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BINOMIAL RINGS: AXIOMATISATION, TRANSFER, AND CLASSIFICATION 3 1 0 2 n QimhRicheyXantcha* a J 16thJanuary2013 5 1 ] A Attheageoftwenty-onehewroteatreatiseupontheBinomialTheorem,which R hashadaEuropeanvogue. . h t SherlockHolmes’sdescriptionofProfessorMoriarty; a ArthurConanDoyle,TheFinalProblem m [ 3 Argument v 1 AdetailedstudyismadeofBinomialRings,ringswithBinomialCo-eHcients,asin- 3 troducedbyHall.TheyareaxiomatisedandprovedidenticaltotheNumericalRings 9 studiedbyEkedahl. ABinomialTransferPrincipleisestablished,enablingcombin- 1 atorialproofsofalgebraicalidentities. ThefinitelygeneratedBinomialRingsare . completelyclassified,andlike-wisethefinitelygenerated,torsion-freemodules. 4 0 2010MathematicsSubjectClassification. Primary:13F99.Secondary:13F20. 1 1 : The ubiquity and utility of Binomial Co-eHcients presumably require no v detailed explanation. Their abstract study seems to have been initiated by i X Hall,[5],whointroducedtheconceptofBinomialRingsinconnexionwithhis r ground-breakingworkonnilpotentgroups. Thedefinitionissimple: aBino- a mial Ring is a commutative, unital ring R which is torsion-free and closed in QbRunderthe“formationofbinomialco-eHcients”: rpr´1q¨¨¨prń`1q r ÞÑ . n! Ekedahl, [2], preferring the axiomatic approach, proposed six axioms in- tended to capture the properties of Binomial Co-eHcients. He appears not to have been familiar with the work of Hall and never proved the two modi *QimhRicheyXantcha,UppsalaUniversity:[email protected] 1 Xantcha BinomialRings: Axiomatisation,Transfer,andClassification operandi to be equivalent. Rectifying this is one object of the present paper. Indeed,wenotonlyjustify,butimproveuponEkedahl’saxioms,givingexpli- citformulæ(wherenoneweregiven),aswellasdroppingtheghastliestaxiom (thesixth): Theorem3. ThefollowingfiveaxiomscharacterisetheclassofBinomialRings: ˆ ˙ ˆ ˙ˆ ˙ ÿ a`b a b I. “ . n p q p`q“n ˆ ˙ ˆ ˙ ˆ ˙ ˆ ˙ ÿn ÿ ab a b b II. “ ¨¨¨ . n m q q m“0 q1`¨¨¨`qm“n 1 m qiě1 ˆ ˙ˆ ˙ ˆ ˙ˆ ˙ˆ ˙ ÿn a a a m`k n III. “ . m n m`k n k k“0 ˆ ˙ 1 IV. “0 whenně2. n ˆ ˙ ˆ ˙ a a V. “1 and “a. 0 1 Themissingsixthaxiomwillbedulycommentedupon. What is known on Binomial Rings stems principally from rather a re- cent paper [3] by Elliott, which in particular aims to elucidate the connexion between binomial rings and λ-rings. Let us compile a list of their most im- portantproperties. 1. TheFreeBinomialRingonthesetX isthering tf PQrXs|fpZXqĎZu. ofinteger-valuedpolynomialsonX. ([3],Proposition2.1.) 2. Thefollowingconditionsonacommutative,unitalringRareequivalent ([3],Theorem4.1,4.2): A. Risthequotientofabinomialring. B. The elements of R satisfy every integer polynomial congruence uni- versallytruefortheintegers. C. apa´1q¨¨¨pań`1qisdivisiblebyn!foreverynPN. D. Fermat’sLittleTheoremholds: ap ”amodpRforeveryprimep. E. TheFrobeniusmapaÞÑap istheidentityonR{pRforeveryprime p. 2 Xantcha BinomialRings: Axiomatisation,Transfer,andClassification F. R{pR is reduced for every prime p, and the residue field of R{pR is isomorphictoZ{p. These transform into criteria for binomial rings if, in each case, an as- sumptiononlackoftorsionbeadded. 