问题标题：
J和K开头的英文单词.J和K开头的关于数学的单词或Phrase,需要definition和一个example列如：s开头的是：Systemofalinearequationdefinition:Twoormorelinearequatiionsinthesamevariables,alsocalledaliears
问题描述：
J和K开头的英文单词.
J和K开头的关于数学的单词或Phrase,需要definition和一个example
列如：s开头的是：
Systemofalinearequation
definition:
Twoormorelinearequatiionsinthesamevariables,alsocalledaliearsystem.
example:
x+2y=7
3x-2y=5
李宗樑回答：
TheJacobsonradical
11.2.1.Definition.LetMbealeftR-module.TheintersectionofallmaximalsubmodulesofMiscalledtheJacobsonradicalofM,andisdenotedbyJ(M).
11.2.2.Definition.LetMbealeftR-module.
ThesubmoduleNofMiscalledessentialorlargeinMifNK(0)forallnonzerosubmodulesKofM.
ThesubmoduleNiscalledsuperfluousorsmallinMifN+KMforallpropersubmodulesKofM.
Kernel
DefinitionLet:R->Sbearinghomomorphism.Theset
{aR|(a)=0}
iscalledthekernelof,denotedbyker().
11.2.3.Proposition.LetNbeasubmoduleofRM.IfKismaximalinthesetofallsubmodulesofMthathavetrivialintersectionwithN,thenN+KisessentialinM,and(N+K)/KisessentialinM/K.
11.2.4.Proposition.Thesocleofanymoduleistheintersectionofitsessentialsubmodules.
11.2.5.Definition.AradicalfortheclassofleftR-modulesisafunctionthatassignstoeachmoduleRMasubmodule(M)suchthat
(i)f((M))(N),forallmodulesRNandallfHomR(M,N);
(ii)(M/(M))=(0).
11.2.6.Definition.LetCbeanyclassofleftR-modules.ForanymoduleRMwemakethefollowingdefinition.
radC(M)=ker(f),
wheretheintersectionistakenoverallR-homomorphismsf:M->X,forallXinC.
11.2.7.Proposition.LetbearadicalfortheclassofleftR-modules,andletFbetheclassofleftR-modulesXforwhich(X)=(0).
(a)(R)isatwo-sidedidealofR.
(b)(R)M(M)forallmodulesRM.
(c)radFisaradical,and=radF.
(d)(R)=Ann(X),wheretheintersectionistakenoverallmodulesXinF.
11.2.8.Lemma.[Nakayama]IfRMisfinitelygeneratedandJ(R)M=M,thenM=(0).
11.2.9.Proposition.LetMbealeftR-module.
(a)J(M)={mM|RmissmallinM}.
(b)J(M)isthesumofallsmallsubmodulesofM.
(c)IfMisfinitelygenerated,thenJ(M)isasmallsubmodule.
(d)IfMisfinitelygenerated,thenM/J(M)issemisimpleifandonlyifitisArtinian.
11.2.10.Theorem.TheJacobsonradicalJ(R)oftheringRisequaltoeachofthefollowingsets:
(1)TheintersectionofallmaximalleftidealsofR;
(2)TheintersectionofallmaximalrightidealsofR;
(3)Theintersectionofallleft-primitiveidealsofR;
(4)Theintersectionofallright-primitiveidealsofR;
(5){xR|1-axisleftinvertibleforallaR};
(6){xR|1-xaisrightinvertibleforallaR};
(7)ThelargestidealJofRsuchthat1-xisinvertibleinRforallxJ.
11.2.11.Definition.TheringRissaidtobesemiprimitiveifJ(R)=(0).
11.2.12.Proposition.LetRbeanyring.
(a)TheJacobsonradicalofRcontainseverynilidealofR.
(b)IfRisleftArtinian,thentheJacobsonradicalofRisnilpotent.
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