TAIWANESE JOURNAL OF MATHEMATICS Vol - PDF document

TAIWANESEJOURNALOFMATHEMATICSVol.11,No.1,pp.267-275,March2007Thispaperisavailableonlineathttp://www.math.nthu.edu.tw/tjm/REDUCEDANDp.q.-BAERMODULESMuhittinBas¸erandAbdullahHarmanciInthispaper,westudyp.q.-Baermodulesandsomepolynomialextensionsofp.q.-Baermodules.Inparticular,weshow:(1)Forareduced ReceivedJanuary4,2005,acceptedMarch21,2005.CommunicatedbyShun-JenCheng.MathematicsSubjectClassification:16D80,16S36.Keywordsandphrases:Reducedmodule,p.q.-Baermodule. MuhittinBas¸erandAbdullahHarmanciIn[12]Lee-Zhouintroducedthefollowingnotation.Foramodule,weex;
]=
si=0mixi:s0,miM,M[[x;
]]==x,xxx,x]]=EachoftheseisanAbeliangroupunderanobviousadditionoperation.More-More-x;
]becomesamoduleoveroverx;
]underthefollowingscalarproductoperation:ForForx;
]andf(x)=ti=0aixiR[x;
]m(x)f(x)=s+tk=0i+j=kmi
i(aj)xk.Similarly,,x;
]]ismoduleoveroverx;
]].Themodulesmodulesx;
]andM[[x;
]]arecalledtheskewpolynomialextensionandtheskewpowerseriesextensionofrespectively.If,thenwithasimilarscalarproduct,product,x,xxx,x)becomesamoduleoveroverx,xxx,x).TheThex,xxx,xarecalledtheskewLaurentpolynomialandtheskewLaurentpowerseriesextensionof,respectively.Firstwerecallthefollowingtheorem.Theorem1.[12,Theorem1.6]Thefollowingareequivalentforamodule-reduced;educed;x;
]R[x;
]isreduced;educed;x;
]]R[[x;
]]isreduced.If,thentheconditionsareequivalenttoeachofofx,xxx,xŠ1;
]isreduced;educed;x,xxx,xŠ1;
]]isreduced. Reducedandp.q.-BaerModulesAccordingtoLee-Zhou[12]amoduleiscalledifthefol-lowingconditions(1)and(2)aresatisfied,andmoduleiscalledofpowerseriestypeifthefollowingconditions(1)and(3)aresatisfied:(1)Forifandonlyif)=0(2)Foranyanyx;
]andf(x)=ti=0aixiR[x;
],m(x)f(x)=0)=0forall(3)Foranyanyx;
]]andf(x)=
i=0aixiR[[x;
]],m(x)f(x)=0)=0forallThemoduleiff-Armendariz;wecalldarizofpowerseriestype-Armendarizofpowerseriestype.If-reducedthen-Armendarizofpowerseriestype.IfofpowerseriestypethenForasubsetofamodule,let.In[12]Lee-ZhouintroducedBaermodules,quasi-Baermodulesand.-modulesasfollows.iscalledif,foranysubsetiscalledif,foranysubmoduleiscalledprincipallyprojective(orsimply)if,foranyInthispaper,westudyp.q.-modulesandthesomepolynomialandpowerseriesextensionsofp.q.-modules.Inparticular,weshow:(1)Forareducedmoduleisap.p.-moduleiffisap.q.-Baermodule.(2)Ifisanmodulewhereisanendomorphismof,thenisap.q.-Baermoduleifffx;
]R[x;
]isap.q.-Baermodule.(3)Foranarbitrarymoduleisap.q.-Baermoduleifandonlyififx]R[x]isap.q.-Baermodule.Webeginwiththefollowingdefinitionwhichisdefinedin[10].Definition2.beamodule.iscalledprincipallyquasi-Baerp.q.)moduleif,foranyItisclearthatisarightp.q.-Baerringiffisap.q.-Baermodule.Ifp.q.-Baerring,thenforanyrightidealisap.q.-Baermodule.Every MuhittinBas¸erandAbdullahHarmancisubmoduleofap.q.-Baermoduleisp.q.-Baermodule.Moreover,everyquasi-Baermoduleisp.q.-Baer,andeveryBaermoduleisquasi-Baer.iscommutativethenp.p.-moduleiffp.q.-Baermodule.Thefollowingexamplesshowthatthereexistsap.q.-Baermodulethatisnotap.p.Example3.[7,Example2(1)]Letbetheringofintegersandfullmatrixringover.Weconsidertheringd,bThenthemodulep.q.