3. The binomial property is preserved under the following constructions: localisation,directandtensorproducts,filteredinductiveandprojective limits. ([3],Propositions5.1,5.4,5.5.) 4. TheinclusionfunctorfromBinomialRingstoRingshasbothaleftand arightadjoint. ([3],Theorems7.1,9.1.) 5. Binomial rings are equivalent to λ-rings with trivial Adams operations. ([7]Proposition1.2,[3]Proposition8.3.) 6. TheBinomialTheorem: LetRbebinomialandletAbeacommutativeal- gebraoverRwhichiscompletewithrespecttotheidealI. Theequation ˆ ˙ ÿ8 r p1`xqr “ xn n n“0 defines an R-module structure on the abelian group p1Ì,¨q. ([3], Pro- position11.1.) As a contribution to the theory, we prove the following TransferPrinciple, formally sanctioning combinatorial proofs of algebraical identities in binomialrings. Itmaybefavourablycomparedtoproperty2Babove. Theorem6: TheBinomialTransferPrinciple. Abinomialpolynomialiden- tityuniversallyvalidinZisvalidineverybinomialring. WealsoprovethefollowingClassificationTheorem. Theorem10:TheStructureTheoremforFinitelyGeneratedBinomialRings. Let R be a finitely generated binomial ring. There exist unique positive, simply compositeintegersm ,...,m suchthat 1 k R–Zrm´1sˆ¨¨¨ˆZrm´1s. 1 k Binomial rings naturally manifest themselves in the theories of integer- valued polynomials, Witt vectors, and λ-rings; to name but a few. We refer the reader to Elliott’s article [3] and the lucid monograph [11] by Yau, where thesetopicshavebeenexpoundedupon. More recently, binomial rings have turned out to form the natural frame- work for discussing polynomial maps and functors of modules; see [8], [9], and[10]. 3 Xantcha BinomialRings: Axiomatisation,Transfer,andClassification §1. DefinitionsandExamples LetusfirststateHall’soriginaldefinition,asfoundin[5]. Definition1([5],Section6). LetRbeacommutativeringwithunity. Itis abinomialringifitistorsion-free1 andclosedinQbRundertheoperations rpr´1q¨¨¨prń`1q r ÞÑ . n! We next present, with minor modifications, Ekedahl’s axioms for numerical rings, with the notable exception of the sixth. The original axioms in [2] wererathernon-explicit,stated,astheywere,intermsofthesamethreemys- teriouspolynomialsoccurringinthetheoryofλ-rings. Ourdefinitionintends toremedythis. Definition2([2],Definition4.1). Anumericalringisacommutativering withunityequippedwithunaryoperations ˆ ˙ r r ÞÑ , nPN; n calledbinomialco-efficientsandsubjecttothefollowingaxioms: ˆ ˙ ˆ ˙ˆ ˙ ÿ a`b a b I. “ . n p q p`q“n ˆ ˙ ˆ ˙ ˆ ˙ ˆ ˙ ÿn ÿ ab a b b II. “ ¨¨¨ . n m q q m“0 q1`¨¨¨`qm“n 1 m qiě1 ˆ ˙ˆ ˙ ˆ ˙ˆ ˙ˆ ˙ ÿn a a a m`k n III. “ . m n m`k n k k“0 ˆ ˙ 1 IV. “0 whenně2. n ˆ ˙ ˆ ˙ a a V. “1 and “a. 0 1 ` ˘ paq Conspicuously absent is a formula for reducing a composition m of bi- n nomial co-eHcients to simple ones, included as Ekedahl’s sixth axiom. Sur- prisingly,suchaformulawillturnouttobeaconsequenceofthefiveaxioms listed. 