On Topological Lattices and an Application to First Submodules

TL;DR

This paper introduces the concept of topological lattices, explores their properties, and applies these ideas to the spectrum of first submodules of modules, linking algebraic and topological structures.

Contribution

It defines and characterizes topological lattices and applies this framework to study the spectrum of first submodules in modules, connecting algebraic and topological perspectives.

Findings

Topological lattice structures are characterized and analyzed.

The spectrum of first submodules is topologized and studied.

Topological properties of modules are linked to algebraic properties.

Abstract

We introduce the notion of a (strongly) topological lattice $\mathcal{L}=(L,\wedge ,\vee)$ with respect to a subset $X\subsetneqq L;$ aprototype is the lattice of (two-sided) ideals of a ring $R,$ which is(strongly) topological with respect to the prime spectrum of $R.$ We investigate and characterize (strongly) topological lattices. Given a non-zero left $R$-module $M,$ we introduce and investigate the spectrum $\mathrm{Spec}^{\mathrm{f}}(M)$ of \textit{first submodules} of $M.$ We topologize $\mathrm{Spec}^{\mathrm{f}}(M)$ and investigate the algebraic properties of $_{R}M$ by passing to the topological properties of the associated space.