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When is CF(X)=MβX∖I(X) ?
Topology and its Applications 194 (2015) 22–25
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
When is CF (X) = M βX\I(X) ? F. Azarpanah a , M. Ghirati b,∗ , A. Taherifar b a b
Department of Mathematics, Shahid Chamran University, Ahvaz, Iran Department of Mathematics, Yasouj University, Yasouj, Iran
a r t i c l e
i n f o
Article history: Received 9 May 2015 Received in revised form 22 July 2015 Accepted 29 July 2015 Available online xxxx MSC: 54C40
a b s t r a c t This short article is in fact an erratum to Theorem 2.5 in Ghirati and Taherifar (2014) [6]. In this paper, we prove that J(C(X)/CF (X)) = 0 or equivalently, the socle CF (X) of C(X) coincides with the intersection of all essential maximal ideals of C(X) if and only if every infinite subset of I(X) contains a closed infinite subset, if and only if every pseudocompact subset of X has at most a finite number of isolated points. This fact shows that part (2) of aforementioned Theorem 2.5 is not correct. Our results in this paper also amend Theorem 5.6 in Azarpanah et al. (2008) [2]. Examples are provided to illustrate and delimit our results. © 2015 Elsevier B.V. All rights reserved.
Keywords: Essential ideal Socle Pseudocompact Round set
1. Introduction We denote by C(X) (C ∗ (X)) the ring of (bounded) real-valued, continuous functions on a completely regular Hausdorff space X. Whenever C(X) = C ∗ (X), then X is called pseudocompact. υX is the Hewitt ˘ realcompactification of X, βX is the Stone–Cech compactification of X and for any p ∈ βX, the maximal p p ideal M (resp., the ideal O ) is the set of all f ∈ C(X) for which p ∈ clβX Z(f ) (resp., p ∈ intβX clβX Z(f )). More generally, for A ⊆ βX, M A (resp. OA ) is the intersection of all M p (resp. Op ) with p ∈ A. If C(X)/M p ∼ = R, then M p is called real, else hyper-real and υX is in fact the set of all p ∈ βX such that M p is real. For each f ∈ C(X), the zero-set Z(f ) is the set of zeros of f , X \ Z(f ) is the cozero-set of f and clX (X\Z(f )) is called the support of f . An ideal I in C(X) (C ∗ (X)) is called fixed if Z[I] = f ∈I Z(f ) = ∅, else is free. A subset A of X is said to be relatively pseudocompact if each member of C(X) is bounded on A. The ideal Cψ (X) in C(X) is the set of all functions with pseudocompact support. It is well-known that Cψ (X) is the set of all functions with relatively pseudocompact support and it also coincides with the intersection of all hyper-real maximal ideals in C(X), i.e., Cψ (X) = M βX\υX , see [9]. * Corresponding author. E-mail addresses: [email protected] (F. Azarpanah), [email protected], [email protected] (M. Ghirati), [email protected], [email protected] (A. Taherifar). http://dx.doi.org/10.1016/j.topol.2015.07.015 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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An ideal in a ring R with identity, is said to be essential if it intersects every nonzero ideal of R nontrivially. It is well-known that an ideal E in C(X) is essential if and only if intX Z[E] = ∅, see [1]. This shows that all free ideals are essential and every fixed maximal ideal Mx , where x is not isolated point, is also an essential ideal in C(X). Hence the intersection of all essential maximal ideals of C(X) is M βX\I(X) , where I(X) is the set of all isolated points of X. Recall that for any ring R with identity the socle of R is the intersection of all essential ideals of R which is also equal to the sum of all minimal ideals of R. We denote the socle of C(X) by CF (X) and it is topologically characterized in [8] as the set of all functions which vanish everywhere except on a finite number of points of X. The reader is referred to [5,7] and [10] for undefined terms and notations. 