The Patch Topology and the Ultrafilter Topology on the Prime Spectrum of a Commutative Ring

Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. We define a topology on Spec(R) by using ultrafilters and demonstrate that this topology is identical to the well known patch or constructible topology. The proof is accomplished by use of a von Neumann regular ring canonically associated with R. Let R be a commutative ring and let Spec(R) denote the collection of prime ideals of R. On Spec(R) we can define a topology known as Zariski’s topology: the collection of all sets V (I) := {P ∈ Spec(R) | I ⊆ P} where I is an ideal of R constitutes the closed sets in this topology. Zariski’s topology has several attractive properties related to the geometric aspects of the study of the set of prime ideals [7, Chapter I]. For example, Spec(R) is always quasicompact (that is, every open covering has a finite refinement). On the other hand, this topology is very coarse. For example, Spec(R) is almost never Hausdorff (that is, two distinct points have nonintersecting neighborhoods). Many authors have considered a finer topology, known as the patch topology [9] and as the constructible topology ([8, pages 337-339] or [1, Chapter 3, Exercises 27, 28 and 30]), which can be defined starting from Zariski’s topology. Consider two collections of subsets of Spec(R). (1) The sets V (I) defined above for I an ideal of R. (2) The sets D(a) := Spec(R) \ V (a) where a ∈ R (where, as usual, V (a) denotes the set V (aR)).