthepolynomialringS). Consequently,themap(†)isanisomorphismofrings.Theirreduciblefactorsofeach polynomialxpαii −1aredistinctsincethispolynomialhasnorootsincommonwithits i derivative(p isodd).ThereforebytheChineseremaindertheorem,eachtensorfactor i is a finite direct product of finite fields of characteristic 2. Since the tensor product distributesoverfinite directproducts,wemayuseLemma2(1)toconcludethatS is ringisomorphictoafinitedirectproductoffinitefieldsofcharacteristic2. For any odd prime p, we now characterize the finite abelian p-groups that are re- alizable as the group of units of a commutativering. Recall that a finite p-group is a finite group whose order is a power of p. An elementary abelian finite p-group is a finitegroupthatisisomorphictoafinitedirectproductofcyclicgroupsoforderp. Corollary4. Letpbeanoddprime.Afiniteabelianp-groupGisthegroupofunits of a commutative ring if and only if G is an elementary abelian p-group and p is a Mersenneprime. Proof. The ‘if’ direction follows from the fact if p = 2n −1 is a Mersenne prime, then (F ×···×F )× ∼= C ×···×C . p+1 p+1 p p 3 Fortheotherdirection,letpbeanoddprimeandletGbeafiniteabelianp-group.If Gisthegroupofunitsofcommutativering,thenbyProposition3,G∼= T× whereT isafinitedirectproductoffinitefieldsofcharacteristic2.Consequently, G ∼= C2n1−1×···×C2nt−1. Since each factor must be a p-group, for each i we have 2ni −1 = pzi for some positiveintegerz .Weclaimthatz = 1foralli.Thisfollowsfrom[3,2.3],butsince i i theargumentisshortweincludeithereforconvenience. Assume to the contrary that z > 1 for some i. Consider the equation pzi +1 = i 2ni.Sincep > 1,wehaven ≥ 2andhencepzi ≡ −1 mod 4.Thismeansp ≡ −1 i mod 4andz isodd.Sincez > 1,wehaveanontrivialfactorization i i 2ni =pzi +1= (p+1)(pzi−1−pzi−2+···−p+1), so pzi−1−pzi−2+···−p+1 mustbe even.On the otherhand,since z andp are i bothodd,workingmodulo2weobtain 0 ≡ pzi−1−pzi−2+···−p+1 ≡ z ≡ 1 mod 2, i a contradiction. Hence z = 1 for all i, so p is Mersenne and G an is elementary i abelianp-group. TheabovecorollarydoesnotholdfortheMersenneprimep = 2;forexample,C = 4 F×.Asfarasweknow,Fuchs’problemforfiniteabelian2-groupsremainsopen. 5 We next provide a simple example demonstrating that every even number is the numberofunitsinacommutativering. Proposition5. Everyevennumberisthenumberofunitsinacommutativering. Proof. Letmbeapositiveintegerandconsiderthecommutativering Z[x] R = . 2m (x2,mx) Every element in this ring can be uniquely represented by an element of the form a+bx, where a is an arbitrary integerand 0 ≤ b ≤ m−1. We will now show that a+bxisaunitinthisringifandonlyifaiseither1or−1;thisimpliestheringhas exactly2munits.(Infact,itcanbeshownthatR× ∼= C ×C .) 2m 2 m If a+bx is a unit in R , there there exits an element a′ +b′x such that (a+ 2m bx)(a′ +b′x) = 1inR .Sincex2 = 0inR ,wemusthaveaa′ = 1inZ;i.e.,a 2m 2m is1or−1.Conversely,ifais1or−1,weseethat(a+bx)(a−bx) = 1inR . 