New methods toward the patch and flat topologies with applications

TL;DR

This paper introduces simplified methods for studying patch and flat topologies on prime spectra of rings, providing new proofs and extending results on minimal and maximal spectra's properties.

Contribution

It offers new, elementary approaches to patch and flat topologies, simplifying existing proofs and deriving finiteness results for minimal and maximal spectra.

Findings

Minimal spectrum is quasi-compact in flat topology

Unified approach to finiteness of minimal primes

Simplified proofs of classical topological results

Abstract

In this paper, we use elementary and simple ideas which are based on the significant applications of the power set ring to rebuild and study the patch topology on the prime spectrum from a completely different and new point of view. Specially, the proof of a major result in the literature on the comparison of topologies greatly simplified and shortened. Then we develop more natural and simple methods to obtain the flat topology and its properties. In particular, it is shown that the minimal spectrum of a commutative ring is quasi-compact with respect to the flat topology. Then as an application of this result, all of the related results of Kaplansky, Anderson, Gilmer-Heinzer, Bahmanpour-Khojali-Naghipour and Naghipour on the finiteness of the minimal primes are deduced as special cases of this result. Dually, a similar result is also obtained for maximal ideals.