Homological Modules And Generalized Homological Dimensions

In the first part, we study some properties of P-flat dimensions. Then we give out a new characterization of von Neumann regular rings by means of P-flat modules and character modules. Next we get a new classification of domain for the weak P dimension of domain is less than 1. At last we characterize weak P dimensions by their endomorphism .In the second part, we investigate generalized injective modules and obtained their several new properties. Next we investigate the relationship among generalized injective modules, generalized R-injective modules and injective modules, and obtain that every ideal of a domain R is generalized injective if and only if it is generalized R-injective if and only if it is injective. If R is a right principal ideal domain, then every right R-module is generalized injective if and only if it is injective. Generalized injective dimensions are also studied.In the last part, we introduce the concept of generalized projective moduleand investigate various properties of it.we show that (?)R_i is a generalized RP-rings if and only if so is R_i for i = 1,2, ... , n. Furthermore, we show that ifR is a generalized RP-rings, so is M_n(R) for all n ∈ N. Finally we introduce the concept of generalized projective dimensions and investigate some dual properties of generalized injective dimensions.