Strongly Simple Projective Modules,Strongly Simple Injective Modules And Their Homological Methods

In this thesis,we mainly study strongly simple projective mod-ules and strongly simple injective modules and the homological dimensions deter-mined by these two classes of modules.Let R be a ring,and M,N be R-modules,for any simple R-module S and positive integer i,if ExtRi(M,S)= 0,then M is calld strongly simple projective module.Correspondingly,if ExtRi(S,N)= 0,then N is calld strongly simple injective module.Let R be a,commutative noetherian ring,we prove every finitely generat-ed strongly simple projective module is projective module;If R is a noethe-rian domain ring and dim(R)= 1,we prove every strongly simple injec-tive module is injective module.Let R be a ring,we define the left simple projective global dimension(/.s.p.dim(R)),and the left simple injective global dimension(l.s.i.dim(R)).we prove l.s.p.dim(R)= 0 if and only if R is left V-ring.Also,s.i.dim(R)= 0 if and only if R is semisimple ring.At last,we define left simple projective hereditary rings and left simple injective hereditary rings,and we give the equivalent characterizations of them,prove left we prove a commutative semihereditary ring is simple projective hereditary;if(R,m)is commutative semihereditary and the maximal ideal m is finitely generated,then R is simple injective hereditary.