1Thewordtorsionwill,hereandelsewhere,betakentomeanZ-torsion. 4 Xantcha BinomialRings: Axiomatisation,Transfer,andClassification ` ˘ It follows easily from Axioms I, IV, and V that, when the functions ´ n are evaluated on multiples of unity, we retrieve the ordinary binomial co- eHcients,namely ˆ ˙ m¨1 mpm´1q¨¨¨pmń`1q “ ¨1, mPN. n n! ` ˘ ` ˘ Since n¨1 “ 1, but 0 “ 0 unless n “ 0, a numerical ring has necessarily n n characteristic0. Our present objective will be shewing that numerical and binomial rings co-incide. It follows that the numerical structure on a given ring is always unique. Example1. InanyQ-algebra,binomialco-eHcientsmaybedefinedbythe usualformula: ˆ ˙ r rpr´1q¨¨¨prń`1q “ . n n! (cid:52) Example 2. For any integer m, the ring Zrm´1s is numerical. Since it in- herits the binomial co-eHcients from Q, it is simply a question of verifying closureundertheformationofbinomialco-eHcients. Because ˆ ˙ a fapfa ´1q¨¨¨pfa ´pn´1qq apa´fq¨¨¨pa´pn´1qfq f “ “ , n n! n!fn itwillsuHcetoprovethatwheneverpi |n!,butp(cid:45)b,then pi |pa`bqpa`2bq¨¨¨pa`nbq. Tothisend,let n“cmpm`¨¨¨`c1p`c0, 0ďci ďp´1, bethebaseprepresentationofn. Forfixedkand0ďd ăc ,thenumbers k a`pcmpm`¨¨¨`ck`1pk`1`dpkìqb, 1ďiďpk, (1) willformasetofrepresentativesforthecongruenceclassesmodulopk,aswill ofcoursethenumbers cmpm`¨¨¨`ck`1pk`1`dpkì, 1ďiďpk. (2) Note that if x ” ymodpk and j ď k, then pj | x iv pj | y. Hence there are at leastasmanyfactorspamongthenumbers(1)asamongthenumbers(2). The claimnowfollows. (cid:52) 5 Xantcha BinomialRings: Axiomatisation,Transfer,andClassification Example 3. As the special case m “ 1 of the preceding example, Z itself is numerical. For this ring there is another, more direct, way of proving the numerical axioms. Let us indicate how they may be arrived at as solutions to problemsofEnumerativeCombinatorics. AxiomI. Wehavetwotypesofballs: roundballs,square2 balls. If we have a round balls and b square balls, in how many ways may wechoosenballs? Letpbethenumberofroundballschosen,and qthenumberofsquareballs. AxiomII. Wehaveachocolateboxcontainingarectangularaˆb arrayofpralines,andwewishtoeatnofthese. Inhowmanyways can this be done? Suppose the pralines we choose to feast upon are located in m of the a rows, and let q be the number of chosen i pralinesinrownumberiofthesem. AxiomIII. ThereareaMathematicians,ofwhichmdoGeometry and n Algebra. Naturally, there may exist people who do both or neither. How many distributions of skills are possible? Let k be thenumberofMathematicianswhodoonlyAlgebra. AxiomIV. We are the owner of a single dog. In how many ways canwechoosenofourdogstotakeforawalk? AxiomV. Snuvy the dog has a blankets. In how many ways may hechoose0(inthesummer)or1(inthewinter)ofhisblanketsto keephimwarminbed? (cid:52) ` ˘ Example 4. Being given by rational polynomials, the operations r ÞÑ r n give continuous maps Qp Ñ Qp in the p-adic topology. It should be well known that Z is