-Baer,butitisnotap.p.Theorem4.beamodulesuchthatforanymRa.Thenisap.p.-moduleifandonlyifisap.q.-Baermodule.Proof..Ifandbyassumption,mRaandso.Then.Butholds.Consequently,.Hencetheclaimfollows. Ournextresultextends[7,Lemma1].Corollary5.beareducedmodule.Thenisap.p.-moduleifandonlyifisap.q.-Baermodule.Proof.isareducedmodule.ThenM,aR,mamRaby[12,Lemma1.2].TheclaimfollowsfromTheorem4. Corollary6.[7,Lemma1]beareducedring.Thenisarightp.p.ifandonlyifisarightp.q.-Baerring.Theorem7.beanendomorphismofandassumethat,for)=0.Thenthefollowinghold:(1)((x;
]R[x;
]isap.q.-Baermodule,thenisap.q.-Baermodule.Theconverseholdsifinadditionreduced.educed.x;
]]R[[x;
]]isp.q.-Baer,thenisp.q.-Baer..x,xxx,xŠ1;
]isap.q.-Baermodule,thenisap.q.-Baermodule.Theconverseholdsifinadditionreduced. Reducedandp.q.-BaerModulesModulesx,xxx,xŠ1;
]]isap.q.-Baermodule,thenisap.q.-Baermodule.Proof.(1)(a)Similartotheproofof(1)(b).Converseof(1)(a):Assumethatisan-reducedmoduleandp.q.module.ForanymRa.ThenbyTheoremisap.p.-module.Sinceisan-reducedmodule,By[12,Theorem2.11.(1)(a)],1.(1)(a)],x;
]R[x;
]isp.p.-module.SinceSincex;
]R[x;
]isreducedbyTheorem1.ByCorollary5,5,x;
]R[x;
]isap.q.Baermodule.(1)(b)SupposeSupposex;
]]R[[x;
]]isap.q.-Baermodule.Forwehavehavex;
]](mR[[x;
]])==x;
]]wheref(x)2=f(x)R[[x;
]].ThusThusx;
]]rR[[x;
]](mR)=rR(mR)[[x;
]].ForForx;
]],mRbforallandhencemR
)=0forall,byassumption.Foranyanyx;
]],u(x)g(x)=ijui
i(bj)xi+j=0.Sog(x)rR[[x;
]]((mR)[[x;
]]))x;
]]==x;
]].Write,whereallThen,foranyforsomesomex;
]]sof(x)a=f(x)f(x)h(x)=f(x)h(x)=a.Itfollowsthatforall.Sop.q.-Baermodule.Nowtherestisclear(2)Similartotheproofof(1). Corollary8.ThefollowingholdforamoduleIfanyoneofofx]R[x],M[[x]]R[[x]],M[x,xxx,xŠ1]andM[[x,xxx,xŠ1]]isap.q.-Baermodule,thensoisbereduced.Ifisap.q.-Baer,thenbothbothx]R[x]andM[x,xxx,xŠ1]arep.q.-Baer.Corollary9.ThefollowingholdforaringIfanyoneofofx],R[[x]],R[x,xxx,xisarightp.q.-Baerring,thensoisbeareducedring.Ifisrightp.q.-Baer,thenbothbothx]andR[x,xarep.q.-Baerring. MuhittinBas¸erandAbdullahHarmanciExample10.Thereisareducedp.q.-Baermodulesuchthatthatx]]R[[x]]isnotap.q.-Baermodule.Proof.beafieldandbetheringiseventuallyconstantwhichisthesubringof,where,....Letdenotethe.Weclaimisap.q.-Baermoduleandreduced.ButButx]]R[[x]]isnotp.q.-Baermodule.Itiswellknownthatisap.q.-Baermoduleandreduced.denotethe”unitvector”,...,,...andlet,...···x]]R[[x]].Assumethatthatx]]R[[x]]isap.q.Baermodule.ThenThenx]](m(x)R[[x]])==x]]forsomeidempotentidempotentx]].Sinceiscommutativering,everyidempotentintheringringx]]belongstobyLemma8in[9].Hencebelongsto,sayNowitiseasytocheckthatthatx]](m(x)R[[x]])==x]]impliesrR(X)=f0R.ThisisnotpossiblebyExample7.54in[11].ThusThusx]]R[[x]]isnotp.q.module.Sinceisreducedreducedx]]R[[x]]isreducedbyTheorem1.ThereforeThereforex]]R[[x]]isnotp.q.-BaermodulebyCorollary5. Recallfrom[4],anidempotent),forall.Equivalently,isleft(resp.right)semicentralif)isanidealof.Ifisap.q.-Baermodule,thenisgeneratedbyaleftsemicentralidempotentbecauseisanideal.Weuseforthesetofallleftsemicentralidempotents.ThenexttheoremimprovedCorollary8forthepolynomialextensioncase.Theorem11.isap.q.