Abstract

In this article, we use quite elementary and simple ideas to rebuild and study the patch and flat topologies on the prime spectrum from a natural point of view (this new approach is based on the significant applications of the power set ring). Especially, the proof of a major result in the literature on the comparison of topologies greatly simplified and shortened. Also a new characterization for the finiteness of the minimal primes of a ring is given. Then as an application, all of the related results of Kaplansky, Anderson, Gilmer-Heinzer, Bahmanpour-Khojali-Naghipour and Naghipour on the finiteness of the minimal primes are easily deduced as special cases of this result. Another finiteness result due to Matlis is also easily obtained which states that a given ring has finitely many minimal primes if and only if no minimal prime is contained in the union of the remaining minimal primes.

摘要

在本文中，我们使用非常基本和简单的想法从自然的角度重建和研究素谱上的贴片和平坦拓扑（这种新方法基于幂集环的重要应用）。特别是，关于拓扑比较的文献中的主要结果的证明大大简化和缩短。还给出了环的最小素数的有限性的新表征。然后作为应用，Kaplansky、Anderson、Gilmer-Heinzer、Bahmanpour-Khojali-Naghipour 和 Naghipour 关于极小素数的有限性的所有相关结果很容易推导出为这个结果的特例。