Gorenstein Projective, Injective And Flat Modules

Abstract

In basic homological algebra, projective,injective and flat modules play an important and fundamental role. Therefore, through-out this document we discuss some properties of Gorenstein projective, injective and flat modules and explain some relationships between Gorenstein projective, injective and flat modules. We also investigate some connections between Gorenstein projective, injective and flat modules under commutative noetherian rings. And also one of the open problems in Gorestein homological algebra is: when is the class of Gorenstein projective, Gorenstein injective and Gorenstein flat modules closed under arbitrary direct sums and arbitrary rings? Our main result gives a sufficient condition for this to happen.We state and prove that when the ring R is commutative and Noetherian ring and such that every R-module has finite Gorenstein projective, injective and flat dimension, every direct sum of Gorenstein projective, injective and flat modules are still Gorenstein projective, injective and flat module.