PI-Injective Modules And Strongly Copure-Injective Modules

This thesis is made up of three chapters:In Chapter 1, the notions of PI-injective module and PI-projective module are introduced. With the help of these objects, we prove that the class of PI-injective mod-ules and the class of PI-projective modules can be constructed to be a completely cotor-sion pair. Furthermore, we define the weakly perfect rings and give some new charac-terizations for some well-known rings, such as von Neumann regular rings, noetherian rings and semisimple Artinian rings.In Chapter 2, the cotorsion pair constructed in Chapter 1 is hereditary if and only if every pure right ideal I of R is PI-projective is proved at first. Furthermore, PI-projective dimension and PI-injective dimension for modules and rings are defined respectively and it turns out that PI-projective dimension and PI-injective dimension measure how far away a ring is from being von Neumann regular and weakly perfect respectively provided that every pure right ideal of R is PI-projective.In Chapter 3, the existence of the strongly copure injective cover is proved over a commutative artinian ring at first. Furthermore, we discuss the copure injective dimen-sions of modules and rings in terms of the left derived functor of Hom.