Ext-FI-injective Modules And MPI-injective And MPI-flat Modules

We introduce and study several new classes of modules over a ring R, and give their characterizations and properties. These extend some classical results.In the first part, we first defines Gorenstein F I-injective and strongly Goren-stein F I-injective modules, we get that the class of Gorenstein F I-injective mod-ules is injectively resolving and the class of strongly Gorenstein F I-injective mod-ules is not injectively resolving. Then we introduce and investigate injectively resolving class between injective modules and strongly Gorenstein F I-injective modules, which is called Ext-F I-injective modules, based on the ralations in-jective modules C strongly Gorenstein F I-injective modules C Gorenstein FI-injective modules. Then we introduce the concept of weak Gorenstein F I-injective module and give their properties.In the second part, we define MPI-injective modules and MPI-flat mod-ules based on max-P-injective modules and max-P-flat modules, and study their properties under max-P-coherent rings and max-P-hereditary rings.