Hilbert basis theorem for generalized Noetherian rings
UDC 512.552 We explore the Hilbert Basis Theorem for uniformly $S$-Noetherian rings, $S$-$w$-Noetherian rings, and uniformly $S$-$w$-Noetherian rings. Moreover, we also show that uniformly $S$-($w$-)Noetherian rings may be not ($w$-)Noetherian even in the case where $S$ consists of nonzero divisors.
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| Date: | 2026 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2026
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/9329 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1866663700544356352 |
|---|---|
| author | Zhang, Xiaolei Zhang, Xiaolei |
| author_facet | Zhang, Xiaolei Zhang, Xiaolei |
| author_institution_txt_mv | [
{
"author": "Xiaolei Zhang",
"institution": "School of Mathematics and Statistics, Tianshui Normal University, Tianshui, China"
}
] |
| author_sort | Zhang, Xiaolei |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2026-05-30T12:44:48Z |
| description | UDC 512.552
We explore the Hilbert Basis Theorem for uniformly $S$-Noetherian rings, $S$-$w$-Noetherian rings, and uniformly $S$-$w$-Noetherian rings. Moreover, we also show that uniformly $S$-($w$-)Noetherian rings may be not ($w$-)Noetherian even in the case where $S$ consists of nonzero divisors. |
| doi_str_mv | 10.3842/umzh.v78i5-6.9329 |
| first_indexed | 2026-05-30T01:00:36Z |
| format | Article |
| fulltext | |
| id | umjimathkievua-article-9329 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-05-31T01:00:28Z |
| publishDate | 2026 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | |
| spelling | umjimathkievua-article-93292026-05-30T12:44:48Z Hilbert basis theorem for generalized Noetherian rings Hilbert basis theorem for generalized Noetherian rings Zhang, Xiaolei Zhang, Xiaolei Hilbert Basis Theorem uniformly S-Noetherian ring S-w-Noetherian ring commutative rings UDC 512.552 We explore the Hilbert Basis Theorem for uniformly $S$-Noetherian rings, $S$-$w$-Noetherian rings, and uniformly $S$-$w$-Noetherian rings. Moreover, we also show that uniformly $S$-($w$-)Noetherian rings may be not ($w$-)Noetherian even in the case where $S$ consists of nonzero divisors. УДК 512.552 Теорема Гільберта про базис для узагальнених нетерових кілець Досліджено теорему Гільберта про базис для рівномірно $S$-нетерових кілець, $S$-$w$-нетерових кілець і рівномірно $S$-$w$-нетерових кілець. Крім того, також показано, що рівномірно $S$-($w$-) нетерові кільця можуть не бути ($w$-) нетеровими навіть тоді, коли $S$ складається з ненульових дільників. Institute of Mathematics, NAS of Ukraine 2026-05-29 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/9329 10.3842/umzh.v78i5-6.9329 Ukrains’kyi Matematychnyi Zhurnal; Vol. 78 No. 5-6 (2026); 382 Український математичний журнал; Том 78 № 5-6 (2026); 382 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/9329/10665 Copyright (c) 2026 Xiaolei Zhang |
| spellingShingle | Zhang, Xiaolei Zhang, Xiaolei Hilbert basis theorem for generalized Noetherian rings |
| title | Hilbert basis theorem for generalized Noetherian rings |
| title_alt | Hilbert basis theorem for generalized Noetherian rings |
| title_full | Hilbert basis theorem for generalized Noetherian rings |
| title_fullStr | Hilbert basis theorem for generalized Noetherian rings |
| title_full_unstemmed | Hilbert basis theorem for generalized Noetherian rings |
| title_short | Hilbert basis theorem for generalized Noetherian rings |
| title_sort | hilbert basis theorem for generalized noetherian rings |
| topic_facet | Hilbert Basis Theorem uniformly S-Noetherian ring S-w-Noetherian ring commutative rings |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/9329 |
| work_keys_str_mv | AT zhangxiaolei hilbertbasistheoremforgeneralizednoetherianrings AT zhangxiaolei hilbertbasistheoremforgeneralizednoetherianrings |