Commutative Algebra [Lecture notes]

77 Pages·2006·0.526 MB·English

by  Wolfram Decker

Commutative Algebra
Prof. Dr. Wolfram Decker
(LATEX-version by Felix Boos)
TUKaiserslautern
WS2012/2013
9. März 2016
Inhaltsverzeichnis
0 Introduction 3
1 RingsandIdeals 4
1.1 BasicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 FirstExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 OperationsonIdeals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Furtherterminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 TheChineseRemainderTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 PrimeIdealsandMaximalIdeals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Localrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.8 NilradicalandJacobsonRadical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.9 MoreExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 Modules 16
2.1 BasicDefinitionsandExamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 FreeModules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 FinitelygeneratedModules,theCayley-HamiltonTheoremandNakayama’sLemma . . . . . 19
2.4 TensorProductsofModules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5 R-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Exactsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Localization 30
3.1 LocalizationofRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 LocalizationofModules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 LocalProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4 ChainConditions 37
4.1 NoetherianRingsandModules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 FreeResolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Modulesoffinitelength,ArtinianModules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 ArtinianRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5 PrimaryDecomposition 46
5.1 DefinitionandExistenceinNoetherianRings . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.2 Uniqueness-Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6 IntegralRingExtensions 52
6.1 BasicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.2 LyingOver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
6.3 GoingDown . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7 Krull Dimension, Noether Normalization, Hilbert’s Nullstellensatz and Krull’s Principle Ideal
Theorem 61
7.1 DefinitionofKrullDimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.2 NoetherNormalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
7.3 PropertiesofKrullDimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.4 Hilbert’sNullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
7.5 Krull’sPrincipalIdealTheoremandRegularLocalRings . . . . . . . . . . . . . . . . . . . . 68
8 ValuationRingsandDedekindDomains 72
8.1 ValuationRings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
8.2 DedekindDomains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
2
0 Introduction
Historical roots
The Commutative Algebra presented in this lecture relies historically on two fields of mathematical rese-
arch. The first one is Algebraic Number Theory, especially Fermat’s last theorem respectively the methods
developed to prove it, the search for unique prime factorization, which leads to Dedekind domains and
primary decomposition in polynomial rings. The second one is Algebraic Geometry, which deals with the
expressionofgeometricproblemsintermsofidealsofpolynomialringsorrespectivelyringsofpolynomial
(rational)functions.
Basic objects
Wewillworkmainlywithrings,idealsofringsandmodulesoverrings.Asthenameofthelecturesuggests,
wewillfocusoncommutativerings,sowhenevertheword„ring“isusedinthelecture,wemeanbythata
commutativeringwithmultiplicativeidentity1.
Theterms„ideal“and„module“willbedefinedinChapter1.Togetafirstimpressionhowever,idealsare
forringswhatnormalsubgroupsareforgroups,andmodulesoverringsareanalogoustovectorspacesof
fields.
3
1 Rings and Ideals
In this first „reasonable“ chapter, we will recall some basic notions from the lecture „Algebraische Struk-
turen“andeventuallyaddsomenewterminologyonourway.
1.1 Basic Definitions
Definition1.1.1
A (commutative) ring R = (R,+,·) (with 1 = 1 ) is an abelian group (R,+) together with a multiplicative
R
operation·:R×R→R,(a,b)(cid:55)→a·b=ab,suchthatforalla,b,c∈Ritholds
• a(bc)=a(bc)(associativity)
• 1a=a(multiplicativeidentity)
• ab=ba(commutativity)
• a(b+c)=ab+ac(distributivity)
Note that if R is not the trivial ring {0}, the additive identity 0 and the multiplicative identity 1 differ, in
short:0 =1 ⇒R={0}.
R R
Recallthefollowingnotions:
• AunitofaringRisanelementtowhichamultiplicativeinverseexistsinsidethering.Wedenoteby
∗
R thesetofallunitsinR,whichformsagroupwiththemultiplication.
• AfieldisaringforwhichR∗=R\{0}.
