The F- Topology on Space of Zero Dimensional rings

Let R be a subring of a ring T , and let F be a non-principal ultraﬁlter on the natural numbers IN . We consider properties and applications of a countably compact, Hausdorﬀ topology called the “ F− topology” deﬁned on space of all zero-dimensional subring of T that contains a ﬁxed subring R . We show that the F− topology is strictly ﬁner than the Zariski topology. We extend results regarding distinguished spectral topologies on the space of zero-dimensional subring.

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 ). After that, we study the connection between Z(R, T ) and Spec(R), and modernize our understanding of the collection of all zerodimensional rings via the category of spectral spaces. As the Theorem 8 demonstrates, the tools are connected by a continuous surjection γ : Z(R, T ) τ F → Spect(R) τ F .

Preliminiries
We begin by giving the notations and preliminary results. Let R be a subring of a ring T , and let Z(R, T ) the set of all zero-dimensional subrings of T containing R, we know by [4,Proposition 2.2] that the set Z(R, T ) can be empty. The following theorem under conditions for which Z(R, T ) is a nonempty. 3. For each finitely generated ideal I, the set {Ann R (I j ) ∞ j=1 } stabilizes for some m ∈ N. Now, assume that Z(R, T ) = ∅, is that Z(R, T ) closed under arbitrary intersection? In the following theorem, R. Gilmer answers this question.
Theorem 2 ([9, Theorem 2.1]) Let R be a subring of a ring T such that Z(R, T ) = ∅ . Then Z(T ) is closed under arbitrary intersection. Remark 1 Suppose R is a subring of the ring T . If Z(R, T ) = ∅, then Theorem 2 shows that Z(R, T ) = ∅ has a unique minimal element. We denote this element by R 0 , and call it the minimal zero-dimensional extension of R in T . Then for each x in R, assume that x m(x) T is idempotent, and let s x be the pointwise inverse of x m(x) in T . By [9,Theorem 2.5] Now, we are interested in the topological structure on S(R, T ). Let R be a subring of a ring T , the set S(R, T ) endowed with a topological structure defined by taking, as a basis for the open sets, the subsets: . This topology is called the Zariski's topology on S(R, T ). If S := {x 1 , x 2 , ...., x n } with x j ∈ T for each j ∈ {1, ..., n}, then Therefore the collection of subsets B := {S(R[x], T ) : x ∈ T } is a base for the Zariski topology on S(R, T ). It is easily seen that S(R, T ) is a Kolmogorov topological space (T 0 space). Indeed, if R 1 = R 2 ∈ S(R, T ), we can assume, without loss of generality, that there is an element x ∈ R 1 \ R 2 , then the open set S(R[x], T ) contains R 1 and does not contain R 2 .
Corollary 1 Let R be a subring of a ring T , If Z(R, T ) = ∅, then Z(R, T ) is a Kolmogorov topological space.
We will work in at least ZFC, that is, Zermelo-Frankel set theory with the axiom of choice. If I is a set, we recall that F is a filter on I if it is a subset of the power set of I that satisfies the following conditions: A filter F on I is called an ultrafilter if F is maximal with respect to being a filter, or equivalently, if whenever A ⊂ I, then either A ∈ F or I \ A ∈ F. An ultrafilter F is called principal if there exists an element i 0 ∈ I such that F consist of all subsets of I that contain i 0 . In this case, we denoted F by F {i0} . Other ultrafilters are called non-principal. We denote the collection of all ultrafilters on a set I by β(I). Note that the notation β(I) is the same for the Stone-Čech compactification of I, this comes naturally, as en can identify both, for more details see the recent book of Comfort and Negrepontis [1]. Now, let R be a commutative ring, and let Spec(R) denote the set of all prime ideals of R. On Spec(R), we can consider the Zariski topology by taking as open sets the collection of all sets D(a) := {P ∈ Spec(R) : a / ∈ P } for all a ∈ R, then the family {D a : a ∈ R} is a basis for the open sets of Spec(R) zar . Zariski's topology has several attractive properties. For example is quasi-compact, Kolmogorov, but almost never compact. More precisely, Spec(R) zar is Hausdorff ⇔ Spec(R) zar is compact ⇔ dim(R) = 0 (see [7,Theorem 3.6]). 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 ideal. This notion of F−limit of collections of prime ideals has been used with central effect in the construction of the F-topology on Spec(R). If (P n ) n∈IN be a sequence of C ⊂ Spec(R) and, if F be a principal ultrafilter on IN , we have that F − lim n∈IN P n = P k , for some P k ∈ C [6, Section 2]. On the other hand, if F is nonprincipal, then it is not at all clear that the prime ideal F − lim n∈IN P n should lie in C. That motivates the following definition.
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. One of the main results of a recent article by Garcia-Ferreira and Ruza-Montilla is the following.
Let R be a subring of a ring T . Taking as starting point the situation on the prime spectrum of a ring, and under the condition, the next goal is to study the F−topology on space Z(R, T ). We begin by recalling a very useful definition.
Definition 2 Let T be a ring, and let I an infinite set. If R i ∈ S(T ), for each i ∈ I, and F is an ultrafilter on I, we define the F − lim of a sequence of subring R ′ i s as: We note that F − lim i∈I R i is also a subring of T , and we have that: Now, we state without proof some easy and well-known properties (For proof see [6]).

