Abstract
The Eakin–Nagata theorem examines the condition that the Noetherian property passes through each other between subrings and extension rings in 1968. Later, a noncommutative version of Eakin–Nagata theorem was also proved. Lam called this version Eakin–Nagata–Eisenbud theorem. In addition, Anderson and Dumitrescu introduced the $S$-Noetherian concept which is an extended notion of the Noetherian property on commutative rings in 2002. In this paper, we consider the $S$-variant of Eakin–Nagata–Eisenbud theorem for general rings by using $S$-Noetherian modules. We also show that every right $S$-Noetherian domain is right Ore, which is embedded into a division ring. For a right $S$-Noetherian ring, we obtain sufficient conditions for its right ring of fractions to be right $S$-Noetherian or right Noetherian. As applications, the $S$-variant of Eakin–Nagata–Eisenbud theorem is applied to the composite polynomial, composite power series and composite skew polynomial rings.
Funding Statement
The first author acknowledges the support of this research work by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2019R1F1A1059883), the second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2021R1I1A1A01041451), and the third author contributed equally to the work as the co-first author.
Acknowledgments
The authors are very thankful to the referee for the many helpful suggestions and valuable comments.
Citation
Gangyong Lee. Jongwook Baeck. Jung Wook Lim. "Eakin–Nagata–Eisenbud Theorem for Right $S$-Noetherian Rings." Taiwanese J. Math. 27 (2) 237 - 257, April, 2023. https://doi.org/10.11650/tjm/221101
Information
