@ Appl. Gen. Topol. 16, no. 2(2015), 183-207doi:10.4995/agt.2015.3445
c© AGT, UPV, 2015

On the locally functionally countable sub
algebra of C(X)

O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran (karamzadeh@ipm.ir,

namdari@ipm.ir, s-soltanpour@phdstu.scu.ac.ir)

Abstract

Let Cc(X) = {f ∈ C(X) : |f(X)| ≤ ℵ0}, C
F (X) = {f ∈ C(X) :

|f(X)| < ∞}, and Lc(X) = {f ∈ C(X) : Cf = X}, where Cf is the
union of all open subsets U ⊆ X such that |f(U)| ≤ ℵ0, and CF (X)
be the socle of C(X) (i.e., the sum of minimal ideals of C(X)). It is
shown that if X is a locally compact space, then Lc(X) = C(X) if
and only if X is locally scattered. We observe that Lc(X) enjoys most
of the important properties which are shared by C(X) and Cc(X).
Spaces X such that Lc(X) is regular (von Neumann) are characterized.
Similarly to C(X) and Cc(X), it is shown that Lc(X) is a regular ring
if and only if it is ℵ0-selfinjective. We also determine spaces X such
that Soc

(

Lc(X)
)

= CF (X) (resp., Soc
(

Lc(X)
)

= Soc
(

Cc(X)
)

). It
is proved that if CF (X) is a maximal ideal in Lc(X), then Cc(X) =

C
F (X) = Lc(X) ∼=

n
∏

i=1

Ri, where Ri = R for each i, and X has a unique

infinite clopen connected subset. The converse of the latter result is also
given. The spaces X for which CF (X) is a prime ideal in Lc(X) are
characterized and consequently for these spaces, we infer that Lc(X)
can not be isomorphic to any C(Y ).

2010 MSC: Primary: 54C30; 54C40; 54C05; 54G12; Secondary: 13C11;
16H20.

Keywords: functionally countable space; socle, zero-dimensional space;
scattered space; locally scattered space, ℵ0-selfinjective.

Received 9 December 2014 – Accepted 22 April 2015

http://dx.doi.org/10.4995/agt.2015.3445


O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

1. Introduction

C(X) denotes the ring of all real valued continuous functions on a topo-
logical space X. In [10] and [11], Cc(X), the subalgebra of C(X), consisting
of functions with countable image are introduced and studied. It turns out
that Cc(X), although not isomorphic to any C(Y ) in general, enjoys most of
the important properties of C(X). This subalgebra has recently received some
attention, see [10], [23], [24], [4], and [11]. Since Cc(X) is the largest subring
of C(X) whose elements have countable image, this motivates us to consider a
natural subring of C(X), namely Lc(X), which lies between Cc(X) and C(X).
Our aim in this article, similarly to the main objective of working in the con-
text of C(X), is to investigate the relations between topological properties of
X and the algebraic properties of Lc(X). In particular, we are interested in
finding topological spaces X for which Lc(X) = C(X). An outline of this pa-
per is as follows: In Section 2, we show that if X is a locally compact space,
then Lc(X) = C(X) if and only if X is locally scattered, which is somewhat
similar to a classical result due to Rudin in [27], and Pelczynski and Semadeni
in [25] (of course, by no means as significant). This classical result says that
a compact space X is scattered if and only if C(X) = Cc(X). Let us for the
sake of the brevity, call the latter classical result, RPS-Theorem. If X is an
almost discrete space or a P-space, then L1(X) = LF (X) = Lc(X) = C(X),
where LF (X) and L1(X) are the locally functionally finite (resp., constant)
subalgebra of C(X), see Definition 2.7.

