Relative FI-injective Modules And Relative FI-flat Modules

As generalizations of three classes of projective, injective and flat modules, the classes of modules with finite projective dimensions, finite injective dimensions and finite flat dimensions play an important role in modules and ring theory and homologicalalgebra. In particular, these classes of modules and their Ext-orthogonal classes of modules are complete cotorsion pairs respectively, and they catch many eyes in the research of cotorsion theory and theory of envelopes and covers. In this thesis, we use the classes of modules with finite FP-injective dimensions (i.e., FP_n) to obtain Ext-orthogonal and Tor-orthogonal classes of modules by Ext and Tor functors, and study deeply some properties and equivalent characterizations related to these modules. In this basis, we characterize the dimensions of modules and rings, study envelopes and covers of modules by applying cotorsion theory.This thesis consists of four chapters.In chapter one, we introduce the background of the research related to this thesis, sum up the groudwork of this thesis, then list some necessary basic concepts and notions in this thesis.In chapter two, we use the classes of modules with finite FP-injective dimensions{i.e.,FP_n) to introduce the concepts of n-FI-injective modules and n-FI-flat modules for a fixed nonnegative innteger n, and give some basic properties and exchangerelations of these modules on different rings. On the other hand, using the classes of FP-injective modules to define FI-injective dimensions of module and rings by Ext functor, which is different from the definitions of the traditional homologicaldimensions by resolutions. Meanwhile we give basic properties of FI-injective dimension, and obtain equivalent characterizations of FI-injective dimension and FI-injective dimension at most n on coherent ring. In chapter three, we study envelopes and covers of n-FI-injective modules and n-FI-flat modules by the classes of modules with finite FP-injective dimensions (i.e.,FP_n) and the classes of modules with finite flat dimensions (i.e.,F_n), and so on, and give the existing property of F_n-preenvelope on left coherent ring, in this basis, we study the cokernel of F_n-preenvelope, FP_n-precover and FF_n-cover, and give the equivalent characterizations of n-FI-injective modules and n-FI-flat modules.In the last chapter, we study the cotorsion theory of left coherent ring with FP-id(_RR)≤n, prove that (FP_n,FI_n) is a complete cotorsion theory, and (FF_n, F_n~⊥) is a hereditary cotorsion theory. Then we use F_n-envelope and FP_n-cover to give the equivalent characterizations of the reduced n-FI-injective modules and wD(R)≤n of coherent ring by applying cotorsion theory, at last we give some equivalent characterizations of semihereditary ring and semisimple Artinian ring.