MI-Injective Module And MI-Injective Dimensions

The subject of relative homological algebra was introduced by S. Eilenberg and J.C. Moore in 1965. The research about the subject largely developed and extended the classical results about homological algebra. The theory of relative homological dimensions of modules and rings are the main research field in the subject. The existence of covers and envelopes, which introduced by Enochs in 1981, are basic researching objects of the subject.This thesis consists of four chapters.In chapter one, we recall some notions, backgrounds and facts needed in the sequel,and list the main results of this thesis.In chapter two, we first introduce the concept of MI-injective modules and MI-flat modules,and if R is a ring of left PS,the injective modules is equal to the MI-injective modules,the flat modules is equal to the MI-flat modules.If R is a left and right coherent ring,we have several equivalent conditions of the submodule of MI-injective modules is a MI-injective modules,that is the submodule of MI-injective modules is a MI-injective modules,if and only if fd((RR)+)≤1,if and only if every MI-flat module is a strong MI-flat module, if and only if every MI-injective modules is a strong MI-injective module, if and only if every quotient module of any MI-injective left R-module is MI-injective module.In chapter three,Then we investigate the.MI-injective dimensions of mod-ules and rings in terms of MI-injective and MI-flat modules and the derived functors Extn(-,-).Various characterizations of the dimension are given.right MI-dimM≤n, if and only if Extn+k(M, N)=Ofor all left R-module N and allk≥-1, if and only if Extn-1(M, N)=0 for all left R-module N.The left .MI-injective dimensions of modules have same characterizations,that is the left MI-dimN≤n-2,if and only if Extn+k(M, N)=Ofor all left R-module Mand all k≥-1, if and only if Extn-1(M, N)=Ofor all left R-module M.In chapter four, we introduce the concept of n-relative.MI-injective modules and n-relative.MZ-flat modules in terms of chapter two and chapter three,for a left R-module M,we can get that M is a n-relative MI-injective modules if and only if for every exact sequence 0→M→E→L→0 with E∈MIn,E→L is an MIn-precover of L,if and only if M is a kernel of an MIn-precover f:A→B with A injective,if and only if M is injective with respect to every exact sequence 0→A→B→C→0 with C∈MIn.If R is a left min-coherent ring and M a finitely presented right R-module. then M is n-relative.MI-flat if and only if M is a cokernel of an MFn-preenvelope K→F of a right R-module K with F flat.