Attaching topological spaces to a module (I): Sobriety and spatial frames of submodules

TL;DR

This paper explores the topological and frame-theoretic structures associated with modules, focusing on semiprimitive submodules, sobriety, and regularity of related spatial frames, providing new insights into module topology.

Contribution

It introduces semiprimitive submodules and establishes their connection to spatial frames and the topology of maximal submodules, advancing the understanding of module-associated topological structures.

Findings

Semiprimitive submodules form an spatial frame isomorphic to Max(M)

Sobriety of Max(M) is characterized via the point space of the frame

Conditions for regularity of the spatial frame are studied

Abstract

In this paper we study some frames associated to an $R$-module $M$. We define semiprimitive submodules and we prove that they form an spatial frame canonically isomorphic to the topology of $Max(M)$. We characterize the soberness of $Max(M)$ in terms of the point space of that frame. Beside of this, we study the regularity of an spatial frame associated to $M$ given by annihilator conditions.