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1、Chapter1TheCategoryofGradedRings1.1GradedRingsUnlessotherwisestated,allringsareassumedtobeassociativeringsandanyringRhasanidentity1∈R.IfXandYarenonemptysubsetsofaringRthenXYdenotesthesetofallfinitesumsofelementsoftheformxywithx∈Xandy∈Y.ThegroupofmultiplicationinvertibleelementsofRwillbedenote
2、dbyU(R).ConsideramultiplicativelywrittengroupGwithidentityelemente∈G.AringRisgradedoftypeGorR,isG-graded,ifthereisafamily{Rσ,σ∈G}ofadditivesubgroupsRσofRsuchthatR=⊕σ∈GRσandRσRτ⊂Rστ,foreveryσ,τ∈G.ForaG-gradedringRsuchthatRσRτ=Rστforallσ,τ∈G,wesaythatRisstronglygradedbyG.Theseth(R)=∪σ∈GRσisthe
3、setofhomogeneouselementsofR;anonzeroelementx∈Rσissaidtobehomogeneousofdegreeσandwewrite:deg(x)=σ.AnelementrofRhasauniquedecompositionasr=σ∈Grσwithrσ∈Rσforallσ∈G,butthesumbeingafinitesumi.e.almostallrσzero.Thesetsup(r)={σ∈G,rσ=0}isthesupportofrinG.Bysup(R)={σ∈G,Rσ=0}wedenotethesupportoftheg
4、radedringR.Incasesup(R)isafinitesetwewillwritesup(R)<∞andthenRissaidtobeaG-gradedringoffinitesupport.IfXisanontrivialadditivesubgroupofRthenwewriteXσ=X∩Rσforσ∈G.WesaythatXisgraded(orhomogeneous)if:X=σ∈GXσ.Inparticular,whenXisasubring,respectively:aleftideal,arightideal,anideal,thenweobtainthe
5、notionsofgradedsubring,respectively:agradedleftideal,agradedrightideal,gradedideal.IncaseIisagradedidealofRthenthefactorringR/Iisagradedringwithgradationdefinedby:(R/I)σ=Rσ+I/I,R/I=⊕σ∈G(R/I)σ.C.NˇastˇasescuandF.VanOystaeyen:LNM1836,pp.1–18,2004.cSpringer-VerlagBerlinHeidelberg200421TheCatego
6、ryofGradedRings1.1.1PropositionLetR=⊕σ∈GRσbeaG-gradedring.Thenthefollowingassertionshold:1.1∈ReandReisasubringofR.2.Theinverser−1ofahomogeneouselementr∈U(R)isalsohomoge-neous.3.Risastronglygradedringifandonlyif1∈RσRσ−1foranyσ∈G.Proof1.SinceReRe⊆Re,weonlyhavetoprovethat1∈Re.Let1=rσbethedecom
7、positionof1withrσ∈Rσ.Thenforanysλ∈Rλ(λ∈G),wehavethatsλ=sλ.1=σ∈Gsλrσ,andsλrσ∈Rλσ.Consequentlysλrσ=0foranyσ=e,sowehavethatsrσ=0foranys∈R.Inparticularfors=1weobtainthatrσ=0foranyσ=e.Hence1=re∈Re.2.Assumethatr∈U(R)∩R.Ifr−1=(r−1)with(r−1)∈R,λσ∈Gσσσ
