Affiliation:
1. University of Allahabad
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.
Publisher
The International Electronic Journal of Algebra
Subject
Algebra and Number Theory
Reference21 articles.
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.