关于\(n -\)戈伦斯坦环的塞尔型准则及其在中山代数中的应用
A Serre-type criterion for $n$-Gorenstein rings and its application to Nakayama algebras
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中文总结 AI 辅助
研究关于左右诺特环的塞尔型条件\((G_n)\)，证明\(R\)是\(n -\)戈伦斯坦环当且仅当满足\((G_n)\)。通过合冲过滤将该准则应用于中山代数，保持并反映奥斯兰德 - 戈伦斯坦性质，解决了相关猜想。
中文摘要 AI 辅助
设\(R\)为左右诺特环。我们引入一个用不可分解内射模的首次出现和平坦维数表述的塞尔型条件\((G_n)\)，并证明\(R\)是\(n -\)戈伦斯坦环当且仅当它满足\((G_n)\)。然后通过合冲过滤将该准则应用于中山代数。结果表明合冲过滤保持奥斯兰德 - 戈伦斯坦性质，并在\(2 -\)戈伦斯坦中山代数类中反映该性质。结合克拉斯、克莱瑙和马尔钦齐克的结果，表明奥斯兰德 - 戈伦斯坦中山代数上奇数级的单模在其级中是正则的，从而解决了他们的猜想。
英文摘要
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.
