The Zariski topology on the secondary like spectrum of a module

TL;DR

This paper introduces a Zariski topology on a new spectrum of secondary submodules of a module, extending the second spectrum with a specific annihilator condition, and explores its topological properties.

Contribution

It defines the secondary-like spectrum of a module and establishes a Zariski topology on it, generalizing the second spectrum with new structural insights.

Findings

The topology extends the Zariski topology on the second spectrum.

Several topological properties of the new spectrum are studied.

The new spectrum generalizes existing spectral concepts in module theory.

Abstract

Let $R$ be a commutative ring with unity and $M$ be a left $R$-module. We define the secondary-like spectrum of $M$ to be the set of all secondary submodules $K$ of $M$ such that $Ann_R(soc(K))=\sqrt{Ann_R(K)}$, and we denote it by $Spec^L(M)$. In this paper, we introduce a topology on $Spec^L(M)$ having the Zariski topology on the second spectrum $Spec^s(M)$ as a subspace topology, and study several topological structures of this topology.