On Auslanderʹs n-gorenstein rings Original Research Article

According to Auslander, a Noetherian ring R is called n-Gorenstein for n ≥ 1 if in a minimal injective resolution 0 → RR → E0 → E1 → … → En →, …, the flat dimension of each Ei is at most i for i = 0, 1, …, n − 1. We prove that for an n-Gorenstein ring R of self-injective dimension n, the last term En in a minimal injective resolution of RR has essential socle. We also prove that the 1-Gorenstein property is inherited by a maximal quotient ring, and as a related result, we characterize a Noetherian ring of dominant dimension at least 2.