Pseudo-injective Modules And DG-injective Modules

In this thesis, we mainly investigated the principle pseudo-injective modules and the DG-injective, DG-projective and DG-flat modules.The thesis is divided into four chapters. The first chapter is introduction. In this chapter, we introduced the development process of every conception.In chapter 2, we first gave out the equivalent definition of the pseudo-projective modules, thus the property domain is widened and pseudo-projective is closed under the direct sums and the direct summands. Some epimorphosis ended with them split. Some diagram commutes etc. Immediately again we discussed the uniqueness of the pseudo-projective covers in the meaning of isomorphism. The order is from the projective modules to the pseudo-projective modules and then to the modules with the pseudo-projective modules. In the first step, the cover is unique. But in the second step, it is not hold. So the way of the definition is unreasonable.Secondly, we defined the pseudo-injective modules in the way duals with the pseudo-projective modules and gives out several equivalent characterizations of the pseudo-injective modules which means the pseudo-injective modules is real dual conceptions of the pseudo-projective modules. And we also introduced the basic properties of them. We discovered the endomorphism rings of them have the same Jacobson radical with the injective modules and pointed out the basic reason is that every endomorphism epimorphosis splits.Finally, we put forward the conception of the principally pseudo-injective modules and gave some characterization of them use the elements in their endomorphism rings. Use the seven equivalent definitions of them, we discovered the relationship between them and their endomorphism rings.In chapter 3, we mainly investigated the relationships between two functors and found out the new conditions under which the Hom functor keeps the (?) functor.