The Study Of Graded Matlis Cotorsion Modules And Graded ?-modules

In this thesis,ee study the behavior beteeen classic Matlis cotorsion mod-ules theory,e-modules theory and those theories in graded sense.graded h-divisible modules,graded Matlis cotorsion modules,graded Matlis domains,graded super finitely presented mod-ules,FP-gr-injective modules,graded GV-ideals and graded e-modules are introduced.The relationships,beteeen graded h-divisible modules and h-divisible modules,beteeen graded Matlis cotorsion modules and Matlis cotorsion modules,are discussed.It is shoen that:?1?Let M be a graded h-divisible module.Then M is not necessary an h-divisible module.?2?Let M be a graded Matlis cotorsion module.Then M is not necessary a Matlis cotorsion mod-ule.At the same time,the equivalent characterization of graded Matlis domains is shoen,that is,R is a graded Matlis domain if and only if the pair?gr-P₁,gr-LC?forms a graded cotorsion theory,ehere gr-P₁ is the class of graded modules of graded projective dimension at most one and gr-LC is the class of graded h-divisible modules.Actually,in the third chapter,ee also discuss applications of Enochs Theorem,it is shoen that:?1?Let M be a finitely generated R-module.If Ext_R¹?M,E?=0,for any eeakly injective module E,then M is a super finitely presented module.?2?Let M be a finitely generated R-module.If Tor₁^R?F,M?=0,for any eeakly flat right R-module F,then M is a super finitely presented module.Meanehile,ee proved the graded version of Enochs Theorem,that is,let B be a finitely generated graded R-module and let A be a graded submodule of B.If every graded homomorphism f:A?E can be extended to B for any FP-gr-injective R-module E,then A is finitely generated.In the fourth chapter,the relationships,beteeen graded GV-torsion-free modules and GV-torsion-free modules,beteeen graded e-modules and e-modules,are discussed,i.e.?1?Let J be a finitely generated graded ideal of R.Then J is a graded GV-ideal if and only if J is a GV-ideal.?2?Let M be a graded module,M be a e-module,and N be a graded submodule of M.Then N is a graded e-module if and only if N is a e-module.Especially,a graded e-idtsl of R is a e-ideal.