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p>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).</p
@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) = {fC(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) = S0(X) and SCF(X) = S1(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) = {xX: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) = {xR: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
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 xXis 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 xXis a λ-isolated point, then
there exists fSλ(X), such that f(x) = 1.
Corollary 2.5. For any space Xthe following statements hold.
(1) An element xXis 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 xX,Mx+Sλ(X) =
C(X).
(3) The ideal Sλ(X)is a free ideal in C(X)if and only if for all xX,
Mx+Sλ(X) = C(X).
(4) An element xXis non-λ-isolated point if and only if Sλ(X)Mx.
(5) If |X| ≥ λand |Iλ(X)|< λ, then Sλ(X) = TxX\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=AB, 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.
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 Xhas no λ-isolated
point, since 1.g = 0, for each gSλ(X), i.e., Sλ(X) = (0). Conversely, if
Sλ(X) = (0), then Ann(Sλ(X)) = C(X) = (1). So put X=AB, 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 xH, there exists
fSλ(X) such that f(x) = 1. But by assumption, ef = 0, implies e(x) = 0,
i.e., HZ(e) and consequently cl(H)Z(e). Now let xZ(e)\cl(H) and
seek a contradiction. By complete regularity of X, there exists gC(X), such
that g(x) = 1 and g(cl(H)) = (0). On the other hand for each yX\H
and every fSλ(X), we have f(y) = 0, see Theorem 2.4, this implies that
gf = 0, for every fSλ(X), which in turn implies gAnn(Sλ(X)) = (e).
Since xZ(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=ABsuch 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 eC(X) and e2=e. We claim Ann(Sλ(X)) = (e). If fSλ(X)
then |X\Z(f)|< λ and this implies X\Z(f)A=Z(e), i.e., f e = 0
or eAnn(Sλ(X)). It reminds to be shown that if fAnn(Sλ(X)), then
f(e). First, we prove that if fAnn(Sλ(X)), then Z(e)Z(f). To see
this, put H=Iλ(X), since for each xH, we infer that there exists gSλ(X)
such that g(x) = 1. Hence (f g)(x) = 0 implies that f(x) = 0, for every xH.
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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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 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 0.
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 fC(X), then we consider two cases. Let
us first take xZ(f), since Xis a Pλ-space, Z(f) is open and so |X\Z(f)|< λ
implies fSλ(X)R+Sλ(X). Now, we suppose x /Z(f), so there exists
06=rRsuch that f(x) = r. Put g=fr, hence xZ(g) and therefor
gSλ(X). We are done.
Using Corollary 2.5, it is evident that if xXis 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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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 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 xX.
(2) Xis a λ-compact space containing only one non-λ-isolated point.
Proof. ((1) (2)) Evidently, xXis 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=SiIGi, such that Giis an open set in X, for each iI
and |I| ≥ λ. Since xSiIGi, there exists kI, such that xGk. But by
complete regularity of X, there exists fC(X) such that xint(Z(f)) Gk.
Since Xis a Pλ-space, xZ(f) and therefore fMx=Sλ(X). Thus
|X\Gk| ≤ |X\Z(f)|=|coz(f)|< λ, i.e., X= (SjJGj)SGk, where JI
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
FX, such that |F|=βand xF. Now, by complete regularity of X, for
each sFand yF\ {s}, there exists fyC(X), such that fy(s) = 0 and
fy(y) = 1. Therefore sTyF\{s}Z(fy) = Gsand since Xis a pλ-space, Gs
is an open set of X. So X= (X\F)∪ {Gs}sFis an open cover of X. It
goes without saying that GsF={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 MxSλ(X), where xis the only
non-λ-isolated point of X. Let fMx, i.e., xZ(f). Since each point of
Xexcept xis a λ-isolated point we infer that for every yX\Z(f), there
exists a neighborhood of yin X, say Gy, with cardinality less than λ. Hence
(X\Z(f)) SiIGyi, where |I|< λ and yiis a λ-isolated point, for each
iI. Thus |SiIGyi|< 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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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 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={fC(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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Abstract ‎Let Cc(X) (resp‎. ‎CF(X)) denote the subring of C(X) consisting of functions with countable (resp‎. ‎finite) image and CF(X) be the socle of C(X)‎. ‎We characterize spaces X with C∗(X)=Cc(X)‎, ‎which generalizes a celebrated result due to Rudin‎, ‎Pelczynnski and Semadeni‎. ‎Two zero-dimensional compact spaces X‎, ‎Y are homeomorphic if and only if Cc(X)≅Cc(Y) (resp‎. ‎CF(X)≅ CF(Y))‎. ‎The spaces XX for which Cc(X)=CF(X) are characterized‎. ‎The socles of Cc(X)‎, ‎CF(X)‎, ‎which are observed to be the same‎, ‎are topologically characterized and spaces XX for which this socle coincides with CF(X) are determined‎, ‎too‎. ‎A certain well-known algebraic property of C(X)‎, ‎where X is realcompact‎, ‎is extended to Cc(X)‎. ‎In contrast to the fact that CF(X) is never prime in C(X)‎, ‎we characterize spaces X for which CF(X) is a prime ideal in Cc(X)‎. ‎It is observed for these spaces‎, Cc(X) coincides with its own socle (a fact‎, ‎which is never true for C(X)‎, ‎unless X is finite)‎. ‎Finally‎, ‎we show that a space X is the one-point compactification of a discrete space if and only if CF(X) is a unique proper essential ideal in CF(X).
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