Commutative Algebra AndreasGathmann ClassNotesTUKaiserslautern2013/14 Contents 0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2. PrimeandMaximalIdeals. . . . . . . . . . . . . . . . . . . . . 18 3. Modules . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4. ExactSequences . . . . . . . . . . . . . . . . . . . . . . . . 36 5. TensorProducts . . . . . . . . . . . . . . . . . . . . . . . . 43 6. Localization . . . . . . . . . . . . . . . . . . . . . . . . . 52 7. ChainConditions. . . . . . . . . . . . . . . . . . . . . . . . 62 8. PrimeFactorizationandPrimaryDecompositions . . . . . . . . . . . . . 70 9. IntegralRingExtensions . . . . . . . . . . . . . . . . . . . . . 80 10. NoetherNormalizationandHilbert’sNullstellensatz . . . . . . . . . . . . 91 11. Dimension. . . . . . . . . . . . . . . . . . . . . . . . . . 96 12. ValuationRings . . . . . . . . . . . . . . . . . . . . . . . . 109 13. DedekindDomains . . . . . . . . . . . . . . . . . . . . . . . 117 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 0. Introduction 3 0. Introduction Commutative algebra is the study of commutative rings. In this class we will assume the basics of ring theory that you already know from earlier courses (e.g. ideals, quotient rings, the homo- morphism theorem, and unique prime factorization in principal ideal domains such as the integers or polynomial rings in one variable over a field), and move on to more advanced topics, some of whichwillbesketchedinRemark0.14below. ForreferencestoearlierresultsIwillusuallyusemy German notes for the “Algebraic Structures” and occasionally the “Foundations of Mathematics” and “Introduction to Algebra” classes [G1, G2, G3], but if you prefer English references you will certainlyhavenoproblemstofindtheminalmostanytextbookonabstractalgebra. You will probably wonder why the single algebraic structure of commutative rings deserves a full one-semester course for its study. The main motivation for this is its many applications in both algebraicgeometry and(algebraic)numbertheory. Especiallytheconnectionbetweencommutative algebraandalgebraicgeometryisverydeep—infact,toacertainextentonecansaythatthesetwo fieldsofmathematicsareessentiallythesamething,justexpressedindifferentlanguages. Although somealgebraicconstructionsandresultsinthisclassmayseemabitabstract,mostofthemhavean easy(andsometimessurprising)translationintermsofgeometry,andknowingaboutthisoftenhelps tounderstandandrememberwhatisgoingon. Forexample,wewillseethattheChineseRemainder Theorem that you already know [G1, Proposition 11.21] (and that we will extend to more general ringsthantheintegersinProposition1.14)canbetranslatedintotheseeminglyobviousgeometric statementthat“givingafunctiononadisconnectedspaceisthesameasgivingafunctiononeach ofitsconnectedcomponents”(seeExample1.15(b)). However,asthisisnotageometryclass,wewilloftenonlysketchthecorrespondencebetweenal- gebraandgeometry,andwewillneveractuallyusealgebraicgeometrytoproveanything. Although our“CommutativeAlgebra”and“AlgebraicGeometry”classesaredeeplylinked,theyaredeliber- atelydesignedsothatnoneofthemneedstheotherasaprerequisite. ButIwillalwaystrytogive youenoughexamplesandbackgroundtounderstandthegeometricmeaningofwhatwedo,incase youhavenotattendedthe“AlgebraicGeometry”classyet. So let us explain in this introductory