A Zariski-like topology on the ideal spectrum of a ring

TL;DR

This paper introduces a new topology on the spectrum of all proper ideals of a ring, revealing its topological properties and differences from spectral spaces, especially in rings with non-trivial idempotents.

Contribution

It defines a Zariski-like topology on all proper ideals and analyzes its topological features, expanding the understanding of ideal spectra.

Findings

The space is T_0 and quasi-compact.

Every irreducible closed subset has a unique generic point.

If the ring has a non-trivial idempotent, the space contains a closed disconnected subspace.

Abstract

The purpose of this paper is to introduce a Zariski-like topology on the spectrum of all proper ideals of a ring. We show that the space is T_0, quasi-compact, and every irreducible closed subset has a unique generic point. Furthermore, this space is weaker than a spectral space and if the ring has a non-trivial idempotent element then the space has a closed disconnected subspace.