() @ Appl. Gen. Topol. 19, no. 2 (2018), 261-268doi:10.4995/agt.2018.9058 c© AGT, UPV, 2018 On the essentiality and primeness of λ-super socle of C(X) S. Mehran a, M. Namdari b and S. Soltanpour c a Shoushtar Branch, Islamic Azad University, Shoushtar, Iran. (s.mehran@iau-shoushtar.ac.ir) b Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran. (namdari@ipm.ir) c Petroleum University of Technology, Iran. (s.soltanpour@put.ac.ir) Communicated by O. Okunev Dedicated to professor O.A.S. Karamzadeh on the occasion of his retirement and to appreciate his peerless activities in mathematics (especially, popularization of mathematics) for nearly half a century in Iran Abstract Spaces X for which the annihilator of Sλ(X), the λ-super socle of C(X) (i.e., the set of elements of C(X) that cardinality of their cozerosets are less than λ, where λ is a regular cardinal number such that λ ≤ |X|) is generated by an idempotent are characterized. This enables us to find a topological property equivalent to essentiality of Sλ(X). It is proved that every prime ideal in C(X) containing Sλ(X) is essential and it is an intersection of free prime ideals. Primeness of Sλ(X) is characterized via a fixed maximal ideal of C(X). 2010 MSC: Primary: 54C30; 54C40; 54C05; 54G12; Secondary: 13C11; 16H20. Keywords: λ-super socle of C(X); λ-isolated point; λ-disjoint spaces. 1. Introduction Unless otherwise mentioned all topological spaces are infinite Tychonoff and we will employ the definitions and notations used in [11] and [7]. C(X) is the ring of all continuous real valued functions on X. The socle of C(X), denoted by CF (X), is the sum of all minimal ideals of C(X) which plays an important role in the structure theory of noncommutative Noetherian rings, see [12], but Received 07 December 2017 – Accepted 19 June 2018 http://dx.doi.org/10.4995/agt.2018.9058 S. Mehran, M. Namdari and S. Soltanpour O.A.S. Karamzadeh initiated the research regarding the socle of C(X) (see [16]), which is the intersection of all essential ideals in C(X) (recall that, an ideal is essential if it intersects every nonzero ideal nontrivially), see[12] and [16]. Also the minimal ideals and the socle of C(X) are characterized via their corresponding z-filters; see [16]. In [10] and [15], the socle of Cc(X) (the func- tionally countable subalgebra of C(X)), and Lc(X) (the locally functionally countable subalgebra of C(X)), are investigated. The concept of the super so- cle is introduced in [8], denoted by SCF (X), which is the set of all elements f in C(X) such that coz(f) is countable. Clearly, SCF (X) is a z-ideal containing CF (X). Recently, the concept of SCF (X) has been generalized to the λ-super socle of C(X), Sλ(X), where Sλ(X) = {f ∈ C(X) : |X \ Z(f)| < λ}, in which λ is a regular cardinal number with λ ≤ |X|, is introduced and studied in [17]. It is manifest that CF (X) = Sℵ0(X) and SCF (X) = Sℵ1(X). It turns out, in this regard, the ideal CF (X) plays an important role in both concepts. As we know the prime ideals are very important in the context of C(X). It turns out that every prime ideal in C(X) is either an essential ideal or a maximal one, therefore the study of essential ideals in C(X) is worthwhile. It is easy to see that for any ideal I in any commutative ring R, the ideal I + Ann(I), where Ann(I) = {x ∈ X : xI = (0)} is the annihilator of I, is an essential ideal in R. Hence an ideal I in a reduced ring is an essential ideal if and only if Ann(I) = (0) (note: it suffices to recall that R is reduced if and only if Z(R) = {x ∈ R : Ann(x) is essential in R} = (0)). In [16, Proposition 2.1], it is proved that CF (X) is an essential ideal in C(X) if and only if the set of all isolated points of X is dense in X. We note that in this case the socle is the smallest essential ideal in C(X). Also the ideal SCF (X) (the super socle of C(X)) is an essential ideal in C(X) if and only if the set of countably isolated points of X is dense in X, see [8, Corollary 3.2]. Similarly, in what follows, we aim to relate the density of the set of λ-isolated points to an algebraic prop- erty of C(X). In [3, Proposition 2.5], it is shown that the socle of C(X), i.e., CF (X) is never a prime ideal in C(X), but in [8], it is seen that SCF (X) can be a prime ideal (or even a maximal ideal) which this may be considered as an advantage of SCF (X) over CF (X). In this article we will see that Sλ(X) can be a prime ideal, as well. In Section 2, some concepts and preliminary results which are used in the subsequent sections are given. In Section 3, we deal with the essentiality of Sλ(X) and also with the essential ideals containing Sλ(X). In this section, we characterize spaces X for which the annihilator of Sλ(X) is generated by an idempotent. Consequently, this enables us to find an algebraic property equivalent to the density of the set of λ-isolated points in a space X. In contrast to the fact that CF (X) is never a prime ideal in C(X), in Section 4, we characterize spaces X for which Sλ(X) is a prime ideal (even maximal ideal). In the final section, for a class of topological spaces, including maximal λ-compact ones, we prove that the λ-super socle of C(X) is the intersection of the essential ideals Ox containing Sλ(X), where x runs through the set of c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 262 On the essentiality and primeness of λ-super socle of C(X) non-λ-isolated points in X. Also we show that the z-filter corresponding to the λ-super socle of C(X) is the intersection of all essential z-filters containing Sλ(X). 2. Preliminaries First we cite the following results and definitions which are in [14] and [17]. Definition 2.1. An element x ∈ X is called a λ-isolated point if x has a neighborhood with cardinality less than λ. The set of all λ-isolated points of X is denoted by Iλ(X). If every point of X is λ-isolated, then X is called a λ-discrete space, i.e., Iλ(X) = X. Definition 2.2. A topological space X is said to be λ-compact whenever each open cover of X can be reduced to an open cover of X whose cardinality is less than λ, where λ is the least infinite cardinal number with this property. Definition 2.3. X is a Pλ-space if every intersection of a family of cardinality less than λ of open sets (i.e., Gλ-set) is open. We begin with the following well-known result for Sλ(X), see [17, Lemma 2.6]. Theorem 2.4. ⋂ Z[Sλ(X)] is equal to the set of non-λ-isolated points, i.e., ⋂ Z[Sλ(X)] = X \ Iλ(X). In particular, if x ∈ X is a λ-isolated point, then there exists f ∈ Sλ(X), such that f(x) = 1. Corollary 2.5. For any space X the following statements hold. (1) An element x ∈ X is a λ-isolated point if and only if Mx + Sλ(X) = C(X). (2) X is a λ-discrete space if and only if for all x ∈ X, Mx + Sλ(X) = C(X). (3) The ideal Sλ(X) is a free ideal in C(X) if and only if for all x ∈ X, Mx + Sλ(X) = C(X). (4) An element x ∈ X is non-λ-isolated point if and only if Sλ(X) ⊆ Mx. (5) If |X| ≥ λ and |Iλ(X)| < λ, then Sλ(X) = ⋂ x∈X\Iλ(X) Mx. 3. On the essentiality of Sλ(X) in C(X) We begin with the following theorem, which is, in fact, our main result in this section. Theorem 3.1. Ann(Sλ(X)) = (e), where e is an idempotent in C(X) if and only if X = A∪B, where A and B are two disjoint open subsets of X such that the set of λ-isolated points of X is a dense subset of A and B has no λ-isolated points of X . c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 263 S. Mehran, M. Namdari and S. Soltanpour Proof. Let us first get rid of the case that Ann(Sλ(X)) = (1). Clearly, this case holds if and only if Sλ(X) = (0), or equivalently if and if X has no λ-isolated point, since 1.g = 0, for each g ∈ Sλ(X), i.e., Sλ(X) = (0). Conversely, if Sλ(X) = (0), then Ann(Sλ(X)) = C(X) = (1). So put X = A ∪ B, where A = φ and B = X, see Theorem 2.4. Now let Ann(Sλ(X)) = (e), where e is an idempotent in C(X) and H = Iλ(X) be the set λ-isolated points of X. We claim cl(H) = Z(e). In view to Theorem 2.4, for each x ∈ H, there exists f ∈ Sλ(X) such that f(x) = 1. But by assumption, ef = 0, implies e(x) = 0, i.e., H ⊆ Z(e) and consequently cl(H) ⊆ Z(e). Now let x ∈ Z(e) \ cl(H) and seek a contradiction. By complete regularity of X, there exists g ∈ C(X), such that g(x) = 1 and g(cl(H)) = (0). On the other hand for each y ∈ X \ H and every f ∈ Sλ(X), we have f(y) = 0, see Theorem 2.4, this implies that