chapter how algebra enters the field of geometry. For this wehavetointroducethemainobjectsofstudyinalgebraicgeometry: solutionsetsofpolynomial equationsoversomefield,theso-calledvarieties. Convention0.1(Ringsandfields). Inourwholecourse,aring Risalwaysmeanttobeacommu- tative ring with 1 [G1, Definition 7.1]. We do not require that this multiplicative unit 1 is distinct fromtheadditiveneutralelement0,butif1=0thenRmustbethezeroring[G1,Lemma7.5(c)]. Subringsmusthavethesameunitastheambientring,andringhomomorphismsarealwaysrequired to map 1 to 1. Of course, a ring R(cid:54)={0} is a field if and only if every non-zero element has a multiplicativeinverse. Definition 0.2 (Polynomial rings). Let R be a ring, and let n∈N . A polynomial over R in n >0 variablesisaformalexpressionoftheform f = ∑ a xi1· ··· ·xin, i1,...,in 1 n i1,...,in∈N withcoefficientsa ∈Randformalvariablesx=(x ,...,x ),suchthatonlyfinitelymanyofthe i1,...,in 1 n coefficientsarenon-zero(see[G1,Chapter9]howthisconceptof“formalvariables”canbedefined inamathematicallyrigorousway). Polynomialscanbeaddedandmultipliedintheobviousway,andformaringwiththeseoperations. WecallitthepolynomialringoverRinnvariablesanddenoteitbyR[x ,...,x ]. 1 n 4 AndreasGathmann Definition0.3(Varieties). LetK beafield,andletn∈N. (a) Wecall An :={(c ,...,c ):c ∈K fori=1,...,n} K 1 n i theaffinen-spaceoverK. IfthefieldK isclearfromthecontext,wewillwriteAn alsoas K An. Note that An is just Kn as a set. It is customary to use two different notations here since K Kn is also a K-vector space and a ring. We will usually use the notation An if we want to K ignoretheseadditionalstructures:forexample,additionandscalarmultiplicationaredefined on Kn, but not on An. The affine space An will be the ambient space for our zero loci of K K polynomialsbelow. (b) Forapolynomial f ∈K[x ,...,x ]asaboveandapointc=(c ,...,c )∈An wedefinethe 1 n 1 n K valueof f atctobe f(c)= ∑ a ci1· ··· ·cin ∈K. i1,...,in 1 n i1,...,in∈N If there is no risk of confusion we will sometimes denote a point in An by the same letter K xasweusedfortheformalvariables,writing f ∈K[x ,...,x ]forthepolynomialand f(x) 1 n foritsvalueatapointx∈An. K (c) LetS⊂K[x ,...,x ]beasetofpolynomials. Then 1 n V(S):={x∈An : f(x)=0forall f ∈S} ⊂An K K is called the zero locus of S. Subsets of An of this form are called (affine) varieties. If K S=(f ,...,f )isafiniteset,wewillwriteV(S)=V({f ,...,f })alsoasV(f ,...,f ). 1 k 1 k 1 k Example 0.4. Varieties, say over the field R of real numbers, can have many different “shapes”. ThefollowingpictureshowsafewexamplesinA2 andA3. R R (a) V(x2+x2−1)⊂A2 (b) V(x2−x3)⊂A2 (c) V(x3−x )⊂A2 1 2 2 1 1 1 (d) V(x6+x6+x6−1)⊂A3 (e) V(x x ,x x )⊂A3 (f) V(x2+x3−x4−x2x2)⊂A3 1 2 3 1 3 2 3 2 3 3 1 3 Ofcourse,theemptyset0/ andallofAnarealsovarietiesinAn,since0/ =V(1)andAn=V(0). It is the goal of algebraic geometry to find out the geometric properties of varieties by looking at the corresponding polynomials from an algebraic point of view (as opposed to an analytical or numericalapproach). However,itturnsoutthatitisnotaverygoodideatojustlookatthedefining polynomials given initially — simply because they are not unique. For example, the variety (a) abovewasgivenasthezerolocusofthepolynomialx2+x2−1,butitisequallywellthezerolocus 1 2 of (x2+x2−1)2, orof the twopolynomials (x −1)(x2+x2−1)and x (x2+x2−1). In orderto 1 2 1 1 2 2 1 2 0. Introduction 5 removethisambiguity,itisthereforeusefultoconsiderall polynomialsvanishingonX atonce. Let usintroducethisconceptnow. Construction0.5(Ringsandidealsassociatedtovarieties). ForavarietyX ⊂An (andinfactalso K foranysubsetX ofAn)weconsidertheset K I(X):={f ∈K[x ,...,x ]: f(x)=0forallx∈X} 1 n of all polynomials vanishing on X. Note that this is an ideal of K[x ,...,x ] (which we write as 1 n I(X)(cid:69)K[x ,...,x ]): itisclearthat0∈I(X),andiftwopolynomials f andgvanishonX,thenso 1 n do f+gand f·hforanypolynomialh. WecallI(X)theidealofX. Withthisidealwecanconstructthequotientring A(X):=K[x ,...,x ]/I(X) 1 n inwhichweidentifytwopolynomials f,g∈K[x ,...,x ]ifandonlyif f−gisthezerofunctionon 1 n X,i.e.if f andghavethesamevalueateverypointx∈X.Soonemaythinkofanelement f ∈A(X) asbeingthesameasafunction X →K, x(cid:55)→ f(x) that can be given by a polynomial. We therefore call A(X) the ring of polynomial functions or coordinate ring of X. Often we will simply say that A(X) is the ring of functions on X since functions in algebra are always given by polynomials. Moreover, the class of a polynomial f ∈ K[x ,...,x ]insucharingwillusuallyalsobewrittenas f ∈A(X), droppingtheexplicitnotation 1 n forequivalenceclassesifitisclearfromthecontextthatwearetalkingaboutelementsinthequotient ring. Remark 0.6 (Polynomials and polynomial functions). You probably know that over some fields thereisasubtledifferencebetweenpolynomials andpolynomialfunctions: e.g.overthefieldK= Z the polynomial f =x2+x∈K[x] is certainly non-zero, but it defines the zero function on A1 2 K [G1,Example9.13]. InourcurrentnotationthismeansthattheidealI(A1)offunctionsvanishing K at every point of A1 is non-trivial, in fact that I(A1)=(x2+x), and that consequently the ring K K A(A1)=K[x]/(x2+x)ofpolynomialfunctionsonA1 isnotthesameasthepolynomialringK[x]. K K Inthisclasswewillskipoverthisproblementirely,sinceourmaingeometricintuitioncomesfrom thefieldsofrealorcomplexnumberswherethereisnodifferencebetweenpolynomialsandpolyno- mialfunctions.Wewillthereforeusuallyassumesilentlythatthereisnopolynomial f ∈K[x ,...,x ] 1 n vanishingonallofAn,i.e.thatI(An)=(0)andthusA(An)=K[x ,...,x ]. K K K 1 n Example0.7(Idealofapoint). Leta=(a ,...,a )∈An beapoint. WeclaimthatitsidealI(a):= 1 n K I({a})(cid:69)K[x ,...,x ]is 1 n I(a)=(x −a ,...,x −a ). 1 1 n n Infact,thisiseasytosee: “⊂” If f ∈I(a)then f(a)=0. Thismeansthatreplacingeachx bya in f giveszero,i.e.that f i i iszeromodulo(x −a ,...,x −a ). Hence f ∈(x −a ,...,x −a ). 1 1 n n 1 1 n n “⊃” If f ∈(x1−a1,...,xn−an)then f =∑ni=1(xi−ai)fiforsome f1,...,fn∈K[x1,...,xn],and socertainly f(a)=0,i.e. f ∈I(a). Construction 0.8 (Subvarieties). The ideals of varieties defined in Construction 0.5 all lie in the polynomial ring K[x ,...,x ]. In order to get a geometric interpretation of ideals in more general 1 n ringsitisusefultoconsiderarelativesituation: letX ⊂An beafixedvariety. Thenforanysubset K S⊂A(X)ofpolynomialfunctionsonX wecanconsideritszerolocus V (S)={x∈X : f(x)=0forall f ∈S} ⊂X X justasinDefinition0.3(c),andforanysubsetY ⊂X asinConstruction0.5theideal I (Y)={f ∈A(X): f(x)=0forallx∈Y} (cid:69)A(X) X of all functions on X that vanish onY. It is clear that the sets of the formV (S) are exactly the X