Zariski Topologies for Coprime and Second Submodules

TL;DR

This paper explores the topological structures of spectra of second and coprime submodules in modules over rings, analyzing their properties and relationships with algebraic features of the modules.

Contribution

It introduces Zariski-type topologies on spectra of second and coprime submodules and studies their properties and algebraic implications.

Findings

Topologies on spectra are well-defined and exhibit interesting properties.

Relationships between topological features and module algebraic properties are established.

The spectra provide new insights into the structure of modules over rings.

Abstract

Let $M$ be a non-zero module over an associative (not necessarily commutative) ring. In this paper, we investigate the so-called \emph{second} and \emph{coprime} submodules of $M.$ Moreover, we topologize the spectrum $% \mathrm{Spec}^{\mathrm{s}}(M)$ of second submodules of $M$ and the spectrum $% \mathrm{Spec}^{\mathrm{c}}(M)$ of coprime submodules of $M,$ study several properties of these spaces and investigate their interplay with the algebraic properties of $M.$