gf = 0, for every f ∈ Sλ(X), which in turn implies g ∈ Ann(Sλ(X)) = (e). Since x ∈ Z(e) and g = he, g(x) = h(x).e(x) = 0, which is a contradiction. Consequently, cl(H) = Z(e) and so cl(H) is clopen. Now put A = cl(H) and X \ cl(H) = B, thus we are done. Conversely, let X = A ∪ B such that A and B are two disjoint open subsets of X, where A and B have the assumed properties. We may define e(x) = { 0 , x ∈ A 1 , x ∈ B It is clear e ∈ C(X) and e2 = e. We claim Ann(Sλ(X)) = (e). If f ∈ Sλ(X) then |X \ Z(f)| < λ and this implies X \ Z(f) ⊆ A = Z(e), i.e., fe = 0 or e ∈ Ann(Sλ(X)). It reminds to be shown that if f ∈ Ann(Sλ(X)), then f ∈ (e). First, we prove that if f ∈ Ann(Sλ(X)), then Z(e) ⊆ Z(f). To see this, put H = Iλ(X), since for each x ∈ H, we infer that there exists g ∈ Sλ(X) such that g(x) = 1. Hence (fg)(x) = 0 implies that f(x) = 0, for every x ∈ H. So f(cl(H)) = 0 (note, f(cl(H)) ⊆ clf(H) ). So cl(H) = A = Z(e) ⊆ Z(f), and since Z(e) is clopen, Z(e) ⊆ int Z(f) and by [11, Problem 1D], f is a multiple of e, thus f ∈ (e) and we are done. � As previously mentioned, the set of isolated points in a space X is dense if and only if the socle of C(X) is essential. Similarly, in [8, Corollary 3.2], it has shown that the ideal SCF (X) is an essential ideal if and only if the set of countably isolated points of X is dense in X. But in the following corollary, we generalize this result for λ-super socle. Corollary 3.2. The ideal Sλ(X) is an essential ideal in C(X) if and only if the set of λ-isolated points of X is dense in X. Proof. Let Sλ(X) be essential ideal, as the previous result Ann(Sλ(X)) = (0), see[1, Proposition 3.1]. Therefore by the comment preceding Theorem 3.1, e = 0 and A = Z(e) = X, i.e., Iλ(X) is dense in X. Conversely, let cl(Iλ(X)) = X, since int( ⋂ Z[Sλ(X)]) = int((Iλ(X)) c) = (cl(Iλ(X)) c = φ, we infer that Sλ(X) is essential in C(X), see[1, Proposition 3.1]. � c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 264 On the essentiality and primeness of λ-super socle of C(X) Clearly, every essential ideal in any commutative ring R contains the socle of R. Now the following definition is in order. Definition 3.3. An essential ideal in C(X) containing Sλ(X) is called a λ- essential ideal where λ is a cardinal number greater than or equal ℵ0. It is well known that the intersection of the essential ideals in a commutative ring R is equal to the socle of R. More generally, any ideal containing the socle of R is also an intersection of essential ideals, see [13, 3N]. It is obvious that Sλ(X) is the intersection of the λ-essential ideals of C(X). Proposition 3.4. Let X be a λ-discrete space, then the set of λ-essential ideals and the set of free ideals containing Sλ(X) coincide. In particular, Sλ(X) is the intersection of free ideals containing it. Proof. Let X be a λ-discrete space and E be a free ideal containing Sλ(X), it is well known that every free ideal in C(X) is an essential ideal, see [2, Proposition 2.1] and the comment preceding it, hence E is a λ-essential ideal which implies that the set of λ-essential ideals and the set of free ideals containing Sλ(X) coincide. � It is clear that every maximal ideal containing the socle of any commutative ring is essential, see [16]. So each maximal ideal M containing Sλ(X) is λ- essential, since CF (X) ⊆ Sλ(X). We also recall that every prime ideal in C(X) is either essential or it is a maximal ideal which is generated by idempotent and it is a minimal prime too, see [4]. In view of these facts and using the above proposition and the fact that Sλ(X) is a z-ideal (hence it is an intersection of prime ideals), we immediately have the following proposition. Proposition 3.5. Every prime ideal P in C(X) containing Sλ(X) (or even CF (X)) is an essential ideal. In particular if X is a λ-discrete space, then Sλ(X) is an intersection of free prime ideals. 