(m,n)-Injective Modules And Copure (m,n)-Injective Modules

This thesis mainly consists of three chapters.In chapter one, we mainly introduce some basic notions, backgrounds and some known facts needed in the sequel, and list the main results of this thesis.In chapter two, let R be a ring and m, n non-negative integers, we prove that if R is left (m,n)-coherent, then every left R-module has an (m,n)-injective cover. We introduce the concepts of (m,n)-pure submodules and (m, n)-pure exact sequence. A submodule T of a left a left R-module N is said to be an (m, n)-pure submodule of N if O→>A(?)T→A(?) N is exact for all (m, n)-presented right R-module A. An exact sequence O→A→B→C→O is called (m, n)-pure exact if A is an (m, n)-pure submodule of B. Next we give a characterization of VN regular rings and some characterizations of (m, n)-pure exact sequence. We prove that every left R-module has an (m, n)-injective cover over a left (m, n)-coherent ring, so every left R-module M has a left Im,n-resolution. It is proved that every left R-module has an (m, n)-injective preenvelope over any rings, so M has a right Im,n-resolution. We can define (m, n)-injective dimensions and left derived functors. Some characterizations of the dimensions are given.In chapter three, let R be a ring and m, n fixed nonnegative integers. Re-call that a left R-module M is called copure (m, n)-injective if Ext1 (N, M)=0 for any (m, n)-injective left R-module N. A right R-module F is called copure (m,n)-flat if Tor1(F,N)=0 for any (m,n)-injective left R-module N. Next we investigate properties of copure (m, n)-injective modules and copure (m, n)-flat modules furtherly. Let R be a ring, we prove that (CFm,n, CFm,n⊥) is a perfect cotorsion theory and (⊥CIm,n,CIm,n) is a complete cotorsion theory.