varietiesinAn containedinX,theso-calledsubvarietiesofX. K 6 AndreasGathmann IfthereisnoriskofconfusionwewillsimplywriteV(S)andI(Y)againinsteadofV (S)andI (Y). X X So in this notation we have now assigned to every variety X a ring A(X) of polynomial functions on X, and to every subvarietyY ⊂X an ideal I(Y)(cid:69)A(X) of the functions that vanish onY. This assignmentofanidealtoasubvarietyhassomenicefeatures: Lemma0.9. LetX beavarietywithcoordinateringA(X). Moreover,letY andY(cid:48) besubsetsofX, andletSandS(cid:48)besubsetsofA(X). (a) IfY ⊂Y(cid:48)thenI(Y(cid:48))⊂I(Y)inA(X);ifS⊂S(cid:48)thenV(S(cid:48))⊂V(S)inX. (b) Y ⊂V(I(Y))andS⊂I(V(S)). (c) IfY isasubvarietyofX thenY =V(I(Y)). (d) IfY isasubvarietyofX thenA(X)/I(Y)∼=A(Y). Proof. (a) AssumethatY ⊂Y(cid:48). If f ∈I(Y(cid:48))then f vanishesonY(cid:48),hencealsoonY,whichmeansthat f ∈I(Y). Thesecondstatementfollowsinasimilarway. (b) Let x∈Y. Then f(x)=0 for every f ∈I(Y) by definition of I(Y). But this implies that x∈V(I(Y)). Again,thesecondstatementfollowsanalogously. (c) By(b)itsufficestoprove“⊃”. AsY isasubvarietyofX wecanwriteY =V(S)forsome S⊂A(X). ThenS⊂I(V(S))by(b),andthusV(S)⊃V(I(V(S)))by(a). ReplacingV(S)by Y nowgivestherequiredinclusion. (d) TheringhomomorphismA(X)→A(Y)thatrestrictsapolynomialfunctiononXtoafunction Y issurjectiveandhaskernelI(Y)bydefinition. Sotheresultfollowsfromthehomomor- phismtheorem[G1,Proposition8.12]. (cid:3) Remark0.10(Reconstructionofgeometryfromalgebra). LetY beasubvarietyofX. ThenLemma 0.9(c)saysthatI(Y)determinesY uniquely. Similarly,knowingtheringsA(X)andA(Y),together withtheringhomomorphismA(X)→A(Y)thatdescribestherestrictionoffunctionsonX tofunc- tionsonY,isenoughtorecoverI(Y)asthekernelofthismap,andthusY asasubvarietyofX by theabove. Inotherwords,wedonotloseanyinformationifwepassfromgeometrytoalgebraand describevarietiesandtheirsubvarietiesbytheircoordinateringsandideals. ThismapA(X)→A(Y)correspondingtotherestrictionoffunctionstoasubvarietyisalreadyafirst specialcaseofaringhomomorphismassociatedtoa“morphismofvarieties”. Letusnowintroduce thisnotion. Construction 0.11 (Morphisms of varieties). Let X ⊂An andY ⊂Am be two varieties over the K K same ground field. Then a morphism from X toY is just a set-theoretic map f :X →Y that can begivenbypolynomials,i.e.suchthattherearepolynomials f ,...,f ∈K[x ,...,x ]with f(x)= 1 m 1 n (f (x),...,f (x))∈Y forallx∈X. Tosuchamorphismwecanassignaringhomomorphism 1 m ϕ :A(Y)→A(X), g(cid:55)→g◦f =g(f ,...,f ) 1 m givenbycomposingapolynomialfunctiononY with f toobtainapolynomialfunctiononX. Note thatthisringhomomorphismϕ ... (a) reversestherolesofsourceandtargetcomparedtotheoriginalmap f :X →Y;and (b) isenoughtorecover f,since f =ϕ(y)∈A(X)ify ,...,y denotethecoordinatesofAm. i i 1 m K Example 0.12. Let X =A1 (with coordinate x) andY =A2 (with coordinates y and y ), so that R R 1 2 A(X)=R[x]andA(Y)=R[y ,y ]byRemark0.6. Considerthemorphismofvarieties 1 2 f :X →Y, x(cid:55)→(y ,y ):=(x,x2) 1 2 0. Introduction 7 whoseimageisobviouslythestandardparabolaZ=V(y −y2)shownin 2 1 thepictureontheright. ThentheassociatedringhomomorphismA(Y)= x R[y ,y ]→R[x]=A(X) of Construction 0.11 is given by composing a 1 2 polynomialfunctioniny andy with f,i.e.byplugginginxandx2 for f 1 2 y andy ,respectively: 1 2 R[y ,y ]→R[x], g(cid:55)→g(x,x2). y 1 2 2 