A Dual Zariski Topology for Modules

TL;DR

This paper introduces a new dual Zariski topology on the spectrum of fully coprime submodules of a duo module over a ring, exploring its properties and relation to module algebraic features.

Contribution

It defines and studies a dual Zariski topology on the spectrum of fully coprime submodules, extending the topological framework for module spectra.

Findings

The dual Zariski topology is well-defined and exhibits interesting topological properties.

The topology's features are closely related to the algebraic structure of the module.

The work provides new insights into the interplay between topology and module theory.

Abstract

We introduce a dual Zariski topology on the spectrum of fully coprime $R$-submodules of a given duo module $M$ over an associative (not necessarily commutative) ring $R$. This topology is defined in a way dual to that of defining the Zariski topology on the prime spectrum of $R$. We investigate this topology and clarify the interplay between the properties of this space and the algebraic properties of the module under consideration.