Proposition 1 Let
A be a set, I an infinite set, F an ultrafilter on I and {S i : i ∈ I} ⊆ S(A), then: 3. Let Γ be an infinite set, and let σ : ∆ → Γ be a surjective function. For The interest in studying the topology on Z(R, T ) comes from the following theorem.
Theorem 4 Let R be a subring of a ring T , and let Z(R, T ) = ∅, If R i ∈ Z(R, T ), for each i ∈ I, and F is an ultrafilter on I, then F − lim i∈I R i is also a zero-dimensional.
Proof According to [12,Proposition 4.2], F − lim i∈I R i is a direct union of zero-dimensional ring. Then the conclusion follows immediately by [11,Introduction].
Proposition 2 Let R be a subring of a ring T . If X ⊆ S(R, T ), I an infinite set, F an ultrafilter on I and {R i : i ∈ I} ⊆ X, let the map Then : π is a surjection if and only if for each F ∈ β(I), X stable by the F − lim.
Proof 1. For each R k , taking the principal ultrafilter F {k} , according to Proposition 1, we have F {k} − lim i∈I R i = R k 2. Noted that, if X = S(R, T ), then for each {R i : i ∈ I} collection of X, if F is an ultrafilter on I, we have F − lim i∈I R i ∈ X, According to Proposition 2, π be a surjection (the surjection comes from the fact that X is stable by F − lim),so to generalize the result, we take X stable by F − lim. Example 1 With the notation of the previous Proposition 2, and by Theorem 4, if we let X = Z(R, T ). Then π is a surjection.
The previous Proposition leads naturally to the following crucial definition of this section. Definition 3 Let R be a subring of a ring T , and let F be an ultrafilter on IN. We say that C ⊆ S(R, T ) is F−closed if for each sequence {R n } n∈IN in C, we have that F − lim n∈IN R n ∈ C We shall define a new topology on the set of subring of a commutative ring. 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.
Remark 2 Note that, by Proposition 2 we have F {k} − lim n∈IN R n = R k for each sequence (R n ) n∈IN of S(R, T ), then we deduce that τ F is the discrete topology.
Lemma 1 Let R be a subring of a ring T , and let F be a nonprincipal ultrafilter on IN. If {R n : n ∈ IN} ⊆ S(R, T ) is an infinite set, then F −lim n∈IN R n is an accumulation point of {R n : n ∈ IN} inside the topology τ F .
As we have introduced, we are interested in the spaces Z(R, T ), A(R, T ) and DU(R, T ), is like Z(R, T ) the largest closed for the topology τ F which contains A(R, T ) and DU(R, T ) (see Theorem 4), we restrict in the following to Z(R, T ). Naturally we start by comparing this topology with the usual topology on Z(R, T ).
Theorem 6 Let R be a subring of a ring T , and let Z(R, T ) = ∅.
1. The F−topology is finer than the Zariski topology on Z(R, T ); 2. The F−topology is hausdorff topology on Z(R, T ); 3. The F−topology is countably compact. , T ) is F−closed for every x ∈ T . Assume, by contradiction, that there exists an ultrafilter F on IN such that F −lim n∈N R n / ∈ C for each sequence (R n ) n∈IN in C. It follows that x ∈ F − lim n∈IN R n , then {n ∈ IN : x ∈ R n } ∈ F, by the definition of C, and the fact that ∅ / ∈ F, we have a contradiction. Hence the confirmation comes from the fact that the F−topology is finer than this topology. 3. In general, a Hausdorff space X is called countably compact if every infinite subset of X has an accumulation point. Then by (2) and Lemma 1, F−topology is countably compact.

Remark 3
The space Z(R, T ) τ F is countably compact but not compact in general.
Theorem 7 Let R be a subring of a ring T such that Z(R, T ) = ∅, and let F r the Frechet ultrafilter on IN . Then, the following conditions are equivalent: Proof 1) ⇒ 2). 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.
Proof According to [4,Remark], we can construct a map γ : Z(R, T ) → Spect(R) sending a zero dimensional ring S ∈ Z(R, T ), with a prime ideal Q, to the prime ideal Q ∩ R of R. If R is a zero dimensional, then γ is a surjection map.
Next, if we consider Z(R, T ) τzar and Spect(R) τzar as topological spaces both endowed with the Zariski topology, we check that γ is continuous, it is enough to show that γ −1 (D The next goal is to study the map γ when Z(R, T ) and Spect(T ) are both equipped with the F−topology.
Theorem 8 Let T be a ring and R a subring of T such that Z(R, T ) = ∅. Then, the surjective map γ : Z(R, T ) τ F → Spect(R) τ F is continuous and closed.
Proof According to Theorem 6, Z(R, T ) is Hausdorff space, by straightforward topological arguments, it is enough to show that is continuous. Let C be a F−closed subset of Spect(R) τ F , and let {S n : n ∈ IN } ⊆ γ −1 (C), F ultrafilter on IN . Then, it suffices to show that F − lim n∈N S n ∈ γ −1 (C). According to Theorem 4, we have F − lim n∈N S n ∈ Z(R, T ), then R ⊆ F − lim n∈N S n . On the other hand, for each (P n ) ∈ C we can also consider the ideal : F − lim n∈N P n = {a ∈ R {n ∈ IN : a ∈ P n } ∈ F} which is a prime ideal of R. By [4], there exists a prime ideal Q of F −lim n∈N S n such that, Q R = F − lim n∈N P n . Since, by [6], C is a F−closed subset, we have γ(F −lim n∈N S n ) = F −lim n∈N P n ∈ C, and so F −lim n∈N S n ∈ γ −1 (C). Therefore, we deduce that γ −1 (C) is closed Z(R, T ) τ F , hence the conclusion.