In Section 3, we introduce zl-ideals in Lc(X) and trivially observe that most
of the facts related to z-ideals are extendable to zl-ideals. In Section 4, topolog-
ical spaces in which points and closed sets are separated by elements of Lc(X),
are called locally countable completely regular space (briefly, lc-completely reg-
ular). Clearly, every zero-dimensional space is lc-completely regular (note, in
the zero-dimensional case, points and closed sets are separated even by the ele-
ments of Cc(X), which is a subring of Lc(X)), see [10, Proposition 4.4]. Spaces
X, for which Lc(X) is regular, are called locally countably P-space (briefly,
LCP-space) and are characterized both algebraically and topologically in this
section. It is shown that P-spaces and LCP-spaces coincide when X is lc-
completely regular. Finally, in this section similar to C(X) and Cc(X), we
prove that Lc(X) is a regular ring if and only if it is ℵ0-selfinjective. The
socle of C(X) (i.e., CF (X)) which is in fact a direct sum of minimal ideals of
C(X) is characterized topologically in [20, Proposition 3.3], and it turns out
that CF (X) is a useful object in the context of C(X), see [20], [1], [2], [8], [3],
and [6]. The socle of Cc(X), denoted by Soc

(
Cc(X)

)
, is studied in [11, Propo-

sition 5.3], and spaces X for which Soc
(
Cc(X)

)
= CF (X) are determined in

[11, Theorem 5.6]. Motivated by the latter facts, we characterize the socle of
Lc(X) both topologically and algebraically, in Section 5. Spaces X for which
Soc

(
Lc(X)

)
= Soc

(
Cc(X)

)
and Soc

(
Lc(X)

)
= CF (X) are also characterized.

In [8, Proposition 1.2], [3, Remark 2.4], it is shown that CF (X) can not be a
prime ideal in C(X), where X is any space. But, in [11, Proposition 6.2], spaces

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 184



On the locally functionally countable sub algebra of C(X)

X such that CF (X) is prime in Cc(X) are characterized. The latter character-
ization is similarly extended to Lc(X). Consequently, this implies that Lc(X)
is not isomorphic to any C(Y ) in general. All topological spaces that appear
in this article are assumed to be infinite completely regular Hausdorff, unless
otherwise mentioned. For undefined terms and notations the reader is referred
to [13], [7].

2. The subalgebra Lc(X) of C(X)

Definition 2.1. Let f ∈ C(X) and Cf be the union of all open sets U ⊆ X,
such that f(U) is countable. We define Lc(X) to be the set of all f ∈ C(X)
such that Cf is dense in X, i.e.,

Cf =
⋃

{U| U is open in X and |f(U)| ≤ ℵ0}

Lc(X) = {f ∈ C(X) : Cf = X}
We shall briefly and easily notice that, Lc(X) is a subalgebra as well as a
sublattice of C(X) containing Cc(X), and we call it the locally functionally
countable subalgebra of C(X).

It is manifest that CF (X) ⊆ CF (X) ⊆ Cc(X) ⊆ Lc(X) ⊆ C(X), where
CF (X) = {f ∈ C(X) : |f(X)| < ∞}, see [10]. The following example shows
that the equality between any two of these objects may not necessarily hold.

Example 2.2. Let the basic neighborhood of x be the set {x}, for each point
x ≥

√
2 and for the rest of the real numbers (i.e., x <

√
2) the basic neigh-

borhoods be the usual open intervals containing x. This is a topology T on R
and in this case we put X = R. Clearly, X is a completely regular Hausdorff
space which is finer than the usual topology of R. The function f : X → R,
where f(x) = 1 for x ≥

√
2, and f(x) = 0 otherwise, is continuous and X\Z(f)

is infinite, hence f ∈ CF (X)\CF (X), see [20, Proposition 3.3]. We define
g : X → R, such that g(x) = x for x ∈ [

√
2, ∞) ∩ Q and g(x) = 0 for

x ∈ ([
√
2, ∞) ∩ Qc) ∪ (−∞,

√
2), hence g ∈ Cc(X)\CF (X). Also we observe

that for the function h : X → R, where h(x) = x for x ≥
√
2, and h(x) =

√
2

otherwise, we have h ∈ Lc(X)\Cc(X). The identity function i : X → R is
continuous and Ci = [

√
2, ∞), see Definition 2.1. Hence i ∈ C(X)\Lc(X).

We note that Cf = X if and only if for every open subset G ⊆ X, there exists
an open subset U ⊆ X such that |f(U)| ≤ ℵ0 and U ∩ G 6= ∅ or equivalently if
and only if for each open subset G ⊆ X, there exists a nonempty open subset
V ⊆ G with |f(V )| ≤ ℵ0.
Lemma 2.3. For the space X the following statements hold.