dense in the ring Zp, and that Zp is closed in Qp. Since the binomialco-eHcientsleaveZinvariant,thesamemustbetrueofZp,whichis thusnumerical. This provides an alternative proof of the fact that Zrm´1s is closed under binomialco-eHcients. Forthisisevidentlytrueofthelocalisations Zppq “QXZp, andthereforealsofor č Zrm´1s“ Z . ppq p(cid:45)m (cid:52) 2ThisisinhonourofDrLars-ChristerBöiersofLund,aneminentteacher,whogaveanex- amplefeaturingroundballsandsquareballsduringhiscourseinDiscreteMathematics. 6 Xantcha BinomialRings: Axiomatisation,Transfer,andClassification §2. ElementaryIdentities Theorem1. Thefollowingformulæarevalidinanynumericalring: ˆ ˙ r rpr´1q¨¨¨prń`1q 1. “ whenr PZ. n n! ˆ ˙ r 2. n! “rpr´1q¨¨¨prń`1q. n ˆ ˙ ˆ ˙ r r 3. n “prń`1q . n n´1 Proof. Themap ˆ ˙ ÿ8 r ϕ: pR,`qÑp1`tRrrtss,¨q, r ÞÑ tn n n“0 is,byAxiomsIandV,agrouphomomorphism. Therefore,whenr PZ, ϕprq“ϕp1qr “p1`tqr, whichexpandsasusual(withordinarybinomialco-eHcients)bytheBinomial Theorem. ThisprovesEquation1. (Aninductiveproofwillalsowork.) ToproveEquations2and3,weproceeddiverently. ByAxiomIII, ˆ ˙ ˆ ˙ˆ ˙ ˆ ˙ˆ ˙ˆ ˙ r r r ÿ1 r n´1`k 1 r “ “ n´1 n´1 1 n´1`k 1 k ˆ ˙ˆ ˙k“ˆ0˙ ˆ ˙ˆ ˙ˆ ˙ r n´1 1 r n 1 “ ` n´1 1 0 n 1 1 ˆ ˙ ˆ ˙ r r “pn´1q `n , n´1 n whichreducestoEquation3. Equation2willthenfollowinductivelyfromEquation3. §3. NumericalversusBinomialRings The crucial step towards shewing the equivalence of binomial and numerical ringsisdemonstratingthelackoftorsioninthelatterclass. ` ˘ Lemma1. Letmbeaninteger. Ifpisprimeandpl |m,butp(cid:45)k,thenpl | m . k Proof. pl dividestheright-handsideof ˆ ˙ ˆ ˙ m m´1 k “m , k k´1 and`th˘ereforealsotheleft-handside. Butpl isrelativelyprimetok,soinfact pl | m . k 7 Xantcha BinomialRings: Axiomatisation,Transfer,andClassification Lemma2. Letm ,...,m benaturalnumbers,andput 1 k m“m `¨¨¨`m . 1 k If n“m `2m `3m `¨¨¨`km 1 2 3 k isprime,then ˆ ˙ m m| , tmu i unlessm “m“n,andallotherm “0. 1 i ř Proof. Letaprimepowerpl |m. Becauseoftherelationn“ mi,then,save i fortheexceptionalcase m “m“p“n 1 givenabove,notallm canbedivisiblebyp. Sayp(cid:45)m;then i j ˆ ˙ ˆ ˙ˆ ˙ m m m´m j “ tmu m tmu i i j i i‰j isdivisiblebypl accordingtoLemma1. Theclaimfollows. Lemma 3. ` C˘onsider a numerical ring R. Let r P R and m,n P N. If nr “ 0, thenalsomn r “0. m Proof. Followsinductively,sinceifnr “0,then ˆ ˙ ˆ ˙ ˆ ˙ r r r mn “npr´m`1q “ńpm´1q . m m´1 m´1 Theorem2. Numericalringsaretorsion-free. Proof. Suppose nr “ 0 in R, and, without any loss of generality, that n is prime. Wecalculateusingthenumericalaxioms: ˆ ˙ ˆ ˙ ˆ ˙ ˆ ˙ ˆ ˙ ÿn ÿ 0 nr r n n 0“ “ “ ¨¨¨ n n m q q m“0 q1`¨¨¨`qm“n 1 m ˆ ˙ qiě1ˆ ˙ ˆ ˙ ÿn r ÿ m ź n mi “ . m ř tmu i m“0 řmi“m i i mii“n Forgivennumbersq,wehaveletm denotethenumberofthesethatareequal i i toi(ofcourseiě1andmi ě0). Conversely,w`hen˘thenumbersmi aregiven, valuesmaybedistributedtothenumbersq in m ways,whichaccountsfor i tmiu themultinomialco-eHcientabove. 