-Baermoduleifandonlyififx]R[x]isap.q.Proof.isap.q.-Baermodule.LetLetx].Thereexistssuchthat,for,...,n....e.ThenThenx]rR[x](m(x)R[x]).ObserveObservex](m(x)R[x])rR[x](m(x)R).Leth(x)rR[x](m(x)R)andg(x)=b0+b1x+...+bkxkR[x].Then Reducedandp.q.-BaerModulesModulesx](m(x)R[x]).Consequently,,x](m(x)R[x])==x](m(x)R).Now,letletx](m(x)R).Since)=0wehavethefollowingsystemofequationswhereisanarbitraryelementofByfirstequation,,where.Letinequation(1).Then.But.Hence,where.Since,thenequation(1)yields.Hence.Takeinequation(2).Then.But=0=.Hence,soandsowehavebyequation(2)Inequation(2)substitutetoobtain.But,so.Thus,thenequation(2)yields.Hence.Summarizingatthispoint,wehaveR,a.Continuingthisprocedureyieldsforall,...,t..x].ConsequentlyConsequentlyx]=rR[x](m(x)R[x]).Conversely,iffx]R[x]isap.q.-Baer,thenisp.q.-BaerbyCorollary8(2). Corollary12.Assumethatisacommutativering.Thenisap.p.ifandonlyififx]R[x]isap.p.Proof.ThisisanimmediateconsequenceofTheorem11,sinceifiscom-mutativethenisap.p.-moduleifandonlyifisap.q.-Baermoduleandiscommutativeifandonlyififx]isacommutative. Corollary13.[4,Theorem3.1]isarightp.q.-Baerringifandonlyififx]isarightp.q.-Baerring. MuhittinBas¸erandAbdullahHarmanciCorollary14.[8,Theorem1.2]isacommutativering.Thenisap.p.-ringifandonlyififx]isap.p.Corollary15.[1,TheoremA]beareducedring.Thenisap.p.ifandonlyififx]isap.p.Corollary16.16.x]]R[[x]]isap.q.-Baermodule,thensoisProof.Thisresultfollowsfrom[4,Proposition2.5]andaproofsimilartothatusedinTheorem11. Corollary17.[4,Proposition3.5.]3.5.]x]]isarightp.q.-Baerring,thensoWewouldliketothankProfessorYiqiangZhoufromMemorialUniversityofNewfoundlandforhisvaluablesuggestionsandcommentsduringhisvisittotheModuleTheoryGroup,HacettepeUniversity,Ankara.WealsothanktoTurkishScientificandResearchCouncilforsupportZhou’svisit.1.E.P.Armendariz,AnoteonextensionsofBaerandp.p.J.AustralianMath.(1974),470-473.2.G.F.Birkenmeier,IdempotentsandCompletelySemiprimeIdeals,Comm.Algebra(1983),567-580.3.G.F.Birkenmeier,J.Y.Kim,J.K.Park,Onextensionsofquasi-Baerandprincipallyquasi-Baerrings,Preprint.4.G.F.Birkenmeier,J.Y.Kim,J.K.Park,OnPolynomialextensionsofprincipallyquasi-Baerrings,KyungpookMath.J.(2000),247-253.5.A.W.Chatters,C.R.Hajarnavis,RingswithChainConditions,Pitman,Boston,6.W.E.Clark,Twistedmatrixunitssemigroupalgebras,DukeMath.J.7.C.Y.Hong,N.K.KimandT.K.Kwak,OreExtensionsofBaerandp.p.PureAppl.Algebra(2000),215-226.8.S.Jøndrup,p.p.-Ringsandfinitelygeneratedflatideals,Proc.Amer.Math.Soc.(1971),431-435. Reducedandp.q.-BaerModules9.N.K.Kim,Y.Lee,ArmendarizRingsandReducedRings,J.Algebra10.M.T.Kos¸an,M.Bas¸erandA.Harmanci,Quasi-ArmendarizModulesandRings,11.T.Y.Lam,LecturesonModulesandRings,Springer-Verlag,NewYork,1999.12.T.K.LeeandY.Zhou,ReducedModules,Rings,modules,algebrasandabeliangroups,365-377,LectureNotesinPureandAppl.Math.,236,Dekker,Newyork,13.T.K.LeeandY.Zhou,ArmendarizandReducedRings,Comm.Alg.MuhittinBas¸erDepartmentofMathematics,FacultyofScienceandArts,KocatepeUniversity,ANSCampusTR-03200,Afyon-TurkeyE-mail:mbaser@aku.edu.trAbdullahHarmanciDepartmentofMathematics,HacettepeUniversity,Ankara-TurkeyE-mail:harmanci@hacettepe.edu.tr