The Generalizations Of Injective Modules And The P-injective Radical Of Module

Modules on ring are genetalizations of ideals and linear spaces. In 19th century, Dirichlet considered the modules on polynomial rings;in 20th century, E.Noether emphasized the importance of modules on the study of algebra. When it came to 1940s, with the springing up of homological algebra, module theory went a step further. In this paper, we further research P-injective modules, define the concept of prime injective modules and prime injective dimension of module, and discuss the properties of them. Moreover, we define the P-injective radical of modules and obtain some properties of them. Injective modules play an important role in module theory and homological algebra. In recent years, lots of authors have generalized the concept of injective module(ref[3],[5],[20],[34]), In [3], P-injective modules are defined, and properties of them are studied, we call P-module M right n-injective module, if for any right ideal Iof R generated by n elements of R and R-morphism α : I→ M can be extended to R-morphism β : R→ M. [3] proved that if M_n(R) is right injective, then R is right n-injective ring .In [5], the concept of P-injective modules is extended.In chapter 2, we focus on the further study of P-injective modules. R is called F-injective ring, if for any finitly generated submodule Mof R-module .R_n ,and R-morphism f : M → R can be extended to , We proved that M_n(R) is right P-injective iff R is F-injective ring .Moreover,we obtianed that a quotient module of a P-injective is p-injective iff R is a right PP-ring. We proved that if Baer-ring R is strong AGP-injective ring, then R is π—regular ringChapter 3 introduces the concept of prime injective modules and prime injective dimension of R-module M,noted by pid(M),and study modules whose dimension is 0 and 1. the relationship between injective modules and prime injective modules arealso discussed here.On the base of [9,10], we define p-injective Radical of a Module X,denotd by L(X), in Chapter 4 ,and discuss the properties of L(X). Some results on von Neummann regular ring are obtianed here,If R is von Neummann regular and X is a /?-modulue,then J{X) = L(X).At last ,we defined the concept of L-rings and prove that semisimple rings are L-Rings.injective module, Baer criterion, p-injective module,p-injective Radical of module, prime injective module, prime injective dimension...