Notethatwiththeimagesofg=y andg=y underthishomomorphism 1 2 wejustrecoverthepolynomialsxandx2definingthemap f. Z If we had considered f as a morphism from X to Z (i.e. restricted the target space to the actual image of f) we would have obtained A(Z)= y1 K[y ,y ]/(y −y2)andthustheringhomomorphism 1 2 2 1 R[y ,y ]/(y −y2)→R[x], g(cid:55)→g(x,x2) 1 2 2 1 instead(whichisobviouslywell-defined). Remark0.13 (Correspondencebetweengeometry andalgebra). Summarizingwhatwe haveseen sofar,wegetthefollowingfirstversionofadictionarybetweengeometryandalgebra: GEOMETRY −→ ALGEBRA variety X ring A(X)of(polynomial)functionsonX subvarietyY ofX ideal I(Y)(cid:69)A(X)offunctionsonX vanishingonY morphism f :X →Y ofvarieties ringhomomorphism A(Y)→A(X), g(cid:55)→g◦f Moreover,passingfromidealstosubvarietiesreversesinclusionsasinLemma0.9(a),andwehave A(X)/I(Y)∼=A(Y) for any subvarietyY of X by Lemma 0.9 (d) (with the isomorphism given by restrictingfunctionsfromX toY). Wehavealsoseenalreadythatthisassignmentofalgebraictogeometricobjectsisinjectiveinthe sense of Remark 0.10 and Construction 0.11 (b). However, not all rings, ideals, and ring homo- morphismsarisefromthiscorrespondencewithgeometry,aswewillseeinRemark1.10,Example 1.25(b), andRemark1.31. Soalthoughthegeometricpictureisveryusefultovisualizealgebraic statements,itcanusuallynotbeusedtoactuallyprovetheminthecaseofgeneralrings. Remark 0.14 (Outline of this class). In order to get an idea of the sort of problems considered in commutativealgebra,letusquicklylistsomeofthemaintopicsthatwewilldiscussinthisclass. • Modules. From linear algebra you know that one of the most important structures related to a field K is that of a vector space over K. If we write down the same axioms as for a vectorspacebutrelaxtheconditiononK toallowanarbitraryring,weobtainthealgebraic structureofamodule,whichisequallyimportantincommutativealgebraasthatofavector spaceinlinearalgebra. WewillstudythisinChapter3. • Localization. IfwehavearingRthatisnotafield,animportantconstructiondiscussedin Chapter6istomakemoreelementsinvertiblebyallowing“fractions”—inthesameway asonecanconstructtherationalnumbersQfromtheintegersZ. Geometrically,wewillsee thatthisprocesscorrespondstostudyingavarietylocallyaroundapoint,whichiswhyitis called“localization”. • Decompositionintoprimes. InaprincipalidealdomainRliketheintegersorapolynomial ringinonevariableoverafield,animportantalgebraictoolistheuniqueprimefactorization ofelementsofR[G1,Proposition11.9]. WewillextendthisconceptinChapter8tomore general rings, and also to a “decomposition of ideals into primes”. In terms of geometry, this corresponds to a decomposition of a variety into pieces that cannot be subdivided any further—e.g.writingthevarietyinExample0.4(e)asaunionofalineandaplane. 8 AndreasGathmann • Dimension. LookingatExample0.4againitseemsobviousthatweshouldbeabletoassign adimensiontoeachvarietyX.Wewilldothisbyassigningadimensiontoeachcommutative ring so that the dimension of the coordinate ring A(X) can be interpreted as the geometric dimension of X (see Chapter 11). With this definition of dimension we can then prove its expected properties, e.g. that cutting down a variety by n more equations reduces its dimensionbyatmostn(Remark11.18). • Ordersofvanishing. Forapolynomial f ∈K[x]inonevariableyouallknowwhatitmeans that it