(1) If f, g ∈ C(X), then Cf+g ⊇ Cf ∩ Cg.
(2) If f, g ∈ C(X), then Cfg ⊇ Cf ∩ Cg.
(3) If f ∈ C(X), then C|f| = Cf .
(4) If f ∈ C(X), then C 1

f
= Cf .

(5) If f, g ∈ Lc(X), then Cf ∩ Cg = X.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 185



O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

Proof. Let Cf =
⋃

U⊆X
|f(U)|≤ℵ0

U and Cg =
⋃

V ⊆X
|g(V )|≤ℵ0

V , where U and V are open

subsets of X, then

Cf ∩ Cg =
⋃

U,V ⊆X
|f(U)|,|g(V )|≤ℵ0

(U ∩ V )

Hence (1), (2), (3), (4) are evident. For part (5) we recall that if Y is a dense
subset of X and G is an open subset of X, then G ∩ Y = G. Since Cf , Cg are
open and dense in X we infer that Cf ∩ Cg = Cf = Cg = X. �

The following examples show that the equalities in (1), (2) of the previous
lemma do not necessarily hold, in general.

Example 2.4. (1) Let i : R → R be the identity function and f : R → R with
f(x) = −x, then Ci = Cf = ∅, but Ci+f = R. Hence Ci+f ) Ci ∩ Cf .

(2) Let i : R\{0} → R be the identity function and f : R\{0} → R with
f(x) = 1/x, then Ci = Cf = ∅, but Cif = R\{0}. Hence Cif ) Ci ∩ Cf .

The following fact shows that Lc(X) is indeed a subalgebra of C(X) such
that whenever Z(f) = ∅ where f ∈ Lc(X), then f is a unit in Lc(X). We
remind the reader that the latter fact is not true for C∗(X).

Corollary 2.5. For the space X the following statements hold.

(1) If f, g ∈ Lc(X), then f + g ∈ Lc(X) and fg ∈ Lc(X).
(2) f ∈ Lc(X) if and only if |f| ∈ Lc(X).
(3) Let f be a unit element in C(X), then f ∈ Lc(X) if and only if 1f ∈

Lc(X).

Corollary 2.6. Lc(X) is a sublattice of C(X).

Definition 2.7. Let f ∈ C(X) and CFf be the union of all open sets U ⊆ X
such that f(U) is finite. We define LF (X) to be the set of all f ∈ C(X) such
that CFf is dense in X, and call it locally functionally finite subalgebra of C(X),
i.e.,

CFf =
⋃

{U| U is open in X and |f(U)| < ∞}

LF (X) = {f ∈ C(X) : CFf = X}
In particular, let f ∈ C(X) and Ccf be the union of all open sets U ⊆ X such
that f(U) is constant. We define L1(X) to be the set of all f ∈ C(X) such
that Ccf is dense in X, and we call it locally functionally constant subalgebra

of C(X), i.e.,

Ccf =
⋃

{U| U is open in X and |f(U)| = 1}

L1(X) = {f ∈ C(X) : Ccf = X}

Clearly, LF (X) and L1(X) are subalgebras of Lc(X). In [26] and [15], E0(X)
is defined, and by the above notation we have E0(X) = L1(X). It is evident
that CF (X) ⊆ LF (X).

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 186



On the locally functionally countable sub algebra of C(X)

Remark 2.8. We note that Lemma 2.3, Corollary 2.5, and Corollary 2.6 are
also valid for LF (X) and L1(X).

Remark 2.9. It is manifest that Cc(X) = R, where X = [0, 1]. But the Can-
tor function f is a monotonic nonconstant continuous function, and Ccf =

[0, 1]\C = [0, 1], where C is the Cantor set, see [9], [5]. Therefore the Cantor
function f belongs to L1([0, 1]), and R ( L1([0, 1]), hence R ( Lc([0, 1]). We
emphasize that Cc(X) = R, but R ( Lc(X), and this can be considered as an
advantage of Lc(X) over Cc(X), in this case.

Remark 2.10. In [15], a first countable compact space X (resp., in [26], a non-
first countable compact space X) is constructed such that L1(X) = R.

We are interested in characterizing topological spaces X for which Lc(X) =
C(X). In the following proposition we have a simple result, which is similar to
RPS-Theorem. Let us recall that in a commutative ring R by an annihilator
ideal I, we mean I = Ann(S) = {r ∈ R : rS = 0}, where S 6= {0} is a
nonempty subset of R.

