Classes Of Modules With Finite Flat Dimensions

As generalizations of three classes of projective, injective and flat modules, theclasses of modules with finite projective dimensions, the classes of modules with finiteinjective dimesions and the classes of modules with finite tint dimensions play animportant role in Ring Theory and Homological Algebra. In particular, these classesof modules and their Ext-orthogonal classes of modules are complete cotorsion pairsrespectively, and they catch many eyes in the research of cotorsion theory, tiltingand cotilting modules, and theory of covers and envelopes. In this paper, we use theclasses of (finitely presented) modules with finite fiat dimensions (i.e.,F_n and F_n^＜∞)to obtain Ext-orthogonal and Tor-orthogonal classes of modules by Ext and Torfunctors, which are used to characterize some important properties of rings, such asdimensions, coherency and hereditarity and so on. We also give a further study onthe existence of relative covers and envelopes of modules.This paper consists of four chapters.In chapter one, we introduce the research related to this paper and their trends,and sum up the groundwork of this paper.In chapter two, we use F_n and F_n^＜∞ to define F_n-injective and F_n^＜∞-injectivemodules and F_n-flat and F_n^＜∞-flat modules and give some basic properties of thesemodules. We characterizc the cases on wD(R)≤n. F_m=F_n, F_n= P by theseclasses of modules. We also study these classes of modules under a weak excellentextension.In chapter three, using the classes F_n and F_n^＜∞ and their Ext-orthogonalmodules and Tor-orthogonal modules, we define right F_n-coherent rings, right F_n-hereditary rings, right weak regular rings and right weak n-noether rings, and thenprovide their characterizations. Meanwhile, these rings are investigated under aweak excellent extension. In the last chapter, we study (pre)envelopes and (pre)covers of modules by theclasses F_n and F_n^＜∞ and their Ext-orthogonal modules and Tor-orthogonal modules,and characterize the exsistences of F_n -injective envelopes (rcsp. F_n-covers) withunique mapping property and those of (epimorphic) (F₁)^T₁-preenvelopes. At last,we prove that each module has a monic weak injective cover if and only if R is aright weak noether and right weak hereditary ring.