Gorenstein Injective Covers and Envelopes over Rings That Satisfy the Auslander Condition

It was recently proved [17] that the class of Gorenstein injective left R-modules is both covering and enveloping over a two-sided noetherian ring R with the property that the character modules of the Gorenstein injective left R-modules are Gorenstein flat. It was also proved that over the same type of rings, the class of Gorenstein flat right R-modules is preenveloping [16]. We prove here that if R is a two-sided noetherian ring R such that R if satisfies the Auslander condition and has finite finitistic left injective dimension, then R has the desired property; the character module of any Gorenstein injective is Gorenstein flat.