Graded modules with Noetherian graded second spectrum

TL;DR

This paper studies the properties of graded modules over G-graded rings with Noetherian graded prime spectra, introducing the graded Zariski socle and analyzing the topological structure of the graded second spectrum.

Contribution

It introduces the concept of the graded Zariski socle and explores the topological properties of the graded second spectrum in Noetherian graded rings.

Findings

Characterization of graded second spectrum as a Noetherian space

Introduction of graded Zariski socle and its properties

Analysis of topological structure of $Spec_G^s(M)$

Abstract

Let $R$ be a $G$ graded commutative ring and $M$ be a $G$-graded $R$-module. The set of all graded second submodules of $M$ is denoted by $Spec_G^s(M)$ and it is called the graded second spectrum of $M$. In this paper, we discuss graded rings with Noetherian graded prime spectrum and obtain some conclusions. In addition, we introduce the notion of the graded Zariski socle of graded submodules and explore their properties. Using these conclusions and properties, we also investigate $Spec_G^s(M)$ with the Zariski topology from the viewpoint of being a Noetherian space and give some related outcomes.