On Injectivity of Certain Gorenstein Injective Modules

In this note, we will be concerned with injectivity of Gorenstein injective modules over certain rings. Speci cally, we will show that if R is a complete local d-Gorenstein ring and M is a Gorenstein injective R-module possessing a syzygy Kn, n d such that ?Kn K? n = Add (Kn) [ Inj(R), then M is injective. This is particularly related to the dual notion of the famous Auslaner-Reiten Conjecture recently posed.