Gorenstein FC-Projective Modules And Gorenstein FP-Flat Modules

In this dissertation,we mainly study Gorenstein homological modules from two aspects base on Gorenstein FP-injective modules.As duals of(strongly)Gorenstein FP-injective modules,we introduce and investigate(strongly)Gorenstein FC-projective modules;as the character modules of Gorenstein FP-injective modules we introduce and investigate Gorenstein FP-flat modules.We characterize some impor-tant rings by these Gorenstein homological modules and investigate several properties under Morita Duality,in cotorsion theories and under almost excellent extensions of rings,respectivelyIn the first chapter,we introduce FC-projective modules and Gorenstein FC-projective modules,and obtain some equivalent characterizations of Gorenstein FC-projective modules.We give some connections between FC-projective modules and Gorenstein FC-projective modules and study the closeness of the class of Gorenstein FC-projective modules.Secondly,we define gFCP-closed rings and prove that a right co-coherent ring R is gFCP-closed if and only if the class of Gorenstein FC-projective modules is projectively resolving.Further,we introduce Gorenstein FC-projective dimensions of rings and modules,obtain some equivalent characterizations of the dimensions,and characterize Gorenstein FC-hereditary rings and Gorenstein FC-semisimple rings.Under an almost excellent extension of rings,we obtain some invariant properties of FC-projective modules,Gorenstein FC-projective modules and dimensions,Gorenstein FC-hereditary rings and Gorenstein FC-semisimple ringsOn the basis of the previous chapter,we introduce and investigate strongly Gorenstein FC-projective modules,study some connections among FC-projective mod-ules and(srongly)Gorenstein FC-projective modules,and show that every FC projec-tive right R-module is strongly Gorenstein FC-projective,and every strongly Gorenst-ein FC-projective right R-module is Gorenstein FC-projective.We also give examples to show that the converses are not true.Under a right gFCP closed right co-cohenrent ring R,we prove that every Gorenstein FC-projective module M is just a direct sum-mand of a strongly Gorenstein FC-projective module.Moreover,we introduce and characterize von Neumann co-regular rings,right strongly Gorenstein FC-semisimple rings and right strongly Gorenstein FC-hereditary rings.Under Morita duality,we ob-tain that FC-projective modules and FP-injective modules,(strongly)Gorenstein FC-projective modules and(strongly)Gorenstein FP-injective modules are Morita-dual pairs,and we further deal with Morita-dual properties of Neumann regular rings and Neumann co-regular rings,(strongly)Gorenstein FC-semisimple rings and(strongly)Gorenstein FP-semisimple rings,respectivelyIn the third chapter,we introduce Gorenstein FP-flat modules whose character modules are Gorenstein FP-injective modules and obtain several equivalent charac-terizations.We discuss the relations between Gorenstein FP-flat modules and each of Gorenstein FP-injecitve modules,flat modules and Gorenstein flat modules.And we prove that the class of Gorenstein FP-flat modules is closed under direct sum-s,pure submodules,pure quotient modules,extensions and direct limits.Moreover,we define gFPF-closed rings,which include strictly both coherent rings and rings with finite weak dimensions,and prove that the class of Gorenstein FP-flat modules is projectively resolving if and only if R is left gFPF-closed,and in this case the class of Gorenstein FP-flat modules is closed under direct summands.Over right gFPL-closed coherent rings7 we study cotorsion theory on the class of Gorenstein FP-flat modules,prove that(gFPF,gFPF?)is a complete and perfect cotorsion theory and obtain the existences of Gorenstein FP-flat covers and Gorenstein FP-flat pre-envelopes.Finally,under an almost excellent extension of rings,we give some invariant properties of Gorenstein FP-flat modules.