$S$-诺特环的模块理论表征

arXiv - MATH - Commutative Algebra Pub Date : 2024-08-27 DOI:arxiv-2408.14781

Xiaolei Zhang

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引用次数: 0

摘要

假设 $R$ 是一个环，$S$ 是 $R$ 的一个乘法子集。在本论文中，我们获得了 $S$-Noetherian 环的 ACC 特性和 Cartan-Eilenberg-Bass 定理。具体地说，我们证明了当且仅当 $R$ 的任何升序枚举链都是 $S$静态的，当且仅当任何注入模块的直接和都是 $S$注入的，当且仅当任何注入模块的直接极限都是 $S$注入的时候，一个环 $R$ 是一个 $S$-Noetherian环。

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A module-theoretic characterization of $S$-Noetherian rings

Let $R$ be a ring and $S$ a multiplicative subset of $R$. In this note, we obtain the ACC characterization and Cartan-Eilenberg-Bass theorem for $S$-Noetherian rings. In details, we show that a ring $R$ is an $S$-Noetherian ring if and only if any ascending chain of ideals of $R$ is $S$-stationary, if and only if any direct sum of injective modules is $S$-injective, if and only if any direct limit of injective modules is $S$-injective.

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arXiv - MATH - Commutative Algebra

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