• AsubringofRisasubsetofRwhichitselfformsaringwiththe„inherited“operationsfromR.
• Aringhomomorphismisamapbetweentworingswhichrespectsadditionandmultiplicationand
whichmapsthemultiplicativeidentity(ofthefirstring)tothemultiplicativeidentity(ofthesecond
ring). By Hom(R,S), we denote the set of all ring homomorphisms from R to S. If a ring homomor-
phismisinjective,wecallitamonomorphism,theattributeofsurjectivitygivesusthenotionepimor-
phismandifahomomorphismisbijectiveitisentitledanisomorphism.Ifthereexistsanisomorphism
betweentworingsRandS,wecallthemisomorphicandwriteR(cid:27)S.
Definition1.1.2
LetRbearing.AnidealofRisanadditivesubgroupI ofRsuchthatforallr ∈Randa∈I itholdsra∈I.
WewriteI (cid:69)R.
IfT isanonemptysubsetofR,thenwedefine
 
(cid:104)T(cid:105) = (cid:88)k r a |k∈N,r ∈R,a ∈T
 i i i i 
i=1
(cid:92)
= I,
T⊆I(cid:69)R
which is the smallest ideal containing T. We call (cid:104)T(cid:105) the ideal generated by T. Sometimes, we will write
(cid:104)T(cid:105) to make clear to which ring the generated ideal belongs. In order to save brackets, we also use the
R
abbreviation(cid:104)f ,...,f (cid:105)=(cid:104){f ,...,f }(cid:105).
1 k 1 k
NotethatinordertocheckifasubsetI ⊆Risanidealitsufficestoshowthat
4
(i) 0∈I
(ii) Ifr,s∈Randa,b∈I,thenra+sb∈I.
Ifwehave1∈I,thenI isalreadythewholering,i.e.I =R.
GivenanidealI (cid:69)R,wehavethefollowingnotions:
• asetofgenerators,
• finitelygenerated,ifitadmitsafinitesetofgenerators,
• principleideal,ifitisgeneratedbyjustoneelement.
Definition1.1.3
LetRbearingandI (cid:69)Ranideal.ThenwehaveanequivalencerelationdefinedbycongruencemoduloI:
a≡b modI ⇔ a−b∈I.
Wewritea=a+I fortheequivalenceclassofa∈Randcallitaresidueclass.ThesetR/I ={a|a∈R}ofall
residueclassesisaringwithalgebraicoperationsa+b=a+banda·b=a·b.Wecallthisringthequotient
ringofRmoduloI.Wehavethecanonicalringepimorphism
R→R/I, a(cid:55)→a.
1.2 First Examples
In this section we will study a few typical examples of rings and their ideals and derive some first state-
mentsabouttheirproperties.
Example1.2.1
Anyfieldisaring,ofcourse.Infact,aringR(cid:44){0}isafieldifandonlyif(cid:104)0(cid:105)andRaretheonlyidealsofR.
Example1.2.2
ThesetZofintegersisaring.TheidealsinZarepreciselythesubsets(cid:104)n(cid:105)wheren∈N(seeSection1.4for
anexplanationofthisfact).AquotientringZ =Z/(cid:104)n(cid:105)(n≥1)isafieldifandonlyifnisaprimenumber.
n
Example1.2.3(Polynomialrings)
LetRbearing.Weusemultiindicesα=(α ,...,α )∈Nn towritepolynomialsinnvariablesx=(x ,...,x )
1 n 1 n
withcoefficientsinR:
• Amonomialisaproductofpowersofvariables,i.e.anexpressionlikexα =xα1·...·xαn.
1 n
• Atermisamonomialmultipliedwithaconstant(acoefficient)a·xα forsomea∈Randα∈Nn.
(cid:80)
• Finally,apolynomialisafinitesum a xα ofterms.
α
α
(cid:80)
Thedegreeofapolynomialf = f xα isdefinedas
α
α
(cid:40)
−∞ forf =0
degf =
max{|α||f (cid:44)0} otherwise
α
where|α|=α +...+α .
