The patch topology and the ultrafilter topology on the prime spectrum of a commutative ring

TL;DR

This paper introduces a topology on the prime spectrum of a commutative ring using ultrafilters and proves its equivalence to the classical patch topology, employing von Neumann regular rings for the proof.

Contribution

It establishes the ultrafilter topology on Spec(R) as identical to the patch topology, providing a new perspective and proof method using von Neumann regular rings.

Findings

Ultrafilter topology coincides with the patch topology on Spec(R)

The proof uses a canonical von Neumann regular ring associated with R

Provides a new approach to understanding the topology on prime spectra

Abstract

Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. We define a topology on Spec(R) by using ultrafilters and demonstrate that this topology is identical to the well known patch or constructible topology. The proof is accomplished by use of a von Neumann regular ring canonically associated with $R$.