Proper spaces are spectral

TL;DR

This paper demonstrates that the set of proper ideals of a ring, equipped with the coarse lower topology, forms a spectral space, contributing to the understanding of spectral spaces in algebraic geometry.

Contribution

It establishes that proper ideals with the coarse lower topology constitute a spectral space, extending Hochster's work on spectral spaces.

Findings

Proper ideals form a spectral space under the coarse lower topology.

The result connects algebraic structures with topological properties.

Enhances understanding of spectral spaces in ring theory.

Abstract

Since Hochster's work, spectral spaces have attracted increasing interest. Through this note we intend to show that the set of proper ideals of a ring endowed with coarse lower topology is a spectral space.