V-Gorenstein injective modules preenvelopes and related dimension

Let R and S be associative rings and S V R a semidualizing (S-R)-bimodule. An R-module N is said to be V-Gorenstein injective if there exists a Hom R (I V (R),.) and Hom R (.,I V (R)) exact exact complex ..I1..d0I0.I0..d0I1.. of V-injective modules I i and I i, i . N0, such that N . Im(I 0 . I 0). We will call N to be strongly V-Gorenstein injective in case that all modules and homomorphisms in the above exact complex are equal, respectively. It is proved that the class of V-Gorenstein injective modules are closed under extension, direct summand and is a subset of the Auslander class A V (R) which leads to the fact that V-Gorenstein injective modules admit exact right I V (R)-resolution. By using these facts, and thinking of the fact that the class of strongly V-Gorenstein injective modules is not closed under direct summand, it is proved that an R-module N is strongly V-Gorenstein injective if and only if N . E is strongly V-Gorenstein injective for some V-injective module E. Finally, it is proved that an R-module N of finite V-Gorenstein injective injective dimension admits V-Gorenstein injective preenvelope which leads to the fact that, for a natural integer n, Gorenstein V-injective injective dimension of N is bounded to n if and only if Ext.n+1IV(R)(I,N)=0 for all modules I with finite I V (R)-injective dimension.