1 n
(cid:40) (cid:41)
The set R[x] = R[x ,...,x ] = (cid:80)a xα |a ∈R of all polynomials with coefficients in R is a ring with
1 n α α
α
5
operations
(cid:88) (cid:88) (cid:88)
a xα+ b xα = (a +b )xα,
α α α α
α α α
     
(cid:88)aαxα·(cid:88)bαxα = (cid:88) (cid:88) aαbβxγ.
α α γ α+β=γ
Example1.2.4(Formalpowerseriesring)
Aspolynomialsweonlyallowedfinitesumsofterms.Ofcourse,wecanalsostudyinfinitesums–havingin
mindthatunlikeinanalysis,wedonotcareaboutnotionslike„convergence“etc.butratherregardinfinite
sumsofterms(orpowerseries)asformalobjects.Wecandefineoperations+and·onformalpowerseries
analogously to those on polynomials in the precedent section, thus giving us a ring R(cid:126)x(cid:127) = R(cid:126)x ,...,x (cid:127).
1 n
Naturally,R[x]⊆R(cid:126)x(cid:127)isasubring.
Obviously,thenotionofdegreedoesnotmakealotofsenseinthecontextofformalpowerseriessincein
most cases we simply don’t have a „maximal exponent“ here. Instead, we introduce the order of a power
seriesasfollows.Givenf =(cid:80)f xα ∈R(cid:126)x(cid:127),weset
α
α
(cid:40)
∞ forf =0
ordf =
min{|α||f (cid:44)0} otherwise
α
Sincethe|α|arenaturalnumbers,theminimumalwaysexists.
Example1.2.5
Let us examine a special type of polynomial rings, namely the ring K[x] of polynomials in one variable
over a field K. Here, the ideals are of type (cid:104)f(cid:105) where f ∈ K[x]. When we „expand“ our interest to formal
power series, we interestingly get an even more special result: The ideals in K(cid:126)x(cid:127) are of type (cid:104)xn(cid:105) where
n≥0.SeeagainSection1.4forfurthercommentsonthattopic.
Example1.2.6(DirectProductsofRings)
Let {Rλ}λ∈Λ be a family of rings. Then we can produce a new ring by endowing the direct product (cid:81) Rλ
λ∈Λ
withcomponentwisedefinedalgebraicoperations.
Example1.2.7(IdealsinQuotientRings)
Let I (cid:69)R be an ideal of the ring R. Consider the canonical epimorphism π :R→R/I,r (cid:55)→r+I =r. There
exists a one-to-one correspondence between ideals J of R containing I and ideals J of R/I, given by J =
π−1(J).
We already know about the homomorphism theorem for groups from the lecture „Algebraische Struktu-
ren“whichallowedustoprovetheexistenceofsomeisomorphisminaveryelegantway.Almostnaturally,
wealsohaveananalogousstatementforrings.
Theorem1.2.8(HomomorphismTheorem)
Letϕ:R→S beahomomorphismofrings.ThethekernelKer(ϕ)isanidealofR,theimageIm(ϕ)isasubring
ofS andtheinducedmapϕ:R/Ker(ϕ)→Im(ϕ),a(cid:55)→ϕ(a)isanisomorphism.
Proof. Thisfollowsdirectlyfromthestatementforgroups.
Notethattheimageofahomomorphismisasubring,butnotnecessarilyanideal(incontrasttothekernel)!
Letusgiveaneasycounterexampletoacceptthis.
Example1.2.9
Ifϕ:Z→Q,z(cid:55)→zistheinclusion,thenforexampleϕ((cid:104)2(cid:105))isnotanidealinQ,andneitherisIm(ϕ),since
theonlyidealsinthefieldQare{0}andQitself.
6
Whatapitythattheimageofahomomorphismdoesn’tgiveusanidealbutonlyasubgroup!Butinorderto
takecomfortinthis,wecanatleastlookatthesmallestidealcontainingtheimage,i.e.theidealgenerated
by the image. And of course, we want to give a name to this construction, as we will do in the following
definition.
Definition1.2.10
Letϕ:R→S beahomomorphismofrings.
