# Maximal Spectra of rings consisting of regulated functions

## TL;DR

This paper explores the properties of rings whose maximal spectra are nearly non-Hausdorff and totally disconnected, building on classical results about the topological structure of spectra of rings.

## Contribution

It investigates rings that nearly have non-Hausdorff and totally disconnected maximal spectra, extending the understanding of spectral topologies in ring theory.

## Key findings

- Constructed examples of rings with nearly non-Hausdorff spectra
- Characterized conditions for maximal spectra to be totally disconnected
- Linked spectral properties to ring-theoretic conditions

## Abstract

In 1969 H\"ochster proved that for every quasi-compact T1-space $X$ we can find a commutative ring $R$ such that $X$ is homeomorphic to the maximal spectrum $\mathrm{Specm}(R)$ of $R$. This result implies the existence of a commutative ring $R$ that admits a non-Hausdorff and totally disconnected maximal spectrum $\mathrm{Specm}(R)$. However, there has not been an example of such a commutative ring yet. The aim of this paper is to investigate rings that almost admit a non-Hausdorff and totally disconnected maximal spectrum.