A note on the $S$-version of Noetherianity
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914011892678656 |
|---|---|
| author | Zhang, Xiaolei |
| author_facet | Zhang, Xiaolei |
| contents | It is well-known that a ring is Noetherian if and only if every ascending chain of ideals is stationary, and an integral domain is a PID if and only if every countably generated ideal is principal. We respectively investigate the similar results on $S$-Noetherian rings and $S$-$\ast_w$-PIDs, where $S$ is a multiplicative subset and $\ast$ is a star operation. In particular, we gave negative answers to the open questions proposed by Hamed and Hizem \cite{hh16}, Kim and Lim \cite{kl18}, and Lim \cite{l18} in terms of valuation domains, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_14781 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on the $S$-version of Noetherianity Zhang, Xiaolei Commutative Algebra It is well-known that a ring is Noetherian if and only if every ascending chain of ideals is stationary, and an integral domain is a PID if and only if every countably generated ideal is principal. We respectively investigate the similar results on $S$-Noetherian rings and $S$-$\ast_w$-PIDs, where $S$ is a multiplicative subset and $\ast$ is a star operation. In particular, we gave negative answers to the open questions proposed by Hamed and Hizem \cite{hh16}, Kim and Lim \cite{kl18}, and Lim \cite{l18} in terms of valuation domains, respectively. |
| title | A note on the $S$-version of Noetherianity |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2408.14781 |