Graded Strongly Simple Injective Modules And Graded Strongly Simple Projective Modules

In this thesis,we mainly study the properties determined by graded strongly simple injective modules and graded strongly simple projective modules.A graded R-module N(resp.M)is said to be graded strongly simple injective(resp.graded strongly simple pro-jective)provided that EXTiR(S,N)=0(resp.EXTi R(M,S)=0)for any graded simple R-module S and positive integer i.In the definition above,if we only consider the case i=1,the graded R-module N(resp.M)is said to be graded simple injective(resp.graded sim-ple projective).Moreover,a series of equivalent characterizations for graded strongly simple injective modules and graded strongly simple projective modules are given.We prove that if R is a left graded Artinian ring or if R is a left graded Noetherian ring with the graded Krull dimension at most one,then a graded module N is graded injective if and only if N is graded simple injective;if and only if N is graded strongly simple injective.In addition,we also prove that l.gr.s.i.dim(R)=0 if and only if R is a graded semisimple ring,l.gr.s.p.dim(R)=0 if and only if R is a left graded V ring.