The Researches On The N-P-injective Modules And The N-flat Modules

The injective modules and the flat modules play an important role in many parts of ring and categories of modules.We introduce,in thesis,the notion of the n-P-injective modules,the n-flat modules,and the n-Pm-injective modules,and study theirs properties.Next,we research the relations among the n-P-injective modules and the n-flat modules and theirs homological dimension.Finally,we use the n-P-injective modules and the n-flat modules to characterize some useful rings.In the first chapter,we introduce the background of the research related to this thesis,and stun up the groudwork of this thesis.In Chapter 2,we study the properties of the n-P-injective modules and the n-flat modules.At first,we investigate the equivalent characterizations of the n-P -injective modules in terms of the P-jnjective modules,and prove the divisibility of the n-P-injective modules.After that,we study several characterizations of the n-flat modules in terms of the P-flat modules and the relation between the n-P-injective modules and the n-flat modules.In Chapter 3,we investigate the relations among the n-P injective modules,the n-flat modules and some rings.At first,We use the divisibility of the n-P-injective modules to give an new equivalent characterization of Dedekind rings.Also we get an equivalent characterization of the n-P-injective on the domain.Further,we study the relation among the n-flat modules and the Pr(u|¨)fer rings,the von Neumann regular rings,the n-coherent rings.In Chapter 4,we give an important generalization of the n-P-injective modules, and introduce the concepts of the n-Pm-injective modules,and then give an equivalent characterization of the generalized P-mP rings in terms of the gpm-injective modules and an equivalent characterization of the n-generalized P-mP rings in terms of the n-Pm-injective modules.Finally,we get an equivalent characterization of the n-regularity rings in terms of the n-Pm-injective modules.In Chapter 5,the homological dimension of n-P-injective modules and n-flat modules are studied.At first,we consider the homological dimensions of n-P-injective modules and n-flat modules,then we point out that the n-flat(n-P-injective) homological dimension is also can be characterized by Tor(Ext)if R is nPQ ring,and we give the equivalent characterization of nPQ ring,then we give the prosperities of n-P-injective and n-flat homological dimension in nPQ rings.