Rings with uniformly S-w-Noetherian spectrum
Fuente:
arXiv
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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911137003470848 |
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| author | Zhang, Xiaolei |
| author_facet | Zhang, Xiaolei |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21403 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rings with uniformly S-w-Noetherian spectrum Zhang, Xiaolei Commutative Algebra 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. |
| title | Rings with uniformly S-w-Noetherian spectrum |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2508.21403 |