Proposition 2.11. If X is an almost discrete space (i.e., I(X), the set of
isolated points of X, is dense in X), then L1(X) = LF (X) = Lc(X) = C(X).
In particular, if every nonzero annihilator ideal of C(X), where X is any space,
contains a nonzero minimal ideal, then the latter equalities hold.

Proof. If f ∈ C(X), then Ccf ⊇
⋃

x∈I(X){x} = I(X). Hence Ccf = X, i.e.,
f ∈ L1(X). Finally, we first recall that C(X) contains many nonzero zero-
divisors (note, for each 0 6= f ∈ C(X), (f − |f|)(f + |f|) = 0. Hence nontrivial
annihilator ideals in C(X) always exist. Consequently, by our assumption the
socle of C(X) is not zero, i.e., CF (X) 6= 0. We now claim that Ann

(
CF (X)

)
=

0. To see this, if I = Ann
(
CF (X)

)
6= 0, then I must contain a nonzero minimal

ideal, hence I ∩CF (X) 6= 0. But, (I ∩CF (X))2 = 0 and since C(X) is reduced,
we infer that I ∩CF (X) = 0, which is absurd. This means that we have already
shown that Ann

(
CF (X)

)
= 0, which by [20, Proposition 2.1] is equivalent to

the density of I(X) in X, hence we are done by the first part. �

Before, presenting the next fact, we evidently note that every scattered space
is an almost discrete space, for if x ∈ X and Ux is a neighborhood of x, then
Ux has an isolated point x0. Since Ux is open, x0 is an isolated point of X,
too. Hence x0 ∈ Ux ∩ I(X) 6= ∅, therefore I(X) = X.

Proposition 2.12. If X is a scattered space, then L1(X) = LF (X) = Lc(X) =
C(X). In particular, if X is a compact scattered space, then the latter rings
coincide with Cc(X).

Proof. By the above comment and RPS-Theorem we are done. �

The following example shows that the converse of the above corollary is not
valid.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 187



O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

Example 2.13. Let for each point x ∈ Q, the basic neighborhood of x be the
singleton {x}, and for each x ∈ Qc, the basic neighborhood of x be the usual
open interval containing x. This constitutes a topology on X = R, and it is
clearly, a Hausdorff normal space which is almost discrete, since I(X) = Q.
Hence Lc(X) = C(X), but X is not scattered.

In view of RPS-Theorem we may naturally define a compact space X to
be scattered if given any f ∈ C(X) and any x ∈ X, there exists a compact
neighborhood Vf of x such that |f(V ◦f )| ≤ ℵ0. Motivated by this we give the
following definition.

Definition 2.14. A space X is called locally scattered if given any f ∈ C(X)
and a nonempty open set G, there exists a compact subset Vf of X in G, with
∅ 6= V ◦f ⊆ G and |f(V ◦f )| ≤ ℵ0.

The space βX where X is discrete is locally scattered. Clearly, every scat-
tered space is a locally scattered space, but the converse is not true. For
example, βN is a locally scattered space which is not scattered, for βN\N has
no isolated point (note, each clopen subset of βN\N has the same cardinality
as βN\N, see [13, 6S(4)]).
Lemma 2.15. Let X be a locally scattered space. Then every open C-embedded
subset of X (e.g., any clopen subset) is also locally scattered.

Proof. Let Y be an open C-embedded subset of X, and G be an open subset
in Y , and f ∈ C(Y ). Since Y is C-embedded in X, we infer that there exists
g ∈ C(X) such that g|Y = f. Clearly, G is open in X and by our assumption,
there exists a compact subset Vg in G such that ∅ 6= V ◦g ⊆ G ⊆ Y , |g(V ◦g )| ≤ ℵ0.
Thus Vg is compact in Y in G with |f(V ◦g )| = |g(V ◦g )| ≤ ℵ0, i.e., Y is locally
scattered. �

Let us recall that a Hausdorff space X is locally compact if and only if each
point in X has a compact neighborhood. Clearly, every compact Hausdorff
space is locally compact. The following result is somewhat similar to RPS-
Theorem.

