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Applied General Topology

c© Universidad Politécnica de Valencia

Volume 12, no. 1, 2011

pp. 15-16

The equality of the Patch topology and the

Ultrafilter topology: A shortcut

Luz M. Ruza and Jorge Vielma

Abstract

In this work R denotes a commutative ring with non-zero identity and

we prove that the patch topology and the ultrafilter topology defined

on the prime spectrum of R are equal, in a different way as the given

by Marco Fontana and K. Alan Loper in ([2]).

2010 MSC: 54E18, 54F65, 13C05.

Keywords: Patch topology, ultrafilter topology, prime spectrum of a ring.

1. Terminology and basic definitions

Let R be a commutative ring with non-zero identity. Spec(R) denotes the
set of all prime ideals of R. For every proper subset I of R, we denote by V (I)
the set of all prime ideals of R containing I, and D0(I) = Spec(R)−V (I). V (a)
will denote the set V (aR) and D0(a) the set D0(aR). The Zariski topology tz
on Spec(R) is the one that has as its closed sets those of the form V (I) ([1]).
The patch topology on Spec(R) is defined as the smallest topology having the
collections V (I) and D0(a) as closed sets. Let C be a subset of Spec(R), and
let Ω be an ultrafilter on C. It was shown in ([2]) that the set PΩ = {a ∈ R :
V (a) ∩ C ∈ Ω} is a prime ideal of R. The set C is said to be ultrafilter-closed
if for every ultrafilter Ω on C, PΩ ∈ C. The ultrafilter-closed sets define a
topology on Spec(R) called the Ultrafilter topology ([2]), and is denoted by τU .
In this work we prove that the patch topology and the ultrafilter topology are
equal, in a different way as the given by Fontana and Loper in ([2]).



16 L. M. Ruza and J. Vielma

2. The shortcut

Theorem 2.1. The Ultrafilter topology τU is compact.

Proof. Let U be a non principal ultrafilter in Spec(R). We want to prove that
U τU -converge to PU . Let θ be a τU -open set containing PU . Suppose that
A = θC belongs to U and let UA = {U ∩ A : U ∈ U} be the ultrafilter on A
induced by U. Since A is τU -closed, then PUA ∈ A. If a ∈ PU , V (a) ∈ U,
then V (a) ∩ A ∈ UA and therefore a ∈ PUA . Also, if b ∈ PUA it follows that
V (b) ∩ A ∈ UA and there exists U ∈ U such that V (b) ∩ A = U ∩ A. Since
U ∩ A ∈ U, then V (b) ∩ A ∈ U which implies that V (b) ∈ U and so b ∈ PU .
Therefore PU = PUA ∈ A which is a contradiction. �

Corollary 2.2. The Ultrafilter topology and the patch topology are equal.

Proof. Since the patch topology is Hausdorff ([3]), weaker than the ultrafilter
topology ([2]) and the well known fact that any compact topology does not
admit a weaker Hausdorff topology unless they are equal, the result follows. �

References

[1] M. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley,
1969.

[2] M. Fontana and K. A. Loper, The patch topology and the ultrafilter topology on the
prime spectrum of a commutative ring, Comm. Algebra 36 (2008), 2917–2922.

[3] M. Hochster, Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142
(1969), 43–60.

(Received November 2008 – Accepted September 2009)

Luz M. Ruza (ruza@ula.ve)
Universidad de los Andes, Departamento de Matemática, Merida, Venezuela

Jorge Vielma (vielma@ula.ve)
Universidad de los Andes, Departamento de Matemática, Merida, Venezuela


	The equality of the Patch topology and the[0.2cm] Ultrafilter topology: A shortcut. By L. M. Ruza and J. Vielma