2. Improvements in two published results In this section, we give a theorem which shows that part (2) of Theorem 2.5 in [6] is incorrect. By our theorem in this section, it turns out that Theorem 5.6 in [2] is not true. Although the authors of the latter reference have amended the error in [3], by replacing υX instead of X, whenever needed, but we improve the theorem by presenting an equivalent condition on X itself. For the sake of completeness, we first recite those two incorrect results. Recall from [9] that a subset A of βX is said to be round if whenever clβX Z contains A, where Z ∈ Z(X), then it is a neighborhood of A (i.e., M A = OA ). By Theorem 3.6 in [11] and Lemma 2.4 in [4], it is well-known that CF (X) = OβX\I(X) . Theorem 2.1. (Theorem 2.5 in [6]) The following statements are equivalent. (1) CF (X) = M βX\I(X) . (2) Every countable subset of I(X) is closed. (3) βX \ I(X) is a round subset of βX. Theorem 2.2. (Theorem 5.6 in [2]) J(C(X)/CF (X)) = (0) if and only if every compact subset of X contains at most a finite number of isolated points of X. The corrected version of Theorem 5.6 which is given in [3] is as follows: Theorem 2.3. J(C(X)/CF (X)) = (0) if and only if every compact subset of υX contains at most a finite number of isolated points of X. In the next theorem, we observe that the statement of part (2) in Theorem 2.1 should be “Every countably infinite subset of I(X) contains a closed infinite subset”, and the second part of Theorem 2.2 should be “every pseudocompact subset of X contains at most a finite number of isolated points of X”, and we show that these new assertions are equivalent. We also give two counterexamples regarding to these points in this section. Whenever I(X) is finite, then CF (X) = M βX\I(X) . For, clearly CF (X) ⊆ M βX\I(X) and if f ∈ βX\I(X) M , then βX \ I(X) ⊆ clβX Z(f ). This shows that βX \ clβX Z(f ) is finite, so X \ Z(f ) is finite, i.e., f ∈ CF (X). The following result which is an erratum to Theorems 2.1 and 2.2, also shows that the converse is not true. Theorem 2.4. The following statements are equivalent. (1) CF (X) = M βX\I(X) . (2) Every countably infinite subset of I(X) contains an infinite closed subset.
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(3) No countably infinite subset of I(X) is relatively pseudocompact in X. (4) Every pseudocompact subset of X has at most a finite number of isolated points. (5) J(C(X)/CF (X)) = (0). Proof. (1) ⇒ (2) Suppose C = {a1 , a2 , · · ·} is a countably infinite subset of I(X) and it has no infinite closed ∞ subset. Let f = n=1 21n χ{an } , where χ{an } is the characteristic function of the singleton set {an }. Then f ∈ C(X) by the Weierstrass M -test and clearly f (an ) = 21n , for each n ∈ N which means that f ∈ / CF (X). Now Z(1 − f g) is a closed subset of C, for each g ∈ C(X), so it must be finite, by our hypothesis and Theorem 2.2 in [6], i.e., f ∈ M βX\I(X) , a contradiction. (2) ⇒ (3) Let C be an infinite countable subset of I(X). By hypothesis, there exists a closed countably infinite subset H of C. Hence H is an infinite closed-open discrete subset of X and this means that C is not relatively pseudocompact in X. (3) ⇒ (4) Assume that K is a pseudocompact subset of X. If K contains an infinite countable subset C of I(X), then C is relatively pseudocompact in X, a contradiction. (4) ⇒ (1) First we note that M βX\I(X) = M βX\υX ∩ M υX\I(X) = Cψ (X) ∩ M υX\I(X) . Next, let f ∈ M βX\I(X) . Then we have X \ Z(f ) ⊆ υX \ clυX Z(f ) ⊆ I(X) and clX (X \ Z(f )) is pseudocompact. Now by hypothesis, clX (X \ Z(f )) ∩ I(X) is finite and hence X \ Z(f ) is finite, i.e., f ∈ CF (X). (1) ⇔ (5) Since every maximal ideal of C(X)/CF (X) is of the form M p /OβX\I(X) , where p ∈ βX \ I(X), the proof is evident. 