Spectrum of A Ring - Zariski Topology

Zariski Topology

For any ideal I of R, define to be the set of prime ideals containing I. We can put a topology on Spec(R) by defining the collection of closed sets to be

This topology is called the Zariski topology.

A basis for the Zariski topology can be constructed as follows. For f∈R, define D_f to be the set of prime ideals of R not containing f. Then each D_f is an open subset of Spec(R), and is a basis for the Zariski topology.

Spec(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in R are precisely the closed points in this topology. However, Spec(R) is always a Kolmogorov space. It is also a spectral space.