Quasi-prime submodules and developed Zariski topology

TL;DR

This paper introduces quasi-prime submodules and a new Zariski topology on modules, exploring their properties and relationships with algebraic and topological characteristics of modules over commutative rings.

Contribution

It defines the developed Zariski topology on quasi-prime spectra of modules and investigates its properties and connections to module algebraic structures.

Findings

Characterizations of modules with $T_0$, irreducible, or Noetherian developed Zariski topology

Relationships between algebraic properties of modules and topological properties of $q\Spec(M)$

Introduction of the developed Zariski topology on quasi-prime submodules

Abstract

Let $R$ be a commutative ring with nonzero identity and $M$ be an $R$-module. Quasi-prime submodules of $M$ and the developed Zariski topology on $q\Spec(M)$ are introduced. We also, investigate the relationship between the algebraic properties of $M$ and the topological properties of $q\Spec(M)$. Modules whose developed Zariski topology is respectively $T_0$, irreducible or Noetherian are studied, and several characterizations of such modules are given.