Flat topology and its dual aspects

Abstract

In this article, a new and natural topology on the prime spectrum is introduced, 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 noetherianess 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.

Keywords:

• Constructible topology
• flat topology
• max-regular ideal

2010 MATHEMATICS SUBJECT CLASSIFICATION::

• 13A99
• 13C11
• 13B10

Acknowledgements

The author would like to give sincere thanks to Professor Marco Fontana for many valuable discussions which the author had with him while writing the present article and also for the information about their earlier work.