has a zero of a certain order at a point. If we now have a different variety, say still locallydiffeomorphictoalinesuchase.g.thecircleX =V(x2+x2−1)⊂A2 inExample 1 2 R 0.4(a),itseemsgeometricallyreasonablethatweshouldstillbeabletodefinesuchvanishing orders of functions on X at a given point. This is in fact possible, but algebraically more complicated—wewilldothisinChapter12andstudytheconsequencesinChapter13. Butbeforewecandiscussthesemaintopicsoftheclasswehavetostartnowbydevelopingmore toolstoworkwithidealsthanwhatyouknowfromearlierclasses. Exercise0.15. ShowthatthefollowingsubsetsX ofAn arenot varietiesoverK: K (a) X =Z⊂A1; R (b) X =A1\{0}⊂A1; R R (c) X ={(x ,x )∈A2 :x =sin(x )}⊂A2; 1 2 R 2 1 R (d) X ={x∈A1 :|x|=1}⊂A1; C C (e) X = f(Y)⊂An foranarbitraryvarietyY andamorphismofvarieties f :Y →X overR. R Exercise 0.16 (Degree of polynomials). Let R be a ring. Recall that an element a∈R is called a zero-divisor ifthereexistsanelementb(cid:54)=0withab=0[G1,Definition7.6(c)],andthatRiscalled an(integral)domain ifnonon-zeroelementisazero-divisor,i.e.ifab=0fora,b∈Rimpliesa=0 orb=0[G1,Definition10.1]. Wedefinethedegree ofanon-zeropolynomial f =∑i1,...,inai1,...,inx1i1· ··· ·xnin ∈R[x1,...,xn]tobe degf :=max{i +···+i :a (cid:54)=0}. 1 n i1,...,in Moreover,thedegreeofthezeropolynomialisformallysetto−∞. Showthat: (a) deg(f·g)≤degf+deggforall f,g∈R[x ,...,x ]. 1 n (b) Equalityholdsin(a)forallpolynomials f andgifandonlyifRisanintegraldomain. 1. Ideals 9 1. Ideals Fromthe“AlgebraicStructures”classyoualreadyknowthebasicconstructionsandpropertiescon- cerningidealsandtheirquotientrings[G1,Chapter8]. Forourpurposeshoweverwehavetostudy idealsinmuchmoredetail—sothiswillbeourgoalforthisandthenextchapter. Letusstartwith somegeneralconstructionstoobtainnewidealsfromoldones. TheidealgeneratedbyasubsetM ofaring[G1,Definition8.5]willbewrittenas(M). Construction1.1(Operationsonideals). LetIandJbeidealsinaringR. (a) Thesumofthetwogivenidealsisdefinedasusualby I+J:={a+b:a∈Iandb∈J}. Itiseasytocheckthatthisisanideal—infact,itisjusttheidealgeneratedbyI∪J. (b) ItisalsoobviousthattheintersectionI∩JisagainanidealofR. (c) WedefinetheproductofI andJastheidealgeneratedbyallproductsofelementsofI and J,i.e. I·J:=({ab:a∈Iandb∈J}). NotethatjustthesetofproductsofelementsofI andJ wouldingeneralnotbeanideal: if wetakeR=R[x,y]andI=J=(x,y),thenobviouslyx2 andy2 areproductsofanelement ofIwithanelementofJ,buttheirsumx2+y2isnot. (d) ThequotientofIbyJisdefinedtobe I:J:={a∈R:aJ⊂I}. Again,itiseasytoseethatthisisanideal. (e) Wecall √ I:={a∈R:an∈Iforsomen∈N} theradicalofI. LetuscheckthatthisanidealofR: √ • Wehave0∈ I,since0∈I. √ • Ifa,b∈ I,i.e.an∈Iandbm∈Iforsomen,m∈N,then n+m(cid:18)n+m(cid:19) (a+b)n+m= ∑ akbn+m−k k k=0 isagainanelementofI,sinceineachsummandwemusthavethatthepowerofaisat leastn(inwhichcaseak∈I)orthepowerofbisatleastm(inwhichcasebn+m−k∈I). √ Hencea+b∈ I. √ • If r∈R and a∈ I, i.e. an ∈I for some n∈N, then (ra)n =rnan ∈I, and hence √ ra∈ I. √ √ Notethatwecertainlyhave I⊃I. WecallI aradicalidealif I=I,i.e.ifforalla∈R √ andn∈Nwithan∈I itfollowsthata∈I. Thisisanaturaldefinitionsincetheradical I √ ofanarbitraryidealIisinfactaradicalidealinthissense: ifan∈ Iforsomen,soanm∈I √ forsomem,thenthisobviouslyimpliesa∈ I. 01 WhetheranidealIisradicalcanalsoeasilybeseenfromitsquotientringR/Iasfollows. Definition1.2(Nilradical,nilpotentelements,andreducedrings). LetRbearing. Theideal (cid:112)(0)={a∈R:an=0forsomen∈N} iscalledthenilradicalofR;itselementsarecallednilpotent. IfRhasnonilpotentelementsexcept 0,i.e.ifthezeroidealisradical,thenRiscalledreduced. 