Research On Some Classes Of Objects And Related Problems In Abelian Category

In the study of algebra,projective modules and injective modules and their generalizations are widely used.Compared with other categories,the abelian category is a more important category,The application of abelian category and homological algebra are very extensive.In this paper,we study the properties of several kinds of objects in abelian category,mainly on the basis of projective objects and injective objects,we study the properties of the objects of Strong FP-injective objects,Strong FP-projective objects,n-FP-injective objects,Strong n-FP-injective objects,FI-injective objects,nFI-injective objects and（n,d）-projective objects.The details are as follows:In the first part,we give the definitions and properties of strong FP-injective objects and strong FP-projective objects in abelian category;and we show that the strong FP-injective object is closed under extension,products and cokernels of monomorphism;then we further discuss the strong FP-injective objects in the coherent abelian category,and prove that if the abelian category is n-coherent,then the FP injective object is equivalent to the strong FP-injective objects;finally,we give the cotorsion theory of the strong FP-projective object.In the second part,we give the definitions of n-FP-injective objects,strongly n-FPinjective objects and（n,d）-projective objects on an abelian category and study their related properties.Firstly,we give the definition and related properties of n-FP-injective objects;then we give the definition of strongly n-FP-injective objects,and discuss the relationship between n-FP-injective objects and strong n-FP-injective objects;at last,the definition of（n,d）-projective objects in an abelian category and some equivalent propositions are given in abelian category.In the third part,we introduce the definitions of FI-injective objects and n-FIinjective objects in an abelian category,and study their related properties.Firstly,we give the definition of FI-injective objecta,and discuss the relationship between FIinjective objecta and injective objecta;secondly,we give the definition of n-FI-injective objects,and give the equivalent proposition about n-FI-injective objects;finally,we introduce the concept of an n-FC category and study the properties of n-FI-injective objects in an n-FC category.