Graded S-Noetherian Modules

Ajim Uddin Ansarı; B. K. Sharma

doi:10.24330/ieja.1229782

Research Article

Graded S-Noetherian Modules

Year 2023, Volume: 33 Issue: 33, 87 - 108, 09.01.2023

Ajim Uddin Ansarı B. K. Sharma

https://doi.org/10.24330/ieja.1229782

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

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