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@Appl. Gen. Topol. 19, no. 2 (2018), 261-268

doi: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 band S. Soltanpour c

aShoushtar Branch, Islamic Azad University, Shoushtar, Iran. (s.mehran@iau-shoushtar.ac.ir)

bDepartment of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran. (namdari@ipm.ir)

cPetroleum 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 Xfor 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 λis 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

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 S CF(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

λis 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 Iin 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 Iin a reduced ring is an essential ideal if and only

if Ann(I) = (0) (note: it suﬃces to recall that Ris 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 Xis 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 S CF(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 Xfor 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 Xfor 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 Oxcontaining Sλ(X), where xruns through the set of

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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∈Xis called a λ-isolated point if xhas a

neighborhood with cardinality less than λ. The set of all λ-isolated points of

Xis denoted by Iλ(X). If every point of Xis λ-isolated, then Xis called a

λ-discrete space, i.e., Iλ(X) = X.

Deﬁnition 2.2. A topological space Xis said to be λ-compact whenever each

open cover of Xcan be reduced to an open cover of Xwhose cardinality is less

than λ, where λis the least inﬁnite cardinal number with this property.

Deﬁnition 2.3. Xis a Pλ-space if every intersection of a family of cardinality

less than λof 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. TZ[Sλ(X)] is equal to the set of non-λ-isolated points, i.e.,

TZ[Sλ(X)] = X\Iλ(X). In particular, if x∈Xis a λ-isolated point, then

there exists f∈Sλ(X), such that f(x) = 1.

Corollary 2.5. For any space Xthe following statements hold.

(1) An element x∈Xis a λ-isolated point if and only if Mx+Sλ(X) =

C(X).

(2) Xis 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∈Xis non-λ-isolated point if and only if Sλ(X)⊆Mx.

(5) If |X| ≥ λand |Iλ(X)|< λ, then Sλ(X) = Tx∈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 eis an idempotent in C(X)if and

only if X=A∪B, where Aand Bare two disjoint open subsets of Xsuch that

the set of λ-isolated points of Xis a dense subset of Aand Bhas no λ-isolated

points of X.

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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 Xhas 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=φand 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∪Bsuch that A

and Bare two disjoint open subsets of X, where Aand Bhave 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., f e = 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 (f g)(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], fis a

multiple of e, thus f∈(e) and we are done.

As previously mentioned, the set of isolated points in a space Xis 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 Xis 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 Xis 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(TZ[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].

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On the essentiality and primeness of λ-super socle of C(X)

Clearly, every essential ideal in any commutative ring Rcontains 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 λis a cardinal number greater than or equal ℵ0.

It is well known that the intersection of the essential ideals in a commutative

ring Ris equal to the socle of R. More generally, any ideal containing the socle

of Ris 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 Xbe 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 Xbe a λ-discrete space and Ebe 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 Eis 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 Mcontaining 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 Pin C(X)containing Sλ(X)(or even

CF(X)) is an essential ideal. In particular if Xis 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 Xis 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

06=r∈Rsuch 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∈Xis the only non-λ-isolated

point of X, then Mxis 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

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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 xis the only non-λ-isolated point. So the space Xhas only one non-λ-

isolated point. Consequently, if Xhas 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 Xis 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 Xbe a Pλ-space, then the following statements are equiv-

alent.

(1) Sλ(X) = Mx, for som x∈X.

(2) Xis a λ-compact space containing only one non-λ-isolated point.

Proof. ((1) ⇒(2)) Evidently, x∈Xis the only non-λ-isolated point in X,

see Corollary 2.5 and Proposition 4.2. Now we show that Xis a λ-compact

space. Put X=Si∈IGi, such that Giis an open set in X, for each i∈I

and |I| ≥ λ. Since x∈Si∈IGi, 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 Xis a Pλ-space, x∈Z(f) and therefore f∈Mx=Sλ(X). Thus

|X\Gk| ≤ |X\Z(f)|=|coz(f)|< λ, i.e., X= (Sj∈JGj)SGk, where J⊆I

and |J|< λ. Now, it is suﬃcient to show that λis the least inﬁnite cardinal

number with this property. To see this we show that there exists an open

cover of Xwith 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|=βand x∈F. Now, by complete regularity of X, for

each s∈Fand y∈F\ {s}, there exists fy∈C(X), such that fy(s) = 0 and

fy(y) = 1. Therefore s∈Ty∈F\{s}Z(fy) = Gsand since Xis a pλ-space, Gs

is an open set of X. So X= (X\F)∪ {Gs}s∈Fis 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 Xwith cardinality less than β. Consequently, Xis

aλ-compact space.

((2) ⇒(1)) It is suﬃcient to show that Mx⊆Sλ(X), where xis the only

non-λ-isolated point of X. Let f∈Mx, i.e., x∈Z(f). Since each point of

Xexcept xis a λ-isolated point we infer that for every y∈X\Z(f), there

exists a neighborhood of yin X, say Gy, with cardinality less than λ. Hence

(X\Z(f)) ⊆Si∈IGyi, where |I|< λ and yiis a λ-isolated point, for each

i∈I. Thus |Si∈IGyi|< λ implies that |X\Z(f)|< λ and we are done.

We note that if Xhas 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 Xis

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].

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On the essentiality and primeness of λ-super socle of C(X)

Proposition 4.4. If Xis 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 Xbe a topological space with |X| ≥ λsuch that |X\

Iλ(X)|>1, then there exists a λ-essential ideal in C(X)which is not a prime

ideal.

Proof. By assumption, there exist two distinct non-λ-isolated points, say x

and y. Now, deﬁne E={f∈C(X) : {x, y} ⊆ Z(f)}, then TZ[E] = {x, y }

and therefore by the criterion for recognizing the essential ideals, Eis essential.

Since x, y ∈TZ[Sλ(X)], by Theorem 2.4 we infer that Sλ(X)⊆E. It is evident

that Eis not a prime ideal, see [11, Theorem 2.11] and we are done.

Acknowledgements. The authors would like to thank professor O.A.S. Karamzadeh

for introducing the topics of this article and for his helpful discussion. The au-

thors are also indebted to the well-informed, meticulous referee for his/her

carefully reading the article and giving valuable and constructive comments.

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