On Coherence Dimensions Of Rings

Serval properties of rings and modules are characterized by finitely presented dimension and n-presented dimension. In this thesis, we generalize the dimensions from two directions as follows. On the one hand, we increase the number of items of finitely generated projective modules in the definition of finitely presented dimen-sions, and obtain finitely generated projective dimensions. On the other hand, we shift the position of finitely presentation to the (n-1)th syzygy and obtain right coherence dimension of rings.This thesis consists of four chapters.In chapter one, we introduce researches related to this thesis and their trends, and sum up the groundwork of this thesis.In chapter two, we define the finitely generated projective dimensions of mod-ules and rings, give some relations of right global dimension rgD(R), weak global dimensions wD(R) and finitely generated projective dimensions rFPD(R) of rings, and combine the n-presented dimensions of rings to give a classification of rings or right n-coherent rings. At last, we study some properties of finitely generated projective dimension under an almost excellent extension of rings.In chapter three, we define right coherent dimension rCoh.D(R) of rings, that is, if every (n-1)th syzygy of right R-modules is2-presented whenever it is finitely presented then rCoh.D(R)≤n. Using the class SnFP and its Ext, Tor-orthogonal modules, we characterize right coherent dimension rCoh.D(R) of rings, and inves-tigate right SnFP-hereditary rings. We also give some properties of SnFP-injective modules. SnFP-fiat modules, right coherent dimension of rings and SnFP-hereditary rings under an almost excellent extension of rings.In chapter four, we use the classes of SnFP-injective modules and SnFP-flat modules to investigate preenvelopes and precovers of modules, and obtain some equivalent conditions between the existence of monic (epic) preenvelopes and pre-covers and right coherent dimensions of rings, SnFP-hereditary rings.