Theorem 2.16. Let X be a compact space. Then Lc(X) = C(X) if and
only if X is locally scattered. In particular, if X is a discrete space and Y
is a non-scattered clopen subset of βX (e.g., X = N and Y = βN), then
Lc(Y ) = C(Y ) = C

∗(Y ) 6= Cc(Y ).
Proof. First, we assume that X is compact and Lc(X) = C(X). Now, for each
f ∈ C(X) we have Cf = X. Hence for any nonempty open subset G in X
there exists an open subset Uf in X such that |f(Uf )| ≤ ℵ0, Uf ∩ G 6= ∅.
Since the open subsets of a locally compact space are locally compact, we
infer that Uf ∩ G is locally compact. Consequently, any neighborhood of a
point x ∈ Uf ∩ G contains a compact neighborhood, Vf say, of x. Hence
x ∈ V ◦f ⊆ Vf ⊆ Uf ∩ G ⊆ X and |f(V ◦f )| ≤ |f(Uf )| ≤ ℵ0, which means that X
is locally scattered and we are done. The converse is evident by Definition 2.14,

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 188



On the locally functionally countable sub algebra of C(X)

and the definition of Lc(X). For the last part, we notice that Y as a closed
subset of βX is compact and by Lemma 2.15, it is locally scattered. Now by
the first part and the compactness of Y , we have Lc(Y ) = C(Y ) = C

∗(Y ).
But, in view of RPS-Theorem and the fact that Y is not scattered, we infer
that Cc(Y ) 6= C(Y ), and we are done. �

The previous proof immediately yields the following fact, too.

Corollary 2.17. Let X be a locally compact space. Then Lc(X) = C(X) if
and only if X is locally scattered.

We recall that if the set of open neighborhoods of a point P in X is closed
under countable intersection, then P is called a P-point. The set of all P-points
of X is denoted by PX and X is called a P-space if PX = X. An interesting
result due to A. W. Hager asserts that a P-space X is functionally countable
(i.e., C(X) = Cc(X)) if and only if it is pseudo-ℵ1-compact (i.e., each locally
finite family of open sets is countable), see [21, Proposition 3.2]. This result is
extended to Cc(X) = C

F (X) in [11, Proposition 4.2]. The following is also a
counterpart of the latter result.

Proposition 2.18. If PX = X (in particular, if X is a P-space), then
L1(X) = LF (X) = Lc(X) = C(X).

Proof. For each f ∈ C(X) and x ∈ PX there exists an open neighborhood Ux of
x such that f is constant on Ux, see [13, 4L(3)]. Therefore C

c
f ⊇

⋃
|f(Ux)|=1

Ux ⊇
PX, hence f ∈ L1(X). �

We note that βN is not a P-space while L1(βN) = LF (βN) = Lc(βN) =
C(βN). By [13, 6V(6)], βN\N has a dense set of P-points, hence L1(βN\N) =
LF (βN\N) = Lc(βN\N) = C(βN\N).
Remark 2.19. Let X be a P-space without isolated points, see [13, 13 P], then
X is not almost discrete. But by Proposition 2.18, L1(X) = LF (X) = Lc(X) =
C(X), see also Proposition 2.11.

Let us borrow the following definition from [16].

Definition 2.20. A topological space X is called locally functionally countable
if every point x ∈ X is countably P-point, in the sense that there exists an open
neighborhood Ux of x such that C(Ux) = Cc(Ux).

The following result implies that if a space X is second countable or a
compact space, then X is locally functionally countable if and only if it is
functionally countable (i.e., C(X) = Cc(X)).

Proposition 2.21. Let X be a Lindelöf space. Then X is locally functionally
countable if and only if it is functionally countable.

Proof. It is evident that every functionally countable space is locally function-
ally countable (note, for each x ∈ X take Ux = X). Conversely, let X be locally
functionally countable, then for each f ∈ C(X), f(X) = f(

⋃
x∈X Ux), where

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O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

Ux is an open neighborhood of x with C(Ux) = Cc(Ux). Since X is Lindelöf
and C(X) ⊆ C(Ux), for each x ∈ X, we infer that f(X) = f(

⋃∞
i=1 Uxi) =⋃∞

i=1 f(Uxi) is countable, and we are done. �

The next result shows that for every locally functionally countable space X,
C(X) coincides with Lc(X). But the converse is not true in general, see Exam-
ple 2.13 (note, R with the topology in this example is not locally functionally
countable, for no irrational number is a countably P-point).

