On the existence of parameterized noetherian rings
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866910910798364672 |
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| author | Zhang, Xiaolei |
| author_facet | Zhang, Xiaolei |
| contents | A ring $R$ is called left strictly $(<\aleph_α)$-noetherian if $\aleph_α$ is the minimum cardinal such that every ideal of $R$ is $(<\aleph_α)$-generated. In this note, we show that for every singular (resp., regular) cardinal $\aleph_α$, there is a valuation domain $D$, which is strictly $(<\aleph_α)$-noetherian (resp., strictly $(<\aleph_α^+)$-noetherian), positively answering a problem proposed in \cite{Marcos25} under some set theory assumption. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_09822 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the existence of parameterized noetherian rings Zhang, Xiaolei Rings and Algebras A ring $R$ is called left strictly $(<\aleph_α)$-noetherian if $\aleph_α$ is the minimum cardinal such that every ideal of $R$ is $(<\aleph_α)$-generated. In this note, we show that for every singular (resp., regular) cardinal $\aleph_α$, there is a valuation domain $D$, which is strictly $(<\aleph_α)$-noetherian (resp., strictly $(<\aleph_α^+)$-noetherian), positively answering a problem proposed in \cite{Marcos25} under some set theory assumption. |
| title | On the existence of parameterized noetherian rings |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2504.09822 |