Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
 Volume 18 (2023), 163 -- 182

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This work is licensed under a Creative Commons Attribution 4.0 International License.

S-NOETHERIAN RINGS, MODULES AND THEIR GENERALIZATIONS

Tushar Singh, Ajim Uddin Ansari and Shiv Datt Kumar

Abstract. 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.

2020 Mathematics Subject Classification: 13E10, 13C13, 13A15
Keywords: S-Noetherian ring, S-Noetherian module, S-Noetherian property.

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Tushar Singh
Department of Mathematics,
Motilal Nehru National Institute of Technology Allahabad,
Prayagraj(UP)- 211004, India.
e-mails: tushar.2021rma11@mnnit.ac.in, sjstusharsingh0019@gmail.com



Ajim Uddin Ansari
Department of Mathematics,
CMP Degree College, University of Allahabad
Prayagraj (UP)-211002, India.
e-mail: ajimmatau@gmail.com


Shiv Datt Kumar (corresponding author)
Department of Mathematics,
Motilal Nehru National Institute of Technology Allahabad,
Prayagraj(UP)- 211004, India.
e-mail: sdt@mnnit.ac.in

http://www.utgjiu.ro/math/sma