C-Projective, C-(FP)-Injective And C-Flat Modules

Semidualizing modules are common generalizations of dualizing modules and free modules of rank one. Foxby, Vasconcelos and Golod initiated the study of semidualiz-ing modules under different names, while Holm and White extended the definition of a semidualizing module to a pair of arbitrary rings. Especially, they defined the so-called C-projective, C-injective and C-flat modules, to characterize the Auslander class Ac(R) and the Bass class Bc(R), with respect to a semidualizing module C. The notion of C-projective (C-injective, C-flat) modules is fundamental and important for the study of relative homological algebra with respect to semidualizing modules. For example, dimensions, (pre)covers and (pre)envelopes induced by these modules are studied by several authors, and these modules are blocks for building the so-called C-Gorenstein projective, C-Gorenstein injective and C-Gorenstein flat modules.In this dissertation, firstly, we characterize some rings with C-projective, C-(FP)-injective and C-flat modules. Then we study (pre)envelopes and (pre)covers by the classes ICn, PCn,ICn⊥and PCn⊥, where ICn and PCn are two classes consisting of modules whose IC-injective dimension and PC-projective dimension are less than or equal to a natural number n, respectively. Finally, we extend the Foxby equivalence to the non-commutative non-noetherian setting, and discuss when these two pairs (ICn,ICn⊥) and (PCn,PCn⊥) are cotorsion theories.This paper is divided into four chapters.In chapter 1, we give the backgrounds and some preliminaries.In chapter 2, rings are assumed to be commutative. We first prove that for a module M, M is C-flat if and only if its character module M*=Homz(M, Q/Z) is C-FP-injective, then we characterize coherent rings and noetherian rings with C-injective and C-flat modules. Moreover we characterize coherent perfect rings and artinian rings with C-injective and C-projective modules.In chapter 3, all rings are still assumed to be commutative. We first show that every module over a noetherian ring admits a special ICn⊥-preenvelope, where ICn is the class of modules with IC-injective dimension less that or equal to n, then we show the existence of special ICn-precovers for every module under some conditions. Moreover, we show that the class ICn is covering when the ground ring is noetherian. After that we study the existence of special PCn⊥-preenvelopes and special PCn-precovers.In chapter 4. rings are arbitrary associative rings and SCR is a semidualizing bi-module. We first extend the Foxby equivalence to the non-commutative non-noetherian setting, then we prove that if SCR is projective over both sides, then (ICn,ICn⊥) is a cotor-sion theory if and only if (In,In⊥) is a cotorsion theory, where In is the class consisting of modules with injcctive dimension less than or equal to n. At the end of chapter we prove that (PCn,PCn⊥) is a cotorsion theory provided the semidualizing bimodule SCR is projective over both sides.