2 Corollary 2.5. If X is first countable, then every countable subset of X consisting entirely of isolated points is closed if and only if J(C(X)/CF (X)) = (0). Proof. If J(C(X)/CF (X)) = (0), then by the revised version of Corollary 5.9 in [2] and [3], I(X) is closed in υX and so it is closed in X. Now, if C is a countable subset of I(X), then C is closed in I(X) and hence it is closed in X. This implies that every cluster point of C is contained in I(X) which is impossible. Hence C has no any cluster point, i.e., C is closed. The converse is clear by equivalence of parts (2) and (5) in Theorem 2.4. 2 Using the corrected version of Theorem 2.2 (i.e., Theorem 2.3) and Theorem 2.4, the following corollary is immediate. Corollary 2.6. If every compact subset of υX has at most a finite number of isolated points, then every pseudocompact subset of X has at most a finite number of isolated points, and vice versa. In the following examples, we show that part (2) of Theorem 2.1 and part (2) of Theorem 2.4 are not equivalent. This example also shows that parts (2) and (3) in Theorem 2.1 are not equivalent. In fact, part (2) of Theorem 2.1 implies part (3) of the same theorem and also implies part (2) of Theorem 2.4. But the following example shows that the converse implications are not true. Example 2.7. We consider the space Σ from [7], 4M as follows: Let U be a free ultrafilter on N, let Σ = N ∪{σ} / N), and we define a topology on Σ such that all points of N are isolated, and the neighborhoods (where σ ∈ of σ are the sets U ∪ {σ} for U ∈ U. If U ∈ U, then N \ U = V ∈ / U. Hence U is not closed in Σ, whereas U is a countable subset of I(Σ) = N, i.e., in this case, part (2) of Theorem 2.1 does not hold. But every infinite countable subset of I(Σ) = N contains an infinite closed subset. To this end, let C ⊆ N be infinite and A, B ⊆ C be two infinite disjoint sets with A ∪ B = C. One of the sets A and B is not in U for A ∩ B = ∅.
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Let A ∈ / U, hence N \ A ∈ U, i.e., (N \ A) ∪ {σ} is open in Σ, so A is a closed infinite subset of C. Hence part (2) of Theorem 2.4 holds in case X = Σ. Now we show that part (3) of Theorem 2.1 holds for X = Σ. Since N is realcompact, then OβN\N = CF (N) = CK (X) = M βN\N , by Theorem 8.19 in [7], Theorem 3.6 in [11] and Lemma 2.4 in [4]. Hence βN \ N is a round subset. But βΣ = βN implies that βΣ \ I(Σ) = βN \ N is also a round subset and we are through. We conclude this section by an example which shows that Theorem 2.2 in the present form is not true. In fact, by Theorem 2.4, if J(C(X)/CF (X)) = (0), then every compact subset of X contains at most a finite number of isolated points of X. But we show that the converse is not true. To this end, it is enough to find a pseudocompact space which is also pseudofinite (a space in which every compact subspace is finite). For such a space, part (3) of Theorem 2.4 does not hold which means that J(C(X)/CF (X)) = (0) but the second part of Theorem 2.2 holds. For