Some Researches On Finitely Flat Presented Modules And Dimensions

In this thesis, we define and investigate F-n-presented modules and F-n-coherent rings, give finitely flat presented dimensions. For a general class (?), we also introduce (?)-presented modules and (?)-FP-injective modules, and characterize left-coherent rings.This thesis consists of three chapters.In Chapter One. we first use flat modules to define F-n-presented modules, give some properties of F-n-presented modules in exact sequences, obtain equivalent char-acterizations of F-n-presented modules. For a left coherent ring R we show that each n-presented module is F-n-presented if and only if the weak global dimension of R≤n. In particular, for an arbitrary ring R each finitely presented module is F-presented if and only if the weak global dimension of R≤1. Second, we use F-n-presented mod-ules to define left F-n-coherent rings and give some equivalent characterizations of a left F-1-coherent ring. For a left F-n-coherent ring, we use F-n-presented modules to characterize the projective dimension of modules. Under an almost excellent ring extension S≥R, we show that a left S-module SM is F-n-presented if and only if so is RM, and S is left F-n-coherent if and only if so is R. Dually, we introduce I-n-copresented modules and I-n-cocoherent rings, show their dualities under a Morita duality. For a IF ring R, we prove that if any n-1of the n modules in an exact sequence is F-presented, so is the remainder one.In Chapter Two, we first define the finitely flat presented dimensions of modules, give some relations with flat dimensions and projective dimensions, and investigate the relations of the finitely flat presented dimensions of modules in a short exact sequence. In particular, one of the modules in the sequence is a special module, we study the shifting of finitely flat presented dimensions. Second, we define the left finitely flat presented dimension of a ring R, obtain some relations among left global dimensions, weak global dimensions and left finitely flat presented dimensions, and give a classification of left F-coherent rings according to these dimensions. Finally, we provide a structure theorem of modules with the finitely flat presented dimension <l,i.e., generalized finitely flat presented modules.In Chapter Three, for a general class t we generalize finitely presented modules to t-presented modules, and investigate their orthogonal modules, i.e.,t-FP-injective modules. We also define left t-coherent rings, and use t-FP-injective modules to provide some equivalent characterizations of left t-coherent rings.