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1、SpringerMonographsinMathematicsFriedrichIschebeckRaviA.RaoIdealsandRealityProjectiveModulesandNumberofGeneratorsofIdealsGI-SpringerFriedrichIschebeckInstitutfurMathematikUniversitatMunsterEinsteinstr.6248149Munster,Germanye-mail:ischebe@math.uni-muenster.deRa
2、viA.RaoSchoolofMathematicsTataInstituteofFundamentalResearchDrHomiBhabhaRoad400005Mumbai,Indiae-mail:ravi@math.tifr.res.inLibraryofCongressControlNumber:2004114476MathematicsSubjectClassification(2000):11C99,13A99,13Cio,13C40,13D15,55R25ISSN1439-7382ISBN3-540
3、-23032-7SpringerBerlinHeidelbergNewYorkThisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,reproductiononmicrofilmorinanyoth
4、erway,andstorageindatabanks.DuplicationofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.ViolationsareliableforprosecutionundertheGe
5、rmanCopyrightLaw.SpringerisapartofSpringerScience+BusinessMediaspringeronline.comOSpringer-VerlagBerlinHeidelberg2005PrintedinGermanyTheuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,evenintheabsenceofaspecificstatem
6、ent,thatsuchnamesareexemptfromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.TypesetbytheauthorsusingaSpringer!+VEXmacropackageProduction:LE-TEXJelonek,Schmidt&VocklerGbR,LeipzigCoverdesign:ErichKirchner,HeidelbergPrintedonacid-freepaper
7、4113142YL-543210DedicatedtoourclosefriendandformercolleagueHartmutLindelwholeftusmuchtooearlyPrefaceBesidesgivinganintroductiontoCommutativeAlgebra-thetheoryofcom-mutativerings-thisbookisdevotedtothestudyofprojectivemodulesandtheminimalnumberofgeneratorsofmod
8、ulesandideals.ThenotionofamoduleoveraringRisageneralizationofthatofavectorspaceoverafieldk.Theaxiomsareidentical.Butwhereaseveryvectorspacepossessesabasis,amoduleneednotalwayshaveone.Modu
