P-flat Modules, P-injective Modules And Some Rings

The flat modules play an important role in many parts of ring and categories of modules, so we introduce the notion of P-flat modules and study theirs properities. Next, some common rings are characterized by P-flat modules. At last, we give the definition of homological dimension of P-flat modules and P-injective modules. In first chapter, it's necessary to give a brief account of flat modules and some definitions which can be used in this paper. Secondly, to verify the introduction the notion of P-flat modules is existed and meaningful, we give an example to explain the P-flat modules is true generalization of flat modules and some equivalent conditions about P-flat modules in §1.2. In §1.3, we study the prosperities of P-flat modules, at same time , due to close relation of P-flat and P-injective modules, we also study the qualities of P-injective modules. As everyone knows, justly by means of deliberation of various modules and theirs homological dimension, people can obtain better description of rings. So in chapter 2, we use P-flat and P-injective modules to characterize some important rings, like SF-rings, Von Neumann regular rings and Coherent rings, etc. In chapter 3, we introduce the homological dimension of P-flat and P-injective modules. For the flat and injective homological dimension can be characterized by Tor and Ext, firstly we prove P-flat and P-injective homological dimension is also can be characterized by Tor and Ext if R is PQ ring which is defined in §3.1. Secondly, we give the prosperities of the P-flat and P-injective homological dimension in PQ rings.