Proposition 2.22. If X is a locally functionally countable space, then Lc(X) =
C(X).

Proof. We must show that for each f ∈ C(X), Cf = X. Let G ⊆ X be an
open set in X and x ∈ G. Since X is locally functionally countable, there
exists an open neighborhood Ux of x such that C(Ux) = Cc(Ux). Clearly
|f(Ux)| = |(f|Ux)(Ux)| ≤ ℵ0. Now, x ∈ Ux

⋂
G 6= ∅ and Ux ⊆ Cf imply that

Cf
⋂
G 6= ∅, hence Cf = X. �

It is clear that if Y is a subset of X such that for each f ∈ C(X), f|Y
is constant, then Y must be a singleton. For otherwise, if y1, y2 ∈ Y and
y1 6= y2, then by complete regularity of X there exists f ∈ C(X) such that
f(y1) 6= f(y2), which is absurd. Hence the following definition, which is also
needed, is now in order.

Definition 2.23. If Y is a subset of a space X, then the set of all f ∈ C(X)
such that f|Y is constant is a subalgebra of C(X), denoted by C1(Y ). Naturally,
we say that Y is constant with respect to a subring A of C(X) if A ⊆ C1(Y ).

We note that for every topological space X, C1(X) = C(X) if and only if
X is singleton. If Y is a proper closed subset of X, then R ( C1(Y ).

The following proposition is evident.

Proposition 2.24. Let X be a topological space and Y be a connected subset of
X, then Cc(X) ⊆ C1(Y ). In particular, if X\Y is countable, then A ⊆ C1(Y )
if and only if A ⊆ Cc(X).

We conclude this section with the following fact whose proof is evident by
the complete regularity of X.

Corollary 2.25. For any subspace Y of X, R ⊆ C1(Y ) ⊆ C(X). Moreover,
C1(Y ) = R if and only if Y is dense in X.

Proof. For the last part we note that if x /∈ Y , then there exists f ∈ C(X)
with f(x) = 0 and f(Y ) = 1, i.e., C1(Y ) 6= R. This implies that Y = X in case
C1(Y ) = R. Conversely, let Y = X and take f ∈ C(X) such that f ∈ C1(Y ),
then f(Y ) = c, where c ∈ R. Consequently, f = c in C(X), for Y is dense in
X, hence we are done. �

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 190



On the locally functionally countable sub algebra of C(X)

3. zl-IDEALS

We remind the reader that many facts in the context of C(X) can be ex-
tended naturally to Lc(X), similarly to Cc(X), see [10]. The proofs of most
of the results in this section follow mutatis mutandis from the proofs of their
corresponding results in [10]. Therefore, we state them without proofs, for the
record, but give pertinent references for their corresponding proofs (note, the
reason that we emphasize on the recording of these facts here is because we do
believe that Lc(X) and Cc(X), are eligible to play appropriate roles as com-
panions of C(X), in the future studies in the context of C(X), see for example,
the comment in the first two lines of the introduction in [4].

Definition 3.1. A space X is said to be locally countably pseudocompact
(briefly, lc-pseudocompact) if L∗c(X) = Lc(X), where L

∗
c(X) = Lc(X)∩C∗(X).

The next three results are the counterparts of [13, Theorem 1.7, Corollary
1.8, and Theorem 1.9].

Proposition 3.2. Every homomorphism ϕ : Lc(X) → Lc(Y ) takes L∗c(X) into
L∗c(Y ).

Corollary 3.3. If Y is not a lc-pseudocompact space, then Lc(Y ) can not be
a homomorphic image of any L∗c(X).

Corollary 3.4. Let ϕ be a homomorphism from Lc(X) into Lc(Y ) whose image
contains L∗c(Y ), then ϕ(L

∗
c(X)) = L

∗
c(Y ).