8 Xantcha BinomialRings: Axiomatisation,Transfer,andClassification We claim the inner sum` is d˘ivisible by mn when m ě 2. Indeed, when 2 ď m ď n´1, then m | m by Lemma 2. Also, there must exist some tmiu ` ˘ 0 ă j ă n such that m ą 0, and for this j, Lemma 1 asserts n | n mj. In the j j casem`“˘n,obviouslyallmi “0foriě2,andm1 “n,andtheinnersumwill equal n n,whichisdivisiblebyn2 “mn. 1 Wecan`n˘owemployLemma3tokillalltermsexceptm“1. Butthisterm issimply r “r,whichisthenequalto0. 1 Theorem3. NumericalandBinomialRingsco-incide. Proof. Clearly,binomialringssatisfythenumericalaxioms. Conversely, numerical rings are torsion-free, and their binomial co-eH- cientsfulfil ˆ ˙ r n! “rpr´1q¨¨¨prń`1q n byTheorem1,andhence ˆ ˙ r rpr´1q¨¨¨prń`1q “ . n n! The appellations numerical and binomial may thus be treated synonym- ously. Let us now resolve the mystery of the missing sixth axiom. In Z, there “exists”aformulaforiteratedbinomialco-eHcients: ˆ` ˘˙ ˆ ˙ r ÿmn r m “ g , (3) n k k k“1 in the sense that there are unique integers g making the formula valid for k everyr PZ. Golombhasexaminedtheseiteratesinsomedetail,andhispaper [4]isbroughttoanendwiththediscouragingconclusion: No simple reduction formulas have yet been found for the most general ` ˘ pnq caseof b . a Note,however,that(3)isapolynomialidentitywithrationalco-eHcients, by which it must hold in every Q-algebra, and therefore in every binomial ring. ThisprovestheredundancyofEkedahl’soriginalsixthaxiom: Theorem4. Theformula ˆ` ˘˙ ˆ ˙ r ÿmn r m “ g n k k k“1 foriteratedbinomialco-efficientsisvalidineverybinomialring. 9 Xantcha BinomialRings: Axiomatisation,Transfer,andClassification §4. BinomialTransfer Let X be a set, and let EpXq be the termalgebra3 based on X. It consists of all finitewordsthatcanbeformedfromthealphabet " ˆ ˙ˇ * ˇ ´ ˇ XY `,´,¨,0,1, ˇnPN , n ` ˘ wherethesymbolsàndärebinary,ánd ´ areunary,and0and1nullary n (constants). ` ˘ Definition3. TheringZ X istheresultofimposinguponthetermalgebra ´ theaxiomsofacommutativeringwithunity,aswellastheNumericalAxioms. Theorem5. Thereisanisomorphism ˆ ˙ X Z –tf PQrXs|fpZXqĎZu, ´ whichisthusthefreebinomialringonthesetX. (Conferproperty1intheintroduct- orysection.) Proof. The Numerical Axioms, together with t`he˘formula for iterated binomialco-eHcients,willreduceanyelementofZ X toabinomialpolynomial. ´ Conversely, it is well known that any integer-valued polynomial is given by a binomialpolynomial. Theorem6: TheBinomialTransferPrinciple. Abinomialpolynomialiden- tityuniversallyvalidinZisvalidineverybinomialring. Proof. Supposeppx ,...,x q“0isvalidforanyintegervaluesofx ,...,x . By 1 k 1 k theprevioustheorem,thereisacanonicalembedding ˆ ˙ Z x1,...,xk ÑZZk ´ ppx ,...,x qÞÑpppn ,...,n qq . 1 k 1 k pn1,...,nkqPZk ` ˘ View p as an element of Z x1,...,xk . It is the zero binomial map, and therefore ´ alsothezerobinomialpolynomial. §5. BinomialIdealsandFactorRings Let us now make a short survey of binomial ideals and the associated factor rings. 3ThedenominationtermalgebraisborrowedfromUniversalAlgebra;conferDefinitionII.10.4 of[1]. 10