(i) IfI ⊆Risanideal,thentheidealIe =(cid:104)ϕ(I)(cid:105)⊆S iscalledtheextensionofI (toS).
(ii) IfJ ⊆S isanideal,thentheidealJc =ϕ−1(J)⊆RiscalledthecontractionofJ (toR).
Thecontractionseemstobesomethinglikethereverseoftheextension.Asthefollowinglemmastates,it
actuallyis–withemphasisontheterm„somethinglike“...
Lemma1.2.11
Letϕ:R→S beahomomorphismofrings,I (cid:69)RandJ (cid:69)S ideals.Theitholds
(i) Iec ⊇I,
(ii) Jce ⊆J,
(iii) Iece =Ie,
(iv) Jcec =Jc.
Proof. Immediatefromthedefinitions.
1.3 Operations on Ideals
Of course, we not only want to look at given ideals and examine their properties, we also want to „work“
with them. For example, we could create new ideals of a ring R form old ones, by combining two or more
idealsinsomeinterestingorreasonablewayorbydoingsomethingwithonlyoneideal.Themostimport-
antsuchoperationsare:
(cid:84)
• theintersection I ,
λ
λ∈Λ
• theproductI ·...·I =(cid:104)a ·...·a |a ∈I ∀i =1,...,k(cid:105),
1 k 1 k i i
(cid:42) (cid:43)
(cid:80) (cid:83)
• thesum I = I ,
λ k
λ∈Λ λ∈Λ
• theidealquotientI :J ={a∈R|ab∈I∀b∈J},
√
• theradical I =rad(I)={a∈R|am∈I forsomem≥1}
Note:Thatthisactuallyisanidealcaneasilybeshownusingthebinomialformula.
• theannihilatorann(I)=ann (I)=(cid:104)0(cid:105):I ={a∈R|aI =(cid:104)0(cid:105)}.
R
√
NotethatI ⊆ I.Ifa∈R,wewriteI :a=I :(cid:104)a(cid:105).
Remark1.3.1 (i) SeeExercise2forsomeformulas.
(ii) Clearly I ·...·I ⊆I ∩...∩I . However, equality does not necessarily occur. For example, (cid:104)2(cid:105)·(cid:104)4(cid:105)=
1 k 1 k
(cid:104)8(cid:105)(cid:44)(cid:104)4(cid:105)=(cid:104)2(cid:105)∩(cid:104)4(cid:105). Actually, that’s kind of good news because otherwise, the notion of the product
wouldbesuperfluous.However,thecaseofequalityisstillimportant,andwewillstudyitinSection
1.5whenitcomestotheChineseRemainderTheorem.
7
√ √
(iii) Neverthelessnotethat I ·...·I = I ∩...∩I ,sotheradicalsofproductandintersectionareequal.
1 k 1 k
In Algebraic Geometry, this is the reason why the geometric interpretation of intersection and pro-
ductarethesamesinceforthat,onelooksattheradicalofanidealratherthanattheidealitself.
Example1.3.2
Letusstudyaspecialandwellknowncase,namelyR=Z.WeknowthatZisaprincipleidealdomain,so
considertheidealsI =(cid:104)n(cid:105),J =(cid:104)m(cid:105)forsomem,n≥1.Thenwehave
• I+J =(cid:104)n,m(cid:105)=(cid:104)gcd(n,m)(cid:105),
• I∩J =(cid:104)lcm(n,m)(cid:105),
• I·J =(cid:104)n·m(cid:105),
(cid:68) (cid:69) (cid:68) (cid:69)
• I :J = n = lcm(n,m) ,
gcd(n,m) m
√
• I =(cid:104)p ,...,p (cid:105),wheren= (cid:81)k pαi istheprimefactorizationofn,
1 k i
i=1
• ann(I)=(cid:104)0(cid:105).
Enjoyably, some of these notions behave exactly as we would intuitively suspect them to do in this easy
case,notablytheproductortheradical.
1.4 Further terminology
Definition1.4.1
LetRbearingandr ∈Rsomeelement.
