Flat and constructible topologies on prime spectrum

In this article, a new and natural topology on the prime spectrum is established which behaves completely as the dual of the Zariski topology. It is called the flat topology. The basic and also some sophisticated properties of the flat topology are proved. Specially, various algebraic characterizations for the noetherianness of the flat topology are given. Using the flat topology, then some facts on the structure of the prime ideals of a ring come to light which are not in the access of the Zariski topology.