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.
