On pseudo-injective and pseudo-projective modules

In this work we obtain characterizations of $QI$ rings and semisimple rings using quasi-injective and pseudo-injective modules respectively. We define and construct the pseudo-injective hull of a module and we give sufficient conditions on a ring to have the following properties: every pseudo-injective module is pseudo-projective and every pseudo-projective module is pseudo-injective. We also give some properties of the big lattice of classes of modules being closed under submodules and quasi-injective hulls.