@ Appl. Gen. Topol. 20, no. 1 (2019), 231-236doi:10.4995/agt.2019.10731
c© AGT, UPV, 2019

When is the super socle of C(X) prime?

S. Ghasemzadeh and M. Namdari

Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran (s.gh8081@gmail.com,

namdari@ipm.ir)

Dedicated to professor O.A.S. Karamzadeh who is not only a “role model” for us, he is

also so for many other mathematics teachers and students, alike, in this country

Communicated by H.-P. A. Künzi

Abstract

Let SCF (X) denote the ideal of C(X) consisting of functions which
are zero everywhere except on a countable number of points of X.
It is generalization of the socle of C(X) denoted by CF (X). Using
this concept we extend some of the basic results concerning CF (X)
to SCF (X). In particular, we characterize the spaces X such that
SCF (X) is a prime ideal in C(X) (note, CF (X) is never a prime ideal
in C(X)). This may be considered as an advantage of SCF (X) over
CF (X). We are also interested in characterizing topological spaces X
such that Cc(X) = R + SCF (X), where Cc(X) denotes the subring of
C(X) consisting of functions with countable image.

2010 MSC: Primary 54C30; 54C40; 54G12; Secondary 13C11; 16H20.

Keywords: super socle of C(X); countably isolated point; countably dis-
crete space; cocountably-disconnected space; one-point Lin-

deĺ’offication.

1. Introduction

We refer the reader to [7] and [12] for necessary background concerning X,
and C(X), the ring of all real-valued continuous functions on a space X. All
topological spaces X in this note are infinite completely regular Hausdorff, un-
less otherwise mentioned. CF (X), the socle of C(X), is the sum of all minimal
ideals of C(X) which is also the intersection of all essential ideals in C(X). An

Received 19 September 2018 – Accepted 05 November 2018

http://dx.doi.org/10.4995/agt.2019.10731


S. Ghasemzadeh and M. Namdari

ideal in a commutative ring is essential if it intersects every nonzero ideal of
the ring nontrivially, see [14], where the socle of C(X) is topologically charac-
terized. It is folklore that one of the main objectives of working in the context
of C(X) is to characterize topological properties of a given space X in terms of
a suitable algebraic properties of C(X). It turns out, in this regard, the ideal
CF (X), plays an appropriate role in the literature, see for examples [2], [3], [5],
[6], [8], [9] [14], [15], and [17]. Motivated by this role of CF (X), the concept
of the super socle, which contains CF (X), is introduced in [11], see also [16].
We are going to extend some of the basic results of the socle of C(X) (i.e.,
CF (X)) to the super socle of C(X) (i.e., SCF (X)). An outline of this article
is as follows:
In Section 2, the concept of the super socle and some preliminary results con-
cerning this ideal, which are frequently used in the subsequent sections, are
given. In the next section, we are going to investigate the primeness of the
super socle in C(X). This may be considered as an advantage of SCF (X)
over CF (X) (note, CF (X) is never a prime ideal in C(X)). We also charac-
terize topological spaces X such that Cc(X) = R + SCF (X), where Cc(X)
denotes the subring of C(X) consisting of functions with countable image, see
[9, Proposition 6.6], [10].

2. Preliminaries

We begin with the definition of the super socle of C(X) which is motivated
by [11, Proposition 3.3].

Definition 2.1. The set S = {f ∈ C(X) : X\Z(f) is countable} is called the
super socle of C(X) and it is denoted by SCF (X).

One can easily see that SCF (X) is a z-ideal in C(X). Clearly CF (X) ⊆ SCF (X),
by [11, Proposition 3.3].
It is trivial to see that a point in a space X is isolated if and only if it has a
finite neighborhood. Motivated by this, the next two definitions are natural
and are also needed.

Definition 2.2. An element p ∈ X is called a countably isolated point if p
has a countable neighborhood. The set of countably isolated points of X is
denoted by Ic(X).

Definition 2.3. If every point of X is countably isolated, then X is called
a countably discrete space. Thus, X is a countably discrete space in case
Ic(X) = X.

We recite the following results which are in [11].

Proposition 2.4 ([11], Proposition 2.4). For any space X, Ic(X) =
⋃
{coz(f) :

f ∈ SCF (X)}.

Corollary 2.5 ([11], Corollary 2.5). For any space X, we have the following:
(a) SCF (X) is not a zero ideal if and only if X has a countably isolated point.
(b) SCF (X) is a free ideal if and only if X is countably discrete.

