S-Noetherian rings, modules and their generalizations
Let R be a commutative ring with identity, M an R-module and S ⊆ R a multiplicative set. Then M is called S-finite if there exist an s ∈ S and a finitely generated submodule N of M such that sM ⊆ N. Also, M is called S-Noetherian if each submodule of M is S-finite. A ring R is called S-Noetherian if...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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University Constantin Brancusi of Targu-Jiu
2023-09-01
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| Series: | Surveys in Mathematics and its Applications |
| Subjects: | |
| Online Access: | https://www.utgjiu.ro/math/sma/v18/p18_13.pdf |
| _version_ | 1797668614175719424 |
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| author | Tushar Singh Ajim Uddin Ansari Shiv Datt Kumar |
| author_facet | Tushar Singh Ajim Uddin Ansari Shiv Datt Kumar |
| author_sort | Tushar Singh |
| collection | DOAJ |
| description | Let R be a commutative ring with identity, M an R-module and S ⊆ R a multiplicative set. Then M is called S-finite if there exist an s ∈ S and a finitely generated submodule N of M such that sM ⊆ N. Also, M is called S-Noetherian if each submodule of M is S-finite. A ring R is called S-Noetherian if it is S-Noetherian as an R-module. This paper surveys the most recent developments in describing the structural properties of S-Noetherian rings, S-Noetherian modules and their generalizations. Some interesting constructed examples of S-Noetherian rings and modules are also presented. |
| first_indexed | 2024-03-11T20:31:55Z |
| format | Article |
| id | doaj.art-08b41aaa5fd7471d8fc19d6390cc32ff |
| institution | Directory Open Access Journal |
| issn | 1843-7265 1842-6298 |
| language | English |
| last_indexed | 2024-03-11T20:31:55Z |
| publishDate | 2023-09-01 |
| publisher | University Constantin Brancusi of Targu-Jiu |
| record_format | Article |
| series | Surveys in Mathematics and its Applications |
| spelling | doaj.art-08b41aaa5fd7471d8fc19d6390cc32ff2023-10-02T07:50:28ZengUniversity Constantin Brancusi of Targu-JiuSurveys in Mathematics and its Applications1843-72651842-62982023-09-0118 (2023)163182S-Noetherian rings, modules and their generalizationsTushar Singh0Ajim Uddin Ansari1Shiv Datt Kumar 2Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj(UP)- 211004, India.Department of Mathematics, CMP Degree College, University of Allahabad Prayagraj (UP)-211002, India.Department of Mathematics, Motilal Nehru National Institute of Technology Allahabad, Prayagraj(UP)- 211004, India.Let R be a commutative ring with identity, M an R-module and S ⊆ R a multiplicative set. Then M is called S-finite if there exist an s ∈ S and a finitely generated submodule N of M such that sM ⊆ N. Also, M is called S-Noetherian if each submodule of M is S-finite. A ring R is called S-Noetherian if it is S-Noetherian as an R-module. This paper surveys the most recent developments in describing the structural properties of S-Noetherian rings, S-Noetherian modules and their generalizations. Some interesting constructed examples of S-Noetherian rings and modules are also presented. https://www.utgjiu.ro/math/sma/v18/p18_13.pdfs-noetherian rings-noetherian modules-noetherian property |
| spellingShingle | Tushar Singh Ajim Uddin Ansari Shiv Datt Kumar S-Noetherian rings, modules and their generalizations Surveys in Mathematics and its Applications s-noetherian ring s-noetherian module s-noetherian property |
| title | S-Noetherian rings, modules and their generalizations |
| title_full | S-Noetherian rings, modules and their generalizations |
| title_fullStr | S-Noetherian rings, modules and their generalizations |
| title_full_unstemmed | S-Noetherian rings, modules and their generalizations |
| title_short | S-Noetherian rings, modules and their generalizations |
| title_sort | s noetherian rings modules and their generalizations |
| topic | s-noetherian ring s-noetherian module s-noetherian property |
| url | https://www.utgjiu.ro/math/sma/v18/p18_13.pdf |
| work_keys_str_mv | AT tusharsingh snoetherianringsmodulesandtheirgeneralizations AT ajimuddinansari snoetherianringsmodulesandtheirgeneralizations AT shivdattkumar snoetherianringsmodulesandtheirgeneralizations |