The Generalizations Of Injective Modules And The Rad-flat Modules

The injective modules and the flat modules play an important role in many parts of ring and categories of modules,so we introduce the notion of generalized direct-injective modules,rad-injective modules,and rad-flat modules and study theirs properties.Next,we research of the relation of the rad-flat modules and rad-injective modules.Finally,we use rad-flat modules and rad-injective modules to characterize some ring properities.In the first chapter,we give a generalization of injective modules,further,we introduce the notion of generalized direct-injective modules,and obtain some favorable properties.For example,we proved that⊕_i=1ⁿ M_i is direct-injective if and only if so is each M_i,(i=1,2,...,n).In chapter 2,we give another generalization of injective modules,we introduce the notion of rad-injective modules,obtain some properties,and proves that if M_R is a projective module,then every quotient of a rad-M-injective right R-module is rad-M-injective iff rad(M)is projective,and showed that a module M is strongly rad-injective iff M can be decomposed as a direct sum of an injective module and a module with zero radical.Finally,it is proved that a ring R is right Noetherian iff every direct sum of strongly rad-injective module is strongly rad-injective.In chapter 3,we give a a generalization of flat modules,and introduce the concepts of rad-flat modules,we will investigate the relation of the rad-flat modules and the rad-injective modules.Finally,we use rad-flat modules and rad-injective modules to characterize several useful rings.