The ℱ-Limit of a Sequence of Prime Ideals

Abstract

In this article, we introduce new topologies on the prime spectrum of a commutative ring by using the -limit of a sequence of prime ideals where is a nonprincipal ultrafilter on the natural numbers . These topologies are strictly finer than the Zariski's Topology, countably compact and Hausdorff. We construct 2 new topologies on Spec which are pairwise non-homeomorphic.

The F- Topology on Space of Zero Dimensional rings

“…Recently, S. Garcia-Ferreira and L.M. Ruza-Montilla in [6] have considered another topology on Spec(R) by using the notion of an ultrafilter. Indeed, let (P n ) n∈IN be a sequence of Spec(R), and let F be an ultrafilter on IN , set…”

“…It is not hard to see that the F−closed subsets of Spec(R) define a topology on the set Spec(R), called F−toplogy on Spec(R) (see [6,Theorem 4.2]). We denote by Spec(R) τ F the set Spec(R) endowed with the F−topology.…”

“…Proof The empty set and S(R, T ) are clearly F−closed subsets. Now, consider two F−closed subsets C 1 , C 2 of S(R, T ), set C := C 1 C 2 , and let (R n ) n∈IN be a sequence in C. By an argument similar to that used in [6,Theorem 4.2], we have F − lim n∈N R n ∈ C. It is evident that the intersection of F−closed subsets is a F−closed subset.…”

“…According to [2, Theorem 3.10.3], every countably compact, countable Hausdorff space is compact, then Z(R, T ) τ F is compact. Inspired by the idea given in as in [6], by [12,Lemma 4.4] Proposition 3 Let R be a subring of a ring T such that Z(R, T ) = ∅. Then γ : Z(R, T ) τzar → Spect(R) τzar is a continuous map.…”

“…The purpose of this paper is twofold: Defined the F−topology on Z(R, T ), and to study the connection between Spec(R) and Z(R, T ) by different topology. In Section 2, we give some basic propriety of Z(R, T ), we also introduce the notion of the ultrafilter and note that S. Garcia-Ferreira and L. M. Ruza-Montilla recently used ultrafilters to define this topology on the set of all prime ideals of a commutative ring, and then prove that this F−topology is a countably compact, Hausdorff topology (see [6]). In Section 3, we define a F−topology on the space Z(R, T ).…”