(i) Wecallr azero-divisorifthereexistssomes∈R\{0}suchthatr·s=0.Thisdefinitionisequivalent
toann(r)(cid:44)(cid:104)0(cid:105).
WecallRanintegraldomainif0istheonlyzero-divisor(and0(cid:44)1).
(ii) Wecallr nilpotentifrm=0forsomem≥1.
Riscalledreducedif0istheonlynilpotentelement.
(iii) Wecallr idempotentifr2=r ⇔r(1−r)=0.
Notethatwehavethefollowingequivalence:
√
R/I reduced ⇐⇒ I = I,i.e.I isaradicalideal.
Example1.4.2 (i) Zisanintegraldomain.
(ii) IfRisanintegraldomain,thensoisR[x].
(iii) R=K[x,y]/(cid:104)xy(cid:105)isnotanintegraldomainsincex isazero-divisor(x·y =xy =0).
(iv) R=K[x]/(cid:104)x2(cid:105)isnotreducedsincex isnilpotent.
(v) IntheringR=Z2,theelement(1,0)isidempotent.
Recallthefollowingnotions:
• A principle ideal domain (PID) is an integral domain in which every ideal is a principle ideal, i.e.
canbegeneratedbyonesingleelement.
8
• AEuclideandomainisanintegraldomaininwhichthereexistsadivisionwithremainder(roughly
speaking).
• Auniquefactorizationdomain(UFD)isanintegraldomaininwhicheverynon-zero,non-unitele-
mentcanbewrittenasfiniteproductofprimeelements.
Therelationshipofthesenotionscanbevisualizedasfollows:
Integraldomains⊃Uniquefactorizationdomains⊃Principleidealdomains⊃Euclideandomains⊃Fields.
This is to be read as e.g. „every Euclidean domain is a principle ideal domain“ (but not vice versa). For a
proofofthesestatementsseethelecture„AlgebraischeStrukturen“.
Example1.4.3 (i) Z and K[x] (where K is a field) are Euclidean domains and hence principle ideal do-
mains.
(ii) K(cid:126)x(cid:127) is a principle ideal domain since every ideal (cid:104)0(cid:105) (cid:44) I of K(cid:126)x(cid:127) is of type (cid:104)xn(cid:105) for some n ∈ N,
whichcanbeseenasfollows.
First note that K(cid:126)x(cid:127)∗ = {f ∈ K(cid:126)x(cid:127) | f(0) (cid:44) 0} (this is to be proved on Exercise sheet 2, e.g. by using
∞
(cid:88)
geometricseries).Nextchoose0(cid:44)g ∈I withord(g)=nisminimal.Theng isoftypeg =xn· g xi−n.
i
i=(cid:32)(cid:32)n(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)
(cid:124) (cid:123)(cid:122) (cid:125)
h
h is a unit in K(cid:126)x(cid:127) by definition of the order and hence we have xn = gh−1 ∈ I. Now if 0 (cid:44) f ∈ I is
arbitrary,thenord(f)≥nbytheverydefinitionofg.Hence,
 ∞ 
f = xn·(cid:88)fixi−n ∈ (cid:104)xn(cid:105).
i=n
1.5 The Chinese Remainder Theorem
In this section, we will give a more general version of the Chinese Remainder Theorem, which is well
knownforintegersandtheirresidueclasses.Asyoupossiblyremember,theChineseRemainderTheorem
for integers requires that the moduli we work with are pairwise coprime. It should not be surprising that
inthegeneralcase,wealsoneedsomekindof„coprime“condition.Soatfirst,wewilldefinesomesuitable
notionforideals.
Definition1.5.1
LetRbearingandI,J (cid:69)Rideals.ThenI andJ arecoprimeifI+J =R.
Notethatbythatdefinition,I andJ arecoprimeifandonlyifthereexistsa∈I andb∈J suchthata+b=1.
Thisisbecause1∈I+J ⇔I+J =R.