4. On the primeness of Sλ(X) in C(X) Our main aim in this section is to investigate the primeness of the λ-super socle. First, we give an example to show that Sλ(X) can be a prime ideal (even a maximal ideal), which is the difference between Sλ(X) and CF (X). Example 4.1. Let X = Y ∪ {x} be one point λ-compactification of a discrete space Y , see [17, Definition 2.11]. We claim that C(X) = R + Sλ(X), i.e., Sλ(X) is a real maximal ideal. Let f ∈ C(X), then we consider two cases. Let us first take x ∈ Z(f), since X is a Pλ-space, Z(f) is open and so |X\Z(f)| < λ implies f ∈ Sλ(X) ⊆ R + Sλ(X). Now, we suppose x /∈ Z(f), so there exists 0 6= r ∈ R such that f(x) = r. Put g = f − r, hence x ∈ Z(g) and therefor g ∈ Sλ(X). We are done. Using Corollary 2.5, it is evident that if x ∈ X is the only non-λ-isolated point of X, then Mx is the unique fixed maximal ideal in C(X) such that Sλ(X) ⊆ Mx. It is well-known that every prime ideal in C(X) is contained in c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 265 S. Mehran, M. Namdari and S. Soltanpour a unique maximal ideal, see [11, Theorem 2.11]. Now let Sλ(X) be a prime ideal in C(X), then Sλ(X) is contained in the unique maximal ideal Mx, such that x is the only non-λ-isolated point. So the space X has only one non-λ- isolated point. Consequently, if X has more than one non-λ-isolated point then Sλ(X) can not be a prime ideal in C(X), see 2.5. Now we have the following results. Proposition 4.2. If X is a topological space with more than one non-λ-isolated point in X, i.e., |X \ Iλ(X)| > 1, then Sλ(X) is not a prime ideal in C(X). Theorem 4.3. Let X be a Pλ-space, then the following statements are equiv- alent. (1) Sλ(X) = Mx, for som x ∈ X. (2) X is a λ-compact space containing only one non-λ-isolated point. Proof. ((1) ⇒ (2)) Evidently, x ∈ X is the only non-λ-isolated point in X, see Corollary 2.5 and Proposition 4.2. Now we show that X is a λ-compact space. Put X = ⋃ i∈I Gi, such that Gi is an open set in X, for each i ∈ I and |I| ≥ λ. Since x ∈ ⋃ i∈I Gi, there exists k ∈ I, such that x ∈ Gk. But by complete regularity of X, there exists f ∈ C(X) such that x ∈ int(Z(f)) ⊆ Gk. Since X is a Pλ-space, x ∈ Z(f) and therefore f ∈ Mx = Sλ(X). Thus |X \ Gk| ≤ |X \ Z(f)| = |coz(f)| < λ, i.e., X = ( ⋃ j∈J Gj) ⋃ Gk, where J ⊆ I and |J| < λ. Now, it is sufficient to show that λ is the least infinite cardinal number with this property. To see this we show that there exists an open cover of X with cardinality β < λ which is not reducible to a subcover with cardinality less than β. By [17, Lemma 2.13], there exists a closed subspace F ⊂ X, such that |F | = β and x ∈ F . Now, by complete regularity of X, for each s ∈ F and y ∈ F \ {s}, there exists fy ∈ C(X), such that fy(s) = 0 and fy(y) = 1. Therefore s ∈ ⋂ y∈F \{s} Z(fy) = Gs and since X is a pλ-space, Gs is an open set of X. So X = (X \ F) ∪ {Gs}s∈F is an open cover of X. It goes without saying that Gs ∩ F = {s} and therefore the above cover cannot reduce to an open cover of X with cardinality less than β. Consequently, X is a λ-compact space. ((2) ⇒ (1)) It is sufficient to show that Mx ⊆ Sλ(X), where x is the only non-λ-isolated point of X. Let f ∈ Mx, i.e., x ∈ Z(f). Since each point of X except x is a λ-isolated point we infer that for every y ∈ X \ Z(f), there exists a neighborhood of y in X, say Gy, with cardinality less than λ. Hence (X \ Z(f)) ⊆ ⋃ i∈I Gyi, where |I| < λ and yi is a λ-isolated point, for each i ∈ I. Thus | ⋃ i∈I Gyi| < λ implies that |X \ Z(f)| < λ and we are done. � We note that if X has at most one non-λ-isolated point, then by criterion for recognizing the essential ideals in C(X), see [1, theorem 3.1], Sλ(X) is essential in C(X) and by Proposition 4.2, it is an essential prime ideal of C(X). If X is the one point λ-compactification of a discrete space, then Sλ(X) is an essential maximal ideal, see Theorem 4.3. The above discussion refers to the following proposition which is proved in [1, Proposition 4.1]. c© AGT, UPV, 2018 Appl. Gen. Topol. 19, no. 2 266 On the essentiality and primeness of λ-super socle of C(X) Proposition 4.4. If X is an infinite space, there is an essential ideal in C(X) which is not a prime ideal. The following theorem is the counterpart of the above proposition. Theorem 4.5. Let X be a topological space with |X| ≥ λ such that |X \ Iλ(X)| > 1, then there exists a λ-essential ideal in C(X) which is not a prime ideal. Proof. 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