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

TL;DR

This paper introduces a Zariski topology on the graded primary spectrum of a graded module over a graded ring, extending the classical prime spectrum topology to a broader setting and analyzing its properties.

Contribution

It defines the graded primary spectrum and establishes a Zariski topology on it, generalizing the prime spectrum topology for graded modules.

Findings

The topology on the graded primary spectrum is well-defined and exhibits interesting topological properties.

The Zariski topology on the graded primary spectrum contains the graded prime spectrum as a subspace.

The paper provides foundational results for the topological structure of graded primary spectra.

Abstract

Let $R$ be a $G$-graded ring and M be a $G$-graded $R$-module. We define the graded primary spectrum of $M$, denoted by $\mathcal{PS}_G(M)$, to be the set of all graded primary submodules $Q$ of M such that $(Gr_M(Q):_R M)=Gr((Q:_R M))$. In this paper, we define a topology on $\mathcal{PS}_G(M)$ having the Zariski topology on the graded prime spectrum $Spec_G(M)$ as a subspace topology, and investigate several topological properties of this topological space.