Gorenstein Pure Injective Model

In this paper, firstly, (strongly)Gorenstein pure injective modules are introduced and invertigated. An R-module M is called (strongly)Gorenstein pure injective if there exists an exact sequence …→E1→Eo→E→E1→… of injective R-modules such that M≌ Im(E0→E0) and Hom(F,-) leaves the se-quence exact whenever F is pure injective (pdR(F)<∞). Some new characterizations of (strongly)Gorenstein pure injective module over special rings are obtained. Secondly, we prove that the class of Gorenstein pure injective modules is injectively resolving. i.e. The class of injective modules is contained in the class of Gorenstein pure injective modules, and for any short exact sequence 0→A→B→C→0 with A ∈gPI, then B ∈gPI if and only if C ∈gPI, where gPI stands for the class of Gorenstein pure injective modules. At last, we define three new homological dimensions GPid(M),rGPid(R) and rIGPid(R). and obtain the relation: rIGPid(R)< rGPid(R)< rD(R) where rD(R) stands for the right global dimension of ring R.