this purpose, we use Example 3.10.19 in [5]. Example 2.8. For every S ⊆ βN let P(S) denote the family of all countably infinite subsets of the set S, and let f be a function assigning to every member A of P(βN) an accumulation point of the set A in the space βN. Letting X0 = N and Xα = ( γ<α Xγ ) ∪ f [P ( γ<α Xγ )] for α < ω1 , we define by transfinite induction a transfinite sequence Xo , X1 , . . . , Xα , . . . , α < ω1 of subsets of βN. Now let X = α<ω1 Xα . Then X is countably compact and hence a pseudocompact space, because every A ∈ P(X) is contained in some Xα and thus has an accumulation point in Xα+1 and, a fortiori, in X. Applying transfinite induction, one easily shows that |Xα | ≤ c · c + (c · c)ℵ0 = c, so that |X| ≤ c. On the other hand, every infinite compact subset of βN is of the cardinal 2c , so if K is an infinite compact subset of X, then it is a compact subset of βN. This implies that 2c ≤ c, a contradiction. Therefore compact subsets in X are finite sets, i.e., X is pseudo-finite, and since N ⊆ X, I(X) is infinite. Acknowledgement The authors would like to thank referees for a careful reading of this article. References [1] F. Azarpanah, Intersection of essential ideals in C(X), Proc. Am. Math. Soc. 125 (1997) 2149–2154. [2] F. Azarpanah, O.A.S. Karamzadeh, S. Rahmati, C(X) vs. C(X) modulo its socle, Colloq. Math. 111 (2) (2008) 315–336. [3] F. Azarpanah, O.A.S. Karamzadeh, S. Rahmati, Erratum to: C(X) vs. C(X) modulo its socle, Colloq. Math. 139 (1) (2015) 147–148. [4] F. Azarpanah, A.R. Olfati, On ideals of ideals of C(X), Bull. Iran. Math. Soc. 41 (1) (2015) 23–41. [5] R. Engelking, General Topology, Sigma Ser. Pure Math., vol. 6, Heldermann Verlag, Berlin, 1989. [6] M. Ghirati, A. Taherifar, Intersections of essential (resp., free) maximal ideals of C(X), Topol. Appl. 167 (2014) 62–68. [7] L. Gillman, M. Jerison, Rings of Continuous Functions, Springer, 1976. [8] O.A.S. Karamzadeh, M. Rostami, On the intrinsic topology and some related ideals of C(X), Proc. Am. Math. Soc. 93 (1) (1985) 179–184. [9] M. Mandelker, Supports of continuous functions, Trans. Am. Math. Soc. 156 (1971) 73–83. [10] J.C. McConnel, J.C. Robson, Noncommutative Noetherian Rings, Wiley-Interscience, New York, 1987. [11] A. Taherifar, Intersections of essential minimal prime ideals, Comment. Math. Univ. Carol. 55 (1) (2014) 121–130.
Contents lists available at ScienceDirect
Topology and its Applications www.elsevier.com/locate/topol
When is CF (X) = M βX\I(X) ? F. Azarpanah a , M. Ghirati b,∗ , A. Taherifar b a b
Department of Mathematics, Shahid Chamran University, Ahvaz, Iran Department of Mathematics, Yasouj University, Yasouj, Iran
a r t i c l e
i n f o
Article history: Received 9 May 2015 Received in revised form 22 July 2015 Accepted 29 July 2015 Available online xxxx MSC: 54C40
a b s t r a c t This short article is in fact an erratum to Theorem 2.5 in Ghirati and Taherifar (2014) [6]. In this paper, we prove that J(C(X)/CF (X)) = 0 or equivalently, the socle CF (X) of C(X) coincides with the intersection of all essential maximal ideals of C(X) if and only if every infinite subset of I(X) contains a closed infinite subset, if and only if every pseudocompact subset of X has at most a finite number of isolated points. This fact shows that part (2) of aforementioned Theorem 2.5 is not correct. Our results in this paper also amend Theorem 5.6 in Azarpanah et al. (2008) [2]. Examples are provided to illustrate and delimit our results. © 2015 Elsevier B.V. All rights reserved.