2m 3. INFINITECARDINALS Propositions3and5solveourproblemforfinitecardi- nals.Forinfinitecardinals,wewillprovethefollowingproposition. Proposition6. Everyinfinitecardinalisthenumberofunitsinacommutativering. OurproofreliesmainlyontheCantor-Bernsteintheorem: Theorem 7 (Cantor-Bernstein). Let A and B be any two sets. If there exist injec- tive mappings f: A −→ B and g: B −→ A, then there exits a bijective mapping h: A −→ B.Inotherwords,if|A| ≤ |B|and|B|≤ |A|,then|A| = |B|. 4 We also use otherstandard facts from set theory which may be found in [9]. For ex- ample,wemakefrequentuseofthe factthatwheneverαandβ areinfinite cardinals withα ≤ β,thenαβ ≤ β.Recallthatℵ denotesthecardinalityofthesetofnatural 0 numbers. ProofofProposition6. LetλbeaninfinitecardinalandletS beasetwhosecardinal- ityisλ.ConsiderF (S),thefieldofrationalfunctionsintheelementsofS.Weclaim 2 that |F (S)×| = λ. By the Cantor-Bernstein theorem,it suffices to provethat |S| ≤ 2 |F (S)×|and|F (S)×|≤ |S|.SinceS ⊆ F (S)×,itisclearthat|S| ≤ |F (S)×|. 2 2 2 2 Forthereverseinequality,firstobservethatifAisafiniteset,then|F [A]| = ℵ . 2 0 This follows by induction on the size of A, because F [x] is countable and R[x] is 2 countablewheneverRiscountable.Wenowhavethefollowing: |F (S)×| ≤ |F (S)| 2 2 ≤ |F [S]×F [S]| (everyrationalfunctionisaratiooftwopolynomials) 2 2 = |F [S]|2 2 = |F [S]| 2 ≤ X |F2[A]| A⊂S,1≤|A|<ℵ0 = X ℵ0 A⊂S,1≤|A|<ℵ0 ∞ ∞ ≤ X|S|iℵ0 = X|S|ℵ0 = |S|ℵ20 = |S|ℵ0 = |S|. i=1 i=1 CombiningPropositions3,5,and6,weobtainourmainresult. Theorem 8. Let λ be a cardinal number. There exists a commutative ring R with |R×| = λifandonlyifλisequalto 1. anoddnumberoftheform t (2ni −1)forsomepositiveintegersn ,...,n , Qi=1 1 t 2. anevennumber,or 3. aninfinitecardinalnumber. ACKNOWLEDGMENTS. WewouldliketothankGeorgeSeelingerforsimplifyingourpresentationofa ringwithanevennumberofunits.Wealsowouldliketothanktheanonymousrefereesfortheircomments. REFERENCES 1. S.K.Chebolu,K.Lockridge,Fuchs’problemforindecomposableabeliangroups,J.Algebra438(2015) 325–336. 2. ———,Fuchs’problemfordihedralgroups,J.PureAppl.Algebra221no.2(2017)971–982. 3. ———,Fieldswithindecomposablemultiplicativegroups,Expo.Math.34(2016)237–242. 4. C.Davis,T.Occhipinti,Which finitesimple groupsareunit groups? J.Pure Appl.Algebra 218no.4 (2014)743–744. 5. ———,Whichalternatingandsymmetricgroupsareunitgroups?J.AlgebraAppl.13no.3(2014). 6. S.Z.Ditor,Onthegroupofunitsofaring,Amer.Math.Monthly78(1971)522–523. 5 7. K.E.Eldridge,Onringstructuresdeterminedbygroups,Proc.Amer.Math.Soc.23(1969)472–477. 8. L.Fuchs,Abeliangroups.InternationalSeriesofMonographsonPureandAppliedMathematics,Perga- monPress,NewYork-Oxford-London-Paris,1960. 9. P.R.Halmos,Naivesettheory. TheUniversitySeriesinUndergraduateMathematics,D.VanNostrand Co.,Princeton,NJ-Toronto-London-NewYork,1960. 10. K.R.Pearson,J.E.Schneider,Ringswithacyclicgroupofunits,J.Algebra16(1970)243–251. DepartmentofMathematics,IllinoisStateUniversity,Normal,IL61790,USA [email protected] DepartmentofMathematics,GettysburgCollege,Gettysburg,PA17325,USA [email protected] 6