ابوالفضل تاری زاده

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.