Keywords: Essential ideal Socle Pseudocompact Round set
1. Introduction We denote by C(X) (C ∗ (X)) the ring of (bounded) real-valued, continuous functions on a completely regular Hausdorff space X. Whenever C(X) = C ∗ (X), then X is called pseudocompact. υX is the Hewitt ˘ realcompactification of X, βX is the Stone–Cech compactification of X and for any p ∈ βX, the maximal p p ideal M (resp., the ideal O ) is the set of all f ∈ C(X) for which p ∈ clβX Z(f ) (resp., p ∈ intβX clβX Z(f )). More generally, for A ⊆ βX, M A (resp. OA ) is the intersection of all M p (resp. Op ) with p ∈ A. If C(X)/M p ∼ = R, then M p is called real, else hyper-real and υX is in fact the set of all p ∈ βX such that M p is real. For each f ∈ C(X), the zero-set Z(f ) is the set of zeros of f , X \ Z(f ) is the cozero-set of f and clX (X\Z(f )) is called the support of f . An ideal I in C(X) (C ∗ (X)) is called fixed if Z[I] = f ∈I Z(f ) = ∅, else is free. A subset A of X is said to be relatively pseudocompact if each member of C(X) is bounded on A. The ideal Cψ (X) in C(X) is the set of all functions with pseudocompact support. It is well-known that Cψ (X) is the set of all functions with relatively pseudocompact support and it also coincides with the intersection of all hyper-real maximal ideals in C(X), i.e., Cψ (X) = M βX\υX , see [9]. * Corresponding author. E-mail addresses: [email protected] (F. Azarpanah), [email protected], [email protected] (M. Ghirati), [email protected], [email protected] (A. Taherifar). http://dx.doi.org/10.1016/j.topol.2015.07.015 0166-8641/© 2015 Elsevier B.V. All rights reserved.
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An ideal in a ring R with identity, is said to be essential if it intersects every nonzero ideal of R nontrivially. It is well-known that an ideal E in C(X) is essential if and only if intX Z[E] = ∅, see [1]. This shows that all free ideals are essential and every fixed maximal ideal Mx , where x is not isolated point, is also an essential ideal in C(X). Hence the intersection of all essential maximal ideals of C(X) is M βX\I(X) , where I(X) is the set of all isolated points of X. Recall that for any ring R with identity the socle of R is the intersection of all essential ideals of R which is also equal to the sum of all minimal ideals of R. We denote the socle of C(X) by CF (X) and it is topologically characterized in [8] as the set of all functions which vanish everywhere except on a finite number of points of X. The reader is referred to [5,7] and [10] for undefined terms and notations. 2. Improvements in two published results In this section, we give a theorem which shows that part (2) of Theorem 2.5 in [6] is incorrect. By our theorem in this section, it turns out that Theorem 5.6 in [2] is not true. Although the authors of the latter reference have amended the error in [3], by replacing υX instead of X, whenever needed, but we improve the theorem by presenting an equivalent condition on X itself. For the sake of completeness, we first recite those two incorrect results. Recall from [9] that a subset A of βX is said to be round if whenever clβX Z contains A, where Z ∈ Z(X), then it is a neighborhood of A (i.e., M A = OA ). By Theorem 3.6 in [11] and Lemma 2.4 in [4], it is well-known that CF (X) = OβX\I(X) . Theorem 2.1. (Theorem 2.5 in [6]) The following statements are equivalent. (1) CF (X) = M βX\I(X) . (2) Every countable subset of I(X) is closed. (3) βX \ I(X) is a round subset of βX. Theorem 2.2. (Theorem 5.6 in [2]) J(C(X)/CF (X)) = (0) if and only if every compact subset of X contains at most a finite number of isolated points of X. The corrected version of Theorem 5.6 which is given in [3] is as follows: Theorem 2.3. J(C(X)/CF (X)) = (0) if and only if every compact subset of υX contains at most a finite number of isolated points of X. In the next theorem, we observe that the statement of part (2) in Theorem 2.1 should be “Every countably infinite subset of I(X) contains a closed infinite subset”, and the second part of Theorem 2.2 should be “every pseudocompact subset of X contains at most a finite number of isolated points of X”, and we show that these new assertions are equivalent. We also give two counterexamples regarding to these points in this section. Whenever I(X) is finite, then CF (X) = M βX\I(X) . For, clearly CF (X) ⊆ M βX\I(X) and if f ∈ βX\I(X) M , then βX \ I(X) ⊆ clβX Z(f ). This shows that βX \ clβX Z(f ) is finite, so X \ Z(f ) is finite, i.e., f ∈ CF (X). The following result which is an erratum to Theorems 2.1 and 2.2, also shows that the converse is not true. Theorem 2.4. The following statements are equivalent. (1) CF (X) = M βX\I(X) . (2) Every countably infinite subset of I(X) contains an infinite closed subset.
