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MathematicsMaterials

Radical

1.Radicalsymbol（√）,asymbolusedtoindicatethesquarerootornthroot

2.Radicalofanideal,animportantconceptinabstractalgebra

Incommutativeringtheory,abranchofmathematics,theradicalofanidealIisanidealsuchthatanelementxisintheradicalifsomepowerofxisinI.Aradicalideal（orsemiprimeideal）isanidealthatisitsownradical（thiscanbephrasedasbeingafixedpointofanoperationonidealscalled'radicalization'）.Theradicalofaprimaryidealisprime.

RadicalidealsdefinedherearegeneralizedtononcommutativeringsintheSemiprimeringarticle.

Definition

TheradicalofanidealIinacommutativeringR,denotedbyRad（I）or

isdefinedas

Intuitively,onecanthinkoftheradicalofIasobtainedbytakingallthepossiblerootsofelementsofI.Rad（I）turnsouttobeanidealitself,containingI.TheeasiestwaytoprovethattheradicalofIofaringAisanidealistonotethatitisthepre-imageoftheidealofnilpotentelementsinA/I.Thisissometimestakenasadefinitionofradical.

IfanidealIcoincideswithitsownradical,thenIiscalledaradicalidealorsemiprimeideal.

Examples

ConsidertheringZofintegers.

Theradicaloftheideal4Zofintegermultiplesof4is2Z.

Theradicalof5Zis5Z.

Theradicalof12Zis6Z.

Ingeneral,theradicalofmZisrZ,whereristheproductofallprimefactorsofm（seeradicalofaninteger）.

Theradicalofaprimaryidealisprime.

Thenilradicalofaring

ConsiderthesetofallnilpotentelementsofR,whichwillbecalledthenilradicalofR（andwillbedenotedbyN（R））.OnecaneasilyseethatthenilradicalofRisjusttheradicalofthezeroideal（0）.Thispermitsanalternativedefinitionforthe（general）radicalofanidealIinR.DefineRad（I）asthepreimageofN（R/I）,thenilradicalofR/I,undertheprojectionmapR→R/I.

ToseethatthetwodefinitionsfortheradicalofIareequivalent,notefirstthatifrisinthepreimageofN（R/I）,thenforsomen,rniszeroinR/I,andhencernisinI.Second,ifrnisinIforsomen,thentheimageofrninR/Iiszero,andhencernisinthepreimageofN（R/I）.

Thisalternativedefinitioncanbeveryuseful,asweshallseerightbelow.See#Propertiesbelowforanothercharacterizationofthenilradical.

Properties

ThissectionwillcontinuetheconventionthatIisanidealofacommutativeringR:

ItisalwaystruethatRad（Rad（I））=Rad（I）.Inwords,thissaysthatRad（I）isaradicalideal.

ForanyidealI,Rad（I）isthesmallestradicalidealcontainingI.

Rad（I）istheintersectionofalltheprimeidealsofRthatcontainI.Ononehand,everyprimeidealisradical,andsotheintersectionJoftheprimeidealscontainingIcontainsRad（I）.SupposerisanelementofRwhichisnotinRad（I）,andletSbetheset{rn|nisanonnegativeinteger}.BythedefinitionofRad（I）,SmustbedisjointfromI.SinceSismultiplicativelyclosedandRhasidentity,anargumentwithZorn'slemmashowsthatthereexistsanidealPinthisringwhichcontainsIandismaximalwithrespecttobeingdisjointfromS.ItiswellknownthatPisnecessarilyaprimeideal.SincePcontainsI,butnotr,thisshowsthatrisnotintheintersectionofprimeidealscontainingI.Thus,theintersectionofprimeidealscontainingIiscontainedinRad（I）,provingequality.

Specializingthelastpoint,thenilradicalisequaltotheintersectionofallprimeidealsofR.ThisshowsthatthenilradicalofRcanalternativelybedefinedastheintersectionoftheprimeidealsofR.

AnidealIinaringRisradicalifandonlyifthequotientringR/Iisreduced.

3.Radicalofaninteger,innumbertheory,theradicalofanintegeristheproductoftheprimeswhichdividethatinteger

Innumbertheory,theradicalofapositiveintegernisdefinedastheproductoftheprimenumbersdividingn:

Continuousfunction

Inmathematics,acontinuousfunctionisafunctionforwhich,intuitively,"small"changesintheinputresultin"small"changesintheoutput.Otherwise,afunctionissaidtobea"discontinuousfunction".Acontinuousfunctionwithacontinuousinversefunctioniscalled"bicontinuous".

Continuityoffunctionsisoneofthecoreconceptsoftopology,whichistreatedinfullgeneralitybelow.Theintroductoryportionofthisarticlefocusesonthespecialcasewheretheinputsandoutputsoffunctionsarerealnumbers.Inaddition,thisarticlediscussesthedefinitionforthemoregeneralcaseoffunctionsbetweentwometricspaces.Inordertheory,especiallyindomaintheory,oneconsidersanotionofcontinuityknownasScottcontinuity.Otherformsofcontinuitydoexistbuttheyarenotdiscussedinthisarticle.