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When is the super socle of C(X) prime?

We remind the reader that an ideal I is said to be regular if for every a ∈ I
there exists b ∈ I such that a = aba.

Theorem 2.6 ([11], Theorem 2.10). The following are equivalent for an un-
countable space X.
(1) X is the one-point Lindelöffication of some uncountable discrete space.
(2) SCF (X) is a regular ideal, and SCF (X) = Ox for some x ∈ X.

3. The primeness of SCF (X) in C(X)

We recall that CF (X) is never a prime ideal in C(X), see [1], [8, Proposition
1.2]. Our aim in this section is to investigate the primeness of the super socle in
C(X). We begin with an example to show that SCF (X) can be a prime ideal
(even a maximal ideal). This may be considered as an advantage of SCF (X)
over CF (X).

Example 3.1. Let X = Y ∪{x0} be the one-point Lindelöffication of a count-
ably discrete space Y , then we claim that C(X) = R+SCF (X) and this shows
that SCF (X) is a maximal ideal in C(X). To see this, let f ∈ C(X), then we
consider two cases. First, let x0 ∈ Z(f), then for all n ∈ N, x0 ∈ f

−1(−1
n
, 1
n
).

Therefore for all n ∈ N, X \ f−1(−1
n
, 1
n
) is countable. So X \ Z(f) is countable

(note, Z(f) =
⋂

n∈N

f−1(− 1
n
, 1
n
)), i.e., f ∈ SCF (X) ⊆ R + SCF (X). Now, let

x0 /∈ Z(f), then there exists 0 6= r ∈ R, such that f(x0) = r. Put g = f − r,
hence x0 ∈ Z(g), then by what we have already proved g ∈ SCF (X) and so
f ∈ R + SCF (X).

In the next theorem, we characterize the spaces X such that SCF (X) is a
prime ideal in C(X). We need the following well-known lemma, which is in
[12, 4I. 4]. Let us first emphasize that since every prime ideal P in C(X) is
contained in a unique maximal ideal, hence it is either free or it is in the fixed
maximal ideal Mx for a unique x ∈ X.

Lemma 3.2. Let P be a fixed prime ideal in C(X). Then there exists x ∈ X
with Ox ⊆ P ⊆ Mx.

Theorem 3.3. Let SCF (X) be a prime ideal in C(X). Then X is either a
countably discrete space or the one-point Lindelöffication of a countably discrete

space.

Proof. If SCF (X) is a free ideal in C(X), then by Corollary 2.5, X is a count-
ably discrete space. If not, then by Lemma 3.2, there exists x ∈ X such that
Ox ⊆ SCF (X). Consequently, in view of Theorem 2.6, X is either countable or
the one-point Lindelöffication of an uncountable discrete space (i.e., a countably
discrete space), hence we are done. �

The following corollary is now immediate.

Corollary 3.4. Let X have at least one non-countably isolated point. Then
SCF (X) is a prime ideal if and only if X is the one-point Lindelöffication of
a countably discrete spaces.

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S. Ghasemzadeh and M. Namdari

Finally, motivated by Example 3.1 and [9, Proposition 6.6], we are interested
in characterizing topological spaces X such that Cc(X) = R + SCF (X), where
Cc(X) denotes the subring of C(X) consisting of functions with countable
image, see [9], [10]. First, we need the following definition.

Definition 3.5. A space X is called cocountably-disconnected if whenever Y
is a clopen subset of X, either Y or X \ Y is countable.

Clearly, connected spaces, one-point Lindelöffication of countably discrete
spaces and X = Y ∪ Ic(X), where Ic(X) = X \ Y is the countable set of
countably isolated points of X and Y is connected (e.g., X = (0, 1

2
) ∪ N as

a subspace of R, or more generally a free union of a connected space with a
countable space) are some examples of cocountably-disconnected spaces.

Finally, we conclude this article with our main result which is the counterpart
of [9, Proposition 6.6]. Before stating this main result we should remind the
reader that in [10], it is shown that to study Cc(X), we may, without loss,
consider X to be zero dimentional space.

Theorem 3.6. Let X be a zero-dimentional space. Then Cc(X) = R+SCF (X)
if and only if X is cocountably-disconnected and if X = Y ∪ Ic(X), where
Ic(X) = X \ Y is the set of countably isolated points, every function in Cc(X)
is constant on Y .