How does that fit to the notion of coprime integers? As you should recall, for a,b ∈Z we have the Bézout
identity which states that there exist x,y ∈ Z with a·x+b·y = gcd(a,b). If a and b are coprime, then
gcd(a,b)=1andtheBézoutidentityreadsa·x+b·y =1.Ofcourse,a·x∈(cid:104)a(cid:105)andb·y ∈(cid:104)b(cid:105),andhencewe
canfinda(cid:48) ∈(cid:104)a(cid:105)andb(cid:48) ∈(cid:104)b(cid:105)suchthata(cid:48)+b(cid:48) =1–andnowyoushouldclearlyseetheconnection.
Nowwearewellpreparedfortheactualtheorem.
9
Theorem1.5.2(ChineseRemainderTheorem)
LetRbearingandI ,...,I (cid:69)Rideals.Considertheringhomomorphism
1 k
(cid:89)k
ϕ:R→ R/I , r (cid:55)→(r,...,r).
i
i=1
(i) IfI ,...,I arepairwisecoprime,thenI ·...·I =I ∩...∩I .
1 k 1 k 1 k
(ii) ϕ issurjectiveifandonlyifI ,...,I arepairwisecoprime.
1 k
(iii) ϕ isinjectiveifandonlyifI ∩...∩I =(cid:104)0(cid:105).
1 k
Proof. (i) ByRemark1.3.1wealreadyknowthattheinclusionI ·...·I ⊆I ∩...∩I holdstrueanyway,
1 k 1 k
sowejustneedtoprovetheotherinclusion,namelyI ∩...∩I ⊆I ·...·I .Weuseinductiononk.
1 k 1 k
Basis(k=2): Since I and I are coprime, we have a+b =1 for some a∈I and b ∈I . If c ∈I ∩I ,
1 2 1 2 1 2
then
c=c·1=c(a+b)= c·a + c·b ∈I ·I .
1 2
(cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125)
∈I ·I ∈I ·I
1 2 1 2
Inductivestep(k−1→k): Wehavea +b =1forsomea ∈I andb ∈I ,where1≤i ≤k−1.Then
i i i i i k
a1·...·ak−1 = (1−b1)·...·(1−bk−1)
= 1+bforsomeb∈I .
k
ThisimpliesthatJ =I1·...·Ik−1 andIk arecoprimesincea1(cid:32)(cid:32)(cid:32)(cid:32)·(cid:32)(cid:32).(cid:32)..·a(cid:32)(cid:32)(cid:32)(cid:32)k(cid:32)(cid:32)−(cid:32)1+ (−b) =1.Usingthebasis
(cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125)
∈J ∈I
k
(k=2)andtheinductionhypothesis,weget
I1·...·Ik =J·Ik ba=sisJ∩Ik I.=H.I1∩...∩Ik−1∩Ik.
(ii) „⇒“ ShowforexamplethatI andI arecoprime(theproofforanyotherpairofidealsiscompletely
1 2
analogous).
Since ϕ is surjective, there exists a∈R such that ϕ(a)=(1,0,...,0). In particular, a≡1 modI
1
anda≡0 modI ,so1=(1−a)+ a ∈I +I .SoI andI arecoprimeasdesired.
2 1 2 1 2
(cid:124)(cid:123)(cid:122)(cid:125) (cid:124)(cid:123)(cid:122)(cid:125)
∈I1 ∈I2
„⇐“ Show, for example, that there exists a ∈ R such that ϕ(a) = (0,...,0,1). With an analogous ar-
gument, we can construct a preimage for every „unit vector“ (0,...,1,...,0) and since together
these„unitvectors“generatethewholering,wetherebyprovedourclaim.
Choose ai and bi as in (i), i.e. ai +bi =1, ai ∈Ii and bi ∈Ik for 1≤i ≤k−1. Set a=a1·...·ak−1.
Then
a ≡ 0 modI forall1≤i ≤k−1and
i
a ≡ 1 modI .
k
Thatis,ϕ(a)=(0,...,0,1).
(iii) ϕ isinjectiveifandonlyifKer(ϕ)=(cid:104)0(cid:105).ButKer(ϕ)=I ∩...∩I .
1 k
10