Asanexample,considerthefunctionh（t）,whichdescribestheheightofagrowingflowerattimet.Thisfunctioniscontinuous.Infact,adictumofclassicalphysicsstatesthatinnatureeverythingiscontinuous.Bycontrast,ifM（t）denotestheamountofmoneyinabankaccountattimet,thenthefunctionjumpswhenevermoneyisdepositedorwithdrawn,sothefunctionM（t）isdiscontinuous.

Real-valuedcontinuousfunctions

Definition

AfunctionfromthesetofrealnumberstotherealnumberscanberepresentedbyagraphintheCartesianplane;thefunctioniscontinuousif,roughlyspeaking,thegraphisasingleunbrokencurvewithno"holes"or"jumps".

Thereareseveralwaystomakethisintuitionmathematicallyrigorous.Thesedefinitionsareequivalenttooneanother,sothemostconvenientdefinitioncanbeusedtodeterminewhetheragivenfunctioniscontinuousornot.Inthedefinitionsbelow,

isafunctiondefinedonasubsetIofthesetRofrealnumbers.ThissubsetIisreferredtoasthedomainoff.PossiblechoicesincludeI=R,thewholesetofrealnumbers,anopeninterval

oraclosedinterval

Here,aandbarerealnumbers.

[edit]Definitionintermsoflimitsoffunctions

Thefunctionfiscontinuousatsomepointcofitsdomainifthelimitoff（x）asxapproachescthroughthedomainoffexistsandisequaltof（c）.[3]Inmathematicalnotation,thisiswrittenas

Indetailthismeansthreeconditions:

first,fhastobedefinedatc.Second,thelimitonthelefthandsideofthatequationhastoexist.Third,thevalueofthislimitmustequalf（c）.

Thefunctionfissaidtobecontinuousifitiscontinuousateverypointofitsdomain.Ifthepointcinthedomainoffisnotalimitpointofthedomain,thenthisconditionisvacuouslytrue,sincexcannotapproachcthroughvaluesnotequalc.Thus,forexample,everyfunctionwhosedomainisthesetofallintegersiscontinuous.

Definitionintermsoflimitsofsequences

Onecaninsteadrequirethatforanysequence

ofpointsinthedomainwhichconvergestoc,thecorrespondingsequence

convergestof（c）.Inmathematicalnotation,

Weierstrassdefinition（epsilon-delta）ofcontinuousfunctions

Illustrationoftheε-δ-definition:

forε=0.5,thevalueδ=0.5satisfiestheconditionofthedefinition.

Explicitlyincludingthedefinitionofthelimitofafunction,weobtainaself-containeddefinition:

GivenafunctionfasaboveandanelementcofthedomainI,fissaidtobecontinuousatthepointcifthefollowingholds:

Foranynumberε>0,howeversmall,thereexistssomenumberδ>0suchthatforallxinthedomainoffwithc−δ

Alternativelywritten,continuityoff:

I→Ratc∈Imeansthatforeveryε>0thereexistsaδ>0suchthatforallx∈I,:

Moreintuitively,wecansaythatifwewanttogetallthef（x）valuestostayinsomesmallneighborhoodaroundf（c）,wesimplyneedtochooseasmallenoughneighborhoodforthexvaluesaroundc,andwecandothatnomatterhowsmallthef（x）neighborhoodis;fisthencontinuousatc.

Inmodernterms,thisisgeneralizedbythedefinitionofcontinuityofafunctionwithrespecttoabasisforthetopology,herethemetrictopology.

Definitionusingoscillation

Thefailureofafunctiontobecontinuousatapointisquantifiedbyitsoscillation.

Continuitycanalsobedefinedintermsofoscillation:

afunctionfiscontinuousatapointx0ifandonlyifitsoscillationatthatpointiszero;[4]insymbols,

Abenefitofthisdefinitionisthatitquantifiesdiscontinuity:

theoscillationgiveshowmuchthefunctionisdiscontinuousatapoint.

Thisdefinitionisusefulindescriptivesettheorytostudythesetofdiscontinuitiesandcontinuouspoints–thecontinuouspointsaretheintersectionofthesetswheretheoscillationislessthanε（henceaGδset）–andgivesaveryquickproofofonedirectionoftheLebesgueintegrabilitycondition.[5]

Theoscillationisequivalenttotheε-δdefinitionbyasimplere-arrangement,andbyusingalimit（limsup,liminf）todefineoscillation:

if（atagivenpoint）foragivenε0thereisnoδthatsatisfiestheε-δdefinition,thentheoscillationisatleastε0,andconverselyifforeveryεthereisadesiredδ,theoscillationis0.Theoscillationdefinitioncanbenaturallygeneralizedtomapsfromatopologicalspacetoametricspace.

Definitionusingthehyperreals

Cauchydefinedcontinuityofafunctioninthefollowingintuitiveterms:

aninfinitesimalchangeintheindependentvariablecorrespondstoaninfinitesimalchangeofthedependentvariable（seeCoursd'analyse,page34）.Non-standardanalysisisawayofmakingthismathematicallyrigorous.Thereallineisaugmentedbytheadditionofinfiniteandinfinitesimalnumberstoformthehyperrealnumbers.Innonstandardanalysis,continuitycanbedefinedasfollows.

Afunctionƒfromtherealstotherealsiscontinuousifitsnatur