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(3) No countably infinite subset of I(X) is relatively pseudocompact in X. (4) Every pseudocompact subset of X has at most a finite number of isolated points. (5) J(C(X)/CF (X)) = (0). Proof. (1) ⇒ (2) Suppose C = {a1 , a2 , · · ·} is a countably infinite subset of I(X) and it has no infinite closed ∞ subset. Let f = n=1 21n χ{an } , where χ{an } is the characteristic function of the singleton set {an }. Then f ∈ C(X) by the Weierstrass M -test and clearly f (an ) = 21n , for each n ∈ N which means that f ∈ / CF (X). Now Z(1 − f g) is a closed subset of C, for each g ∈ C(X), so it must be finite, by our hypothesis and Theorem 2.2 in [6], i.e., f ∈ M βX\I(X) , a contradiction. (2) ⇒ (3) Let C be an infinite countable subset of I(X). By hypothesis, there exists a closed countably infinite subset H of C. Hence H is an infinite closed-open discrete subset of X and this means that C is not relatively pseudocompact in X. (3) ⇒ (4) Assume that K is a pseudocompact subset of X. If K contains an infinite countable subset C of I(X), then C is relatively pseudocompact in X, a contradiction. (4) ⇒ (1) First we note that M βX\I(X) = M βX\υX ∩ M υX\I(X) = Cψ (X) ∩ M υX\I(X) . Next, let f ∈ M βX\I(X) . Then we have X \ Z(f ) ⊆ υX \ clυX Z(f ) ⊆ I(X) and clX (X \ Z(f )) is pseudocompact. Now by hypothesis, clX (X \ Z(f )) ∩ I(X) is finite and hence X \ Z(f ) is finite, i.e., f ∈ CF (X). (1) ⇔ (5) Since every maximal ideal of C(X)/CF (X) is of the form M p /OβX\I(X) , where p ∈ βX \ I(X), the proof is evident. 2 Corollary 2.5. If X is first countable, then every countable subset of X consisting entirely of isolated points is closed if and only if J(C(X)/CF (X)) = (0). Proof. If J(C(X)/CF (X)) = (0), then by the revised version of Corollary 5.9 in [2] and [3], I(X) is closed in υX and so it is closed in X. Now, if C is a countable subset of I(X), then C is closed in I(X) and hence it is closed in X. This implies that every cluster point of C is contained in I(X) which is impossible. Hence C has no any cluster point, i.e., C is closed. The converse is clear by equivalence of parts (2) and (5) in Theorem 2.4. 2 Using the corrected version of Theorem 2.2 (i.e., Theorem 2.3) and Theorem 2.4, the following corollary is immediate. Corollary 2.6. If every compact subset of υX has at most a finite number of isolated points, then every pseudocompact subset of X has at most a finite number of isolated points, and vice versa. In the following examples, we show that part (2) of Theorem 2.1 and part (2) of Theorem 2.4 are not equivalent. This example also shows that parts (2) and (3) in Theorem 2.1 are not equivalent. In fact, part (2) of Theorem 2.1 implies part (3) of the same theorem and also implies part (2) of Theorem 2.4. But the following example shows that the converse implications are not true. Example 2.7. We consider the space Σ from [7], 4M as follows: Let U be a free ultrafilter on N, let Σ = N ∪{σ} / N), and we define a topology on Σ such that all points of N are isolated, and the neighborhoods (where σ ∈ of σ are the sets U ∪ {σ} for U ∈ U. If U ∈ U, then N \ U = V ∈ / U. Hence U is not closed in Σ, whereas U is a countable subset of I(Σ) = N, i.e., in this case, part (2) of Theorem 2.1 does not hold. But every infinite countable subset of I(Σ) = N contains an infinite closed subset. To this end, let C ⊆ N be infinite and A, B ⊆ C be two infinite disjoint sets with A ∪ B = C. One of the sets A and B is not in U for A ∩ B = ∅.
