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arXiv · 2508.21403

Rings with uniformly S-w-Noetherian spectrum

Xiaolei Zhang
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Abstract

In this paper, the notion of rings with uniformly S-w-Noetherian spectrum is introduced. Several characterizations of rings with uniformly S-w-Noetherian spectrum are given. Actually, we show that a ring R has uniformly S-w-Noetherian spectrum with respect to some s 2 S if and only if each ascending chain of radical w-ideals of R is stationary with respect to s 2 S, if and only if each radical (prime) (w-)ideal of R is radically S-w-finite with respect to s, if and only if each countably generated ideal of R is radically S-w-finite with respect to s, if and only if R[X] has uniformly S-w-Noetherian spectrum, if and only if RfXg has uniformly S-Noetherian spectrum. Beside, we show a ring R has Noetherian (resp., uniformly S-Noetherian) spectrum with respect to s if and only if each countably generated ideal of R is radically finite (S-finite with respect to s), which is a new result in the classical case.

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Xiaolei Zhang. 2025-08-29. Rings with uniformly S-w-Noetherian spectrum. https://arxiv.org/abs/2508.21403

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