On torsion Gorenstein injective modules

summary:In this paper, we define Gorenstein injective rings, Gorenstein injective modules and their envelopes. The main topic of this paper is to show that if $D$ is a Gorenstein integral domain and $M$ is a left $D$-module, then the torsion submodule $tGM$ of Gorenstein injective envelope $GM$ of $M$ is also Gorenstein injective. We can also show that if $M$ is a torsion $D$-module of a Gorenstein injective integral domain $D$, then the Gorenstein injective envelope $GM$ of $M$ is torsion