Abstract
According to Auslander, a Noetherian ring R is called n-Gorenstein for n ≥ 1 if in a minimal injective resolution 0 → R R → E 0 → E 1 → … → E n →, …, the flat dimension of each E i 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 E n in a minimal injective resolution of R R 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.
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