The Quasi-Zariski topology on the graded quasi-primary spectrum of a graded module over a graded commutative ring

TL;DR

This paper introduces a new topology called the Quasi-Zariski topology on the graded quasi-primary spectrum of a graded module over a graded commutative ring, exploring its properties and conditions for Noetherian and spectral space structures.

Contribution

It defines the Quasi-Zariski topology on the graded quasi-primary spectrum and investigates its topological properties and conditions for being Noetherian or spectral.

Findings

The topology is well-defined and exhibits interesting properties.

Conditions for the space to be Noetherian are established.

Criteria for the space to be spectral are provided.

Abstract

Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. A proper graded submodule $Q$ of $M$ is called a graded quasi-primary submodule if whenever $r\in h(R)$ and $m\in h(M)$ with $rm\in Q$, then either $r\in Gr((Q:_{R}M))$ or $m\in Gr_{M}(Q)$. The graded quasi primary spectrum $qp.Spec_{g}(M)$ is defined to be the set of all graded quasi primary submodules of $M$. In this paper, we introduce and study a topology on $qp.Spec_{g}(M)$, called the Quasi-Zariski Topology, and investigate properties of this topology and some conditions under which (% $qp.Spec_{g}(M)$, $q.\tau ^{g}$) is a Noetherian, spctral space.