A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras

Let $R$ be a left and right Noetherian ring.

We introduce a Serre-type condition $(G_n)$, formulated in terms of the first occurrence and the flat dimension of indecomposable injective modules, and prove that $R$ is $n$-Gorenstein if and only if it satisfies $(G_n)$.

We then apply the criterion to Nakayama algebras via syzygy filtration.

It is shown that syzygy filtration preserves the Auslander-Gorenstein property and reflects it within the class of $2$-Gorenstein Nakayama algebras.

Combined with a result of Klász, Kleinau and Marczinzik, this shows that simple modules of odd grade over Auslander-Gorenstein Nakayama algebras are regular in their grade, thereby settling their conjecture.