Some Studies On The Representation Theory Of Modules

In the first chapter,we introduce the history and development of the algebraic representation theory and the latest research subject.In the second chapter,We firstly give the concept and related properties of the Jacobson radical,and by using these properties and the method of partitioned matrices,we obtained the form of the radical of incidence algebra.The third chapter of this paper gives some basic conclusions of author on the study of the module category.Here,we firstly discussed the relationship between the module's exact sequence defined in the Abelian category and the one we say normally.Secondly,we have proved that the additive functor between Abelian categories is exact if and only if it is left exact and right exact.In the fourth chapter,we summarize the main characteristics of the dual functors,give structural theorem of the finite dimensional single modules,indecomposable projective modules and indecomposable injective modules.And we get the concept of the socle of module by using the dual functors.In the last chapter of this paper,we introduce the quiver representation theory of module,we proved the quiver representation form of the indecomposable projective modules and injective modules.At the same time,for a given quiver representation of a module,we have obtained the representations of its special submodules.