10 AndreasGathmann Lemma1.3. AnidealI(cid:69)RisradicalifandonlyifR/Iisreduced. Proof. ByConstruction1.1(e),theidealIisradicalifandonlyifforalla∈Randn∈Nwithan∈I it follows that a∈I. Passing to the quotient ring R/I, this is obviously equivalent to saying that an=0impliesa=0,i.e.thatR/Ihasnonilpotentelementsexcept0. (cid:3) Example1.4(Operationsonidealsinprincipalidealdomains). Recallthataprincipalidealdomain (or short: PID) is an integral domain in which every ideal is principal, i.e. can be generated by one element [G1, Proposition 10.38]. The most prominent examples of such rings are probably Euclideandomains,i.e.integraldomainsadmittingadivisionwithremainder[G1,Definition10.21 andProposition10.38],suchasZorK[x]forafieldK [G1,Example10.22andProposition10.23]. WeknowthatanyprincipalidealdomainRadmitsauniqueprimefactorizationofitselements[G1, Proposition11.9]—aconceptthatwewilldiscussinmoredetailinChapter8. Asaconsequence, alloperationsofConstruction1.1canthenbecomputedeasily: ifI andJ arenotthezeroidealwe canwriteI=(a)andJ=(b)fora=pa1·····pan andb=pb1·····pbn withdistinctprimeelements 1 n 1 n p ,...,p anda ,...,a ,b ,...,b ∈N. Thenweobtain: 1 n 1 n 1 n (a) I+J=(pc11·····pcnn)withci=min(ai,bi)fori=1,...,n:another(principal)idealcontains I (resp.J)ifandonlyifitisoftheform(pc11· ··· ·pcnn)withci≤ai (resp.ci≤bi)foralli, sothesmallestidealI+JcontainingIandJisobtainedforc =min(a,b); i i i (b) I∩J=(pc11· ··· ·pcnn)withci=max(ai,bi); (c) I·J=(ab)=(pc11· ··· ·pcnn)withci=ai+bi; (d) √I:J=(pc11· ··· ·pcnn)withci=max(ai−bi,0); (e) I=(pc11· ··· ·pcnn)withci=min(ai,1). Inparticular,wehaveI+J=(1)=Rifandonlyifaandbhavenocommonprimefactor,i.e.ifa andbarecoprime. Weusethisobservationtodefinethenotionofcoprimeidealsingeneralrings: Definition1.5(Coprimeideals). TwoidealsIandJinaringRarecalledcoprimeifI+J=R. Example 1.6 (Operations on ideals in polynomial rings with SINGULAR). In more general rings, the explicit computation of the operations of Construction 1.1 is quite complicated and requires advancedalgorithmicmethodsthatyoucanlearnaboutinthe“ComputerAlgebra”class. Wewill not need this here, but if you want to compute some examples in polynomial rings you can use e.g. the computer algebra system SINGULAR [S]. For example, for the ideals I =(x2y,x√y3) and √J =(x+y) in Q[x,y] the following SINGULAR code computes that I:J =(x2y,xy2) and I·J = I∩J=(x2y+xy2),andchecksthaty3∈I+J: > LIB "primdec.lib"; // library needed for the radical > ring R=0,(x,y),dp; // set up polynomial ring Q[x,y] > ideal I=x2y,xy3; // means I=(x^2*y,x*y^3) > ideal J=x+y; > quotient(I,J); // compute (generators of) I:J _[1]=xy2 _[2]=x2y > radical(I*J); // compute radical of I*J _[1]=x2y+xy2 > radical(intersect(I,J)); // compute radical of intersection _[1]=x2y+xy2 > reduce(y3,std(I+J)); // gives 0 if and only if y^3 in I+J 0 √ √ Inthisexampleitturnedoutthat I·J= I∩J. Infact,thisisnotacoincidence—thefollowing lemmaandexerciseshowthattheproductandtheintersectionofidealsareverycloselyrelated. Lemma1.7(Productandintersectionofideals). ForanytwoidealsIandJinaringRwehave