A Zariski Topology for Modules

TL;DR

This paper introduces a Zariski topology on the spectrum of fully prime submodules of a duo module over a ring, exploring how topological properties relate to algebraic features of the module.

Contribution

It defines a new Zariski topology on the spectrum of fully prime submodules for duo modules over rings and studies its properties.

Findings

The topology reflects algebraic properties of the module.

Connections between topological and algebraic structures are established.

The spectrum's topological features correspond to module characteristics.

Abstract

Given a duo module $M$ over an associative (not necessarily commutative) ring $R,$ a Zariski topology is defined on the spectrum $\mathrm{Spec}^{\mathrm{fp}}(M)$ of {\it fully prime} $R$-submodules of $M$. We investigate, in particular, the interplay between the properties of this space and the algebraic properties of the module under consideration.