FP-projective Modules And Strongly GFP-injective Modules

The thesis has investigated FP-projective modules and strongly GFP-injective modules. M is a R-module and M is called FP-projective if ExtR1(M, N)= 0 for any finitely presented module N; E is a strongly GFP-injective module if homo-morphisms:A→E can be extended to a homomorphism:B→E, where A is any finitely presented submodule of any R-module B. In chapter 2, we give some equiv-alent characterizations of FP-projective and strongly GFP-injective and discuss their basic prosperities. In chapter 3, we prove that every module is FP-projective if and only if every module is strongly GFP-injective; if and only if every finitely presented module is injective. It’s also prove that R is a left Noether ring, then every mod-ule is FP-projective (strongly GFP-injective) if and only if R is a semi-simple ring. However, let R is a left coherent ring, then every module is FP-projective (strongly GFP-injective) if and only if R is a von Neumann regular ring and R is a left self-injective ring. Then we consider the concept of left FP-hereditary rings. We prove that R is a left FP-hereditary ring if and only if every finitely presented module’s injective dimension is at most 1. Finally, we define the strongly left FP-projective dimension of modules and rings and prove that the strongly left FP-projective dimension of R is zero if and only if every module is FP-projective. The strongly left FP-projective dimension of R is at most 1 if and only if R is a left FP-hereditary ring.