FI-gr-injective Modules And FI-gr-flat Modules

In this thesis,we introduce and study Fl-gr-injective modules and Fl-gr-flat modules.Firstly,we give the characterizations of Fl-gr-injective modules and Fl-gr-flat mod-ules.It is proved that graded ring R is a graded QF ring if and only if every gr-injective left R-module is gr-projective,if and only if every graded left R-module is Fl-gr-injective,and that R is a left graded IF ring if and only if every graded pure-injective left R-module is FI-gr-injective,if and only if every graded right R-module is Fl-gr-flat,if and only if every finitely presented graded right R-module is Fl-gr-flat.Then,we give the concepts of strongly FI-gr-injective modules and strongly Fl-gr-flat modules and investigate their basic properties.Finally,we study the Fl-gr-injective dimension and the Fl-gr-flat dimension of graded rings and graded modules and show that R is a graded QF ring if and only if l.gr-fiD(r)=0.In addition,we also prove that graded commutative ring r is a graded IF ring if and only if gr-ffD(r)=0.