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Let A ∈ / U, hence N \ A ∈ U, i.e., (N \ A) ∪ {σ} is open in Σ, so A is a closed infinite subset of C. Hence part (2) of Theorem 2.4 holds in case X = Σ. Now we show that part (3) of Theorem 2.1 holds for X = Σ. Since N is realcompact, then OβN\N = CF (N) = CK (X) = M βN\N , by Theorem 8.19 in [7], Theorem 3.6 in [11] and Lemma 2.4 in [4]. Hence βN \ N is a round subset. But βΣ = βN implies that βΣ \ I(Σ) = βN \ N is also a round subset and we are through. We conclude this section by an example which shows that Theorem 2.2 in the present form is not true. In fact, by Theorem 2.4, if J(C(X)/CF (X)) = (0), then every compact subset of X contains at most a finite number of isolated points of X. But we show that the converse is not true. To this end, it is enough to find a pseudocompact space which is also pseudofinite (a space in which every compact subspace is finite). For such a space, part (3) of Theorem 2.4 does not hold which means that J(C(X)/CF (X)) = (0) but the second part of Theorem 2.2 holds. For this purpose, we use Example 3.10.19 in [5]. Example 2.8. For every S ⊆ βN let P(S) denote the family of all countably infinite subsets of the set S, and let f be a function assigning to every member A of P(βN) an accumulation point of the set A in the space βN. Letting X0 = N and Xα = ( γ<α Xγ ) ∪ f [P ( γ<α Xγ )] for α < ω1 , we define by transfinite induction a transfinite sequence Xo , X1 , . . . , Xα , . . . , α < ω1 of subsets of βN. Now let X = α<ω1 Xα . Then X is countably compact and hence a pseudocompact space, because every A ∈ P(X) is contained in some Xα and thus has an accumulation point in Xα+1 and, a fortiori, in X. Applying transfinite induction, one easily shows that |Xα | ≤ c · c + (c · c)ℵ0 = c, so that |X| ≤ c. On the other hand, every infinite compact subset of βN is of the cardinal 2c , so if K is an infinite compact subset of X, then it is a compact subset of βN. This implies that 2c ≤ c, a contradiction. Therefore compact subsets in X are finite sets, i.e., X is pseudo-finite, and since N ⊆ X, I(X) is infinite. Acknowledgement The authors would like to thank referees for a careful reading of this article. References [1] F. Azarpanah, Intersection of essential ideals in C(X), Proc. Am. Math. Soc. 125 (1997) 2149–2154. [2] F. Azarpanah, O.A.S. Karamzadeh, S. Rahmati, C(X) vs. C(X) modulo its socle, Colloq. Math. 111 (2) (2008) 315–336. [3] F. Azarpanah, O.A.S. Karamzadeh, S. Rahmati, Erratum to: C(X) vs. C(X) modulo its socle, Colloq. Math. 139 (1) (2015) 147–148. [4] F. Azarpanah, A.R. Olfati, On ideals of ideals of C(X), Bull. Iran. Math. Soc. 41 (1) (2015) 23–41. [5] R. Engelking, General Topology, Sigma Ser. Pure Math., vol. 6, Heldermann Verlag, Berlin, 1989. [6] M. Ghirati, A. Taherifar, Intersections of essential (resp., free) maximal ideals of C(X), Topol. Appl. 167 (2014) 62–68. [7] L. Gillman, M. Jerison, Rings of Continuous Functions, Springer, 1976. [8] O.A.S. Karamzadeh, M. Rostami, On the intrinsic topology and some related ideals of C(X), Proc. Am. Math. Soc. 93 (1) (1985) 179–184. [9] M. Mandelker, Supports of continuous functions, Trans. Am. Math. Soc. 156 (1971) 73–83. [10] J.C. McConnel, J.C. Robson, Noncommutative Noetherian Rings, Wiley-Interscience, New York, 1987. [11] A. Taherifar, Intersections of essential minimal prime ideals, Comment. Math. Univ. Carol. 55 (1) (2014) 121–130.