The Zariski Topology on the Graded Primary Spectrum Over Graded Commutative Rings

Abstract

Let G be a group with identity e and let R be a G-graded ring. A proper graded ideal P of R is called a graded primary ideal if whenever rgsh∈P, we have rg∈ P or sh∈ Gr(P), where rg,sg∈ h(R). The graded primary spectrum p.Specg(R) is defined to be the set of all graded primary ideals of R.In this paper, we define a topology on p.Specg(R), called Zariski topology, which is analogous to that for Specg(R), and investigate several properties of the topology.

The Zariski topology on the graded second spectrum of a graded module

“…The Zariski-topology on some graded spectrums of graded rings and graded modules has been studied by several authors (see, for example, [3,4,10,11,14]).…”

Zariski Topologies on Graded Ideals

“…Mainly, the graded radical of a graded commutative ring, which has an important place in the graded ring theory, was characterized by using topological concepts. After that, the Zariski topology on graded modules has also attracted considerable attention of many authors, for example, [1], [3] and [4].…”