# On strongly harmonic and Gelfand modules

## TL;DR

This paper introduces and studies the properties of strongly harmonic and Gelfand modules, generalizing classical ring concepts, and explores their lattice and topological structures, including conditions for compactness and Hausdorff properties.

## Contribution

It defines new classes of modules extending ring-theoretic notions and characterizes their structure via submodules and maximal submodules, connecting algebraic and topological perspectives.

## Key findings

- Strongly harmonic modules have a compact Hausdorff space of maximal submodules under certain conditions.
- The lattice of open sets of the maximal submodules space is isomorphic to a specific frame (M).
- The paper raises open questions about these modules' properties and applications.

## Abstract

We introduce the notions of Strongly harmonic and Gelfand module, as a generalization of the well-known ring theoretic case. We prove some properties of these modules and we give a characterization via their lattice of submodules and their space of maximal submodules. It is also observed that, under some assumptions, the space of maximal submodules of a strongly harmonic module constitutes a compact Hausdorff space whose frame of open sets is isomorphic to the frame $\Psi(M)$ defined in [arXiv:1612.07407]. Finally, we mention some open questions that arose during this investigation.