If f ∈ Lc(X) and f > 0, then there exists g ∈ Lc(X) with f = g2. We also
note that whenever f ∈ Lc(X) and fr ∈ C(X) where r ∈ R, then fr ∈ Lc(X).
We recall that all positive units in Lc(X) have the same number of square roots,
see [13, 1B(1)]. The following proposition and its corollary are the counterparts
of [13, 1D(1)] and [10, Lemma 2.4] for Lc(X). Since the latter facts play a basic
role in the context of C(X), we present sketch of proofs for these counterparts.

Proposition 3.5. If f, g ∈ Lc(X) and Z(f) is a neighborhood of Z(g), then
f = gh for some h ∈ Lc(X).
Proof. We have Zl(g) ⊆ intZl(f). Put

h(x) =

{
0 , x ∈ Zl(f)

f(x)
g(x)

, x /∈ intZl(f)

therefore h ∈ C(X), and Ch ⊇ Cf ∩ C1/g = Cf ∩ Cg = X. Hence h ∈ Lc(X)
and f = gh. �

Corollary 3.6. If f, g ∈ Lc(X), and |f| ≤ |g|r, r > 1, then f = gh for some
h ∈ Lc(X). In particular, if |f| ≤ |g|, then whenever fr is defined for r > 1,
fr is a multiple of g.

Proof. Let

h(x) =

{
0 , x ∈ Zl(g)

f(x)
g(x)

, x /∈ Zl(g)

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 191



O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

then h ∈ C(X), Ch ⊇ Cf ∩ Cg = X. Hence h ∈ Lc(X) and f = gh. �

Proposition 3.7. If f ∈ Lc(X), then there exists a positive unit u ∈ Lc(X)
with (−1 ∨ f) ∧ 1 = uf.

Proof. Put

u(x) =

{
1 , − 1 ≤ f(x) ≤ 1

1
|f(x)|

, 1 ≤ |f(x)|
Clearly Cu = Cf , hence u ∈ Lc(X) and (−1 ∨ f) ∧ 1 = uf. So if f ∈ Lc(X),
then f and (−1 ∨ f) ∧ 1 belongs to an ideal of Lc(X) �

Remark 3.8. The previous results are also true if we replace Lc(X) by either
LF (X) or L1(X).

Convention. Let us put Zl(X) = {Z(f) : f ∈ Lc(X)}, ZF (X) = {Z(f) : f ∈
LF (X)}, and Z1(X) = {Z(f) : f ∈ L1(X)}, where X is a topological space.

Definition 3.9. Two subsets A and B of a topological space X are said to
be locally countably separated (briefly, lc-separated) in X if there is an element
f ∈ Lc(X) such that f(A) = 1, f(B) = 0.

The following result is the counterpart of [13, Theorem 1.15], [10, Theorem
2.8] .

Theorem 3.10. Two subsets A, B of a space X are lc-separated if and only if
they are contained in disjoint members of Zl(X). Moreover, lc-separated sets
have disjoint zero-set neighborhoods in Zl(X).

Clearly, if a < b and f ∈ Lc(X) such that f(x) ≤ a, ∀x ∈ A, and f(x) ≥ b,
∀x ∈ B, where A, B are subsets of X, then A, B are lc-separated in X.

Corollary 3.11. If A, B are lc-separated in X, then there are zero-sets Z1, Z2
in Zl(X) with A ⊆ X \ Z1 ⊆ Z2 ⊆ X \ B.

Definition 3.12. ∅ 6= F ⊆ Zl(X) is called a zl-filter on X if F satisfies the
following conditions.

(1) ∅ /∈ F .
(2) Z1, Z2 ∈ F , then Z1 ∩ Z2 ∈ F .
(3) Z ∈ F , Z′ ∈ Zl(X) with Z′ ⊇ Z, then Z′ ∈ F .

Prime zl-filter and zl-ultrafilter are defined similarly to their counterparts
in [13]. If I is an ideal of Lc(X), then Zl[I] = {Z(f) : f ∈ I} is a zl-filter on
X. Conversely, if F is a zl-filter on X, then Z

−1[F ] = {f ∈ Lc(X) : Z(f) ∈ F}
is an ideal in Lc(X). Moreover, every zl-filter F is of the form F = Zl[I] for
some ideal I in Lc(X) and for any ideal J in Lc(X), Z

−1[Zl[J]] is an ideal
in Lc(X) containing J. In Example 2.13, we consider the identity function
i : (R, T ) → R, clearly i ∈ Lc(R) = C(R). Now, put I = (i), then Zl(I) = {0}.
Clearly, f(x) = x1/3 ∈ Lc(R), f ∈ Z−1[Zl[I]]\I. Hence the following definition
is in order.

