Graded S-Noetherian Modules

Abstract

Let $G$ be an abelian group and $S$ a given multiplicatively closed subset of a commutative $G$-graded ring $A$ consisting of homogeneous elements. In this paper, we introduce and study $G$-graded $S$-Noetherian modules which are a generalization of $S$-Noetherian modules. We characterize $G$-graded $S$-Noetherian modules in terms of $S$-Noetherian modules. For instance, a $G$-graded $A$-module $M$ is $G$-graded $S$-Noetherian if and only if $M$ is $S$-Noetherian, provided $G$ is finitely generated and $S$ is countable. Also, we generalize some results on $G$-graded Noetherian rings and modules to $G$-graded $S$-Noetherian rings and modules.

Keywords

• Noetherian module
• graded Noetherian module
• $S$-Noetherian module
• graded $S$-Noetherian module
• graded strong $S$-Noetherian module

References

1. D. D. Anderson and T. Dumitrescu, $S$-Noetherian rings, Comm. Algebra, 30(9) (2002), 4407-4416.
2. A. U. Ansari and B. K. Sharma, $G$-graded $S$-Artinian modules and graded $S$-secondary representations, Palest. J. Math., 11(3) (2022), 175-193.
3. J. Baeck, G. Lee and J. W. Lim, $S$-Noetherian rings and their extensions, Taiwanese J. Math., 20 (2016), 1231-1250.
4. Z. Bilgin, M. L. Reyes and Ü. Tekir, On right $S$-Noetherian rings and $S$-Noetherian modules, Comm. Algebra, 46(2) (2018), 863-869.
5. S. Goto and K. Yamagishi, Finite generation of Noetherian graded rings, Proc. Amer. Math. Soc., 89 (1983), 41-44.
6. A. Hamed and H. Sana, Modules satisfying the $S$-Noetherian property and $S$-ACCR, Comm. Algebra, 44(5) (2016), 1941-1951.
7. A. Hamed, $S$-Noetherian spectrum condition, Comm. Algebra, 46(8) (2018), 3314-3321.
8. R. Hazrat, Graded Rings and Graded Grothendieck Groups, London Math. Soc. Lecture Notes Series, v. 435, Cambridge University Press, Cambridge, 2016.

9. B. P. Johnson, Commutative Rings Graded by Abelian Groups, PhD Thesis. The University of Nebraska-Lincoln, 2012.
10. D. K. Kim and J. W. Lim, When are graded rings graded S-Noetherian rings, Mathematics, 8(9) (2020), 1532 (11pp).
11. D. K. Kim and J. W. Lim, The Cohen type theorem and the Eakin-Nagata type theorem for $S$-Noetherian rings revisited, Rocky Mountain J. Math., 50 (2020), 619-630.
12. M. J. Kwon and J. W. Lim, On nonnil-S-Noetherian rings, Mathematics, 8(9) (2020), 1428 (14pp).
13. . W. Lim, A note on $S$-Noetherian domains, Kyungpook Math. J., 55 (2015), 507-€“514.
14. J. W. Lim and D. Y. Oh, $S$-Noetherian properties on amalgamated algebras along an ideal, J. Pure Appl. Algebra, 218 (2014), 1075-1080.
15. J. W. Lim and D. Y. Oh, $S$-Noetherian properties of composite ring extensions, Comm. Algebra, 43 (2015), 2820-2829.
16. J. W. Lim and D. Y. Oh, Chain conditions on composite Hurwitz series rings, Open Math., 15 (2017), 1161-1170.
17. C. N\v{a}st\v{a}sescu and F. Van Oystaeyen, Graded Ring Theory, North-Holland Mathematical Library, 28, North-Holland Publishing Co., Amsterdam-New York, 1982. C. N\v{a}st\v{a}sescu and F. Van Oystaeyen, Graded rings with finiteness conditions II, Comm. Algebra, 13(3) (1985), 605-618.
18. C. N\v{a}st\v{a}sescu and F. Van Oystaeyen, Methods of Graded Rings, Lecture Notes in Math., 1836, Springer-Verlag, Berlin, 2004.
19. E. Noether, Idealtheorie in ringbereichen, Math. Ann., 83 (1921), 24-66.
20. M. Özen, O. A. Nazi, Ü. Tekir and K. P. Shum, Characterization theorems of $S$-Artinian modules, C. R. Acad. Bulgare Sci., 74(4) (2021), 496-505.
21. E. S. Sevim, Ü. Tekir and S. Koç, $S$-Artinian rings and finitely $S$-cogenerated rings, J. Algebra Appl., 19(3) (2020), 2050051 (16 pp).