Proof. First, let X be a cocountably-disconnected space with the above prop-
erties. We are to show that Cc(X) ⊆ R + SCF (X). If X is connected,
SCF (X) = (0), Cc(X) = R, and we are done. Hence we may assume that
X is disconnected, which in turn, implies that the set of countably isolated
points of X is nonempty, for X is cocountably-disconnected. Now we consider
two cases. Case I: Let Ic(X) be finite. Since X is cocountably-disconnected, we
infer that Y = X \ Ic(X) must be connected. Therefore for each f ∈ Cc(X), f
is constant, say r, on Y (note, since f ∈ Cc(X), it is constant on Y automati-
cally and no need to make use of our assumption that f is constant on Y ). Now
let f ∈ Cc(X) and define g ∈ SCF (X) with g(Y ) = 0, g(x) = r − f(x) for all
x ∈ Ic(X). Hence f = r+g and we are done. Case II: Let us assume that Ic(X)
is an infinite set. Let f ∈ SCF (X), hence f(X) = {r1, r2, · · · , rn, · · · } ⊆ R.

Thus X =
∞⋃

i=1

Ai, where Ai = f
−1(ri) for each i ∈ N. By above hypothesis,

f(Y ) = rk, for some k ∈ N and since X is a zero-dimensional space, there
exists a clopen set Uk such that Uk ⊆ X \ Ak. Hence Y ⊆ Ak ⊆ X \ Uk
and we infer that Uk is countable (note, X \ Uk is uncountable). Now if we
define g ∈ SCF (X) with g(X \ Uk) = 0, g(x) = f(x) − rk for all x ∈ Uk,
we have f = g + rk and we are through in this case, too. Conversely, let
Cc(X) = R + SCF (X). If X is connected, we are trivially done. Therefore we
put X = A ∪ B, where A, B are two disjoint clopen subsets of X. We claim
that Ic(X) 6= ∅, for otherwise SCF (X) = (0), hence Cc(X) = R. But the
function f, with f(A) = {0}, f(B) = {1} which is in Cc(X) \ R leads us to

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When is the super socle of C(X) prime?

a contradiction, hence we must have SCF (X) 6= (0). Now we claim that X is
cocountably-disconnected. To see this, let X = A ∪ B, where A, B are two
uncountable disjoint clopen subsets of X and obtain a contradiction. But if f
is the function as above, i.e., f(A) = {0}, f(B) = {1}, then f ∈ Cc(X) but
f /∈ R + SCF (X) (note, if f = r + g with r ∈ R, g ∈ SCF (X), then either
g(A) 6= 0 or g(B) 6= 0 which is impossible, for g must vanish everywhere except
on a countable subset of X). Finally, we must show that each f ∈ SCF (X)
is constant on Y . To see this, let x1, x2 ∈ Y with f(x1) 6= f(x2) and obtain
a contradiction. Since Cc(X) = R + SCF (X), we must have f = r + g, for
some r ∈ R, g ∈ SCF (X). But by Proposition 2.4, g is non-zero only on some
countably isolated points. Whereas either g(x1) 6= 0 or g(x2) 6= 0, which is
absurd. �

In [11, Theorem 2.10], it is observed that, in fact, SCF (X) is a maximal ideal
in C(X) when X is the one-point Lindelöffication of an uncountable discrete
space. We conclude this note with the following related remark.

Remark 3.7. Let X be a zero-dimensional cocountably-disconnected space,
with X \ Ic(X) a singleton and f(Ic(X)) a countable set for any f ∈ C(X).
Then in view of this theorem and Corollary 3.4, X is the one-point Lindelöffi-
cation of a countably discrete space. Moreover in this case SCF (X) becomes
a maximal ideal in C(X), too (note, Cc(X) = C(X) = R + SCF (X)). In
comparison to the latter equality, let us recall that if X is the one-point com-
pactification of a discrete space then CF (X) = R + CF (X), see [9, Theorem
6.7]. Moreover, in this case CF (X) is a unique proper essential ideal in C

F (X)
(consequently, a maximal ideal in CF (X)), where CF (X) is the subalgebra
of Cc(X), a fortiori of C(X), whose elements have finite images, see [13] for
general rings with the latter property .

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

Karamzadeh for introducing the concept of super socle of C(X) and for his
helpful suggestions. The authors are also indebted to the well-informed, metic-

ulous referee for reading the article carefully and giving valuable and construc-

tive comments.

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