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On the locally functionally countable sub algebra of C(X)

Definition 3.13. An ideal I in Lc(X) is called a zl-ideal if whenever Z(f) ∈
Zl[I] and f ∈ Lc(X), then f ∈ I. Similarly, zF -ideal and z1-ideal are defined,
see the previous convention.

Clearly, every zl-ideal is an intersection of prime ideals in Lc(X). Similarly,
every zF -ideal and z1-ideal is an intersection of prime ideals in LF (X) and
L1(X).

We emphasize again that the proofs of the following results are the same as
the proofs of their counterparts in C(X) and Cc(X), see [10], and [13]. One
can easily observe that these results up to Proposition 3.24 and including it are
also valid for Lc(X) and LF (X). The next theorem is the counterpart of [13,
Theorem 2.9], [10, Theorem 2.13].

Theorem 3.14. Let P be any zl-ideal in Lc(X). Then the following statements
are equivalent.

(1) P is a prime ideal in Lc(X).
(2) P contains a prime ideal in Lc(X).
(3) For all f, g ∈ Lc(X), if fg = 0, then f ∈ P or g ∈ P.
(4) For each f ∈ Lc(X), there exists a zero-set in Zl[P ] on which f does

not change sign.

Corollary 3.15. Every prime ideal in Lc(X) is contained in a unique maximal
ideal in Lc(X).

Clearly if P is a prime ideal in Lc(X), then Zl[P ] is a prime zl-filter, and if
F is a prime zl-filter, then Z

−1
l [F ] is a prime zl-ideal. It is evident that every

prime zl-filter is contained in a unique zl-ultrafilter. The following lemma is
the counterpart of [10, Lemma 3.1], also see [28].

Lemma 3.16. Let f, g, l ∈ Lc(X), Z(f) ⊇ Z(g) ∩ Z(l) and define

h(x) =

{
0 , x ∈ Z(g) ∩ Z(l)
fg2

g2+l2
, x /∈ Z(g) ∩ Z(l)

, k(x) =

{
0 , x ∈ Z(g) ∩ Z(l)
fl2

g2+l2
, x /∈ Z(g) ∩ Z(l)

Then we have the following conditions.

(1) |k| ∨ |h| ≤ |f|.
(2) f = h + k.
(3) fl2 = k(g2 + l2), fg2 = h(g2 + l2).
(4) h, k ∈ Lc(X).
(5) Ch ⊇ Cf ∩ Cg ∩ Cl and Ck ⊇ Cf ∩ Cg ∩ Cl.

The following results are the counterparts of [10, Corollary 3.2 to Corollary
3.8].

Lemma 3.17. Let A, B be two zl-ideals in Lc(X). Then either A+B = Lc(X)
or A + B is a zl-ideal.

Corollary 3.18. Let F = {Ai}i∈I be a collection of zl-ideals in Lc(X). Then
either

∑
i∈I

Ai = Lc(X) or
∑
i∈I

Ai is a zl-ideal.

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O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

Proposition 3.19. Every minimal prime ideal in Lc(X) is a zl-ideal.

Corollary 3.20. Let F = {Pi}i∈I be a collection of minimal prime ideals in
Lc(X). Then either

∑
i∈I

Pi = Lc(X) or P =
∑
i∈I

Pi is a prime ideal in Lc(X).

Proposition 3.21. A prime ideal P in Lc(X) is absolutely convex.

Proposition 3.22. The sum of a collection of semiprime ideals in Lc(X) is a
semiprime ideal or is the entire ring Lc(X).

Proposition 3.23. Let P be a prime ideal in Lc(X). Then the ring Lc(X)/P
is totally ordered and its prime ideals are comparable.

The next corollary is much stronger than Corollary 3.20 whose proof is
similar to [10, Corollary 3.9].

Proposition 3.24. Let {Pi}i∈I be a collection of semiprime ideals in Lc(X)
such that at least one of Pi’s is prime, then

∑
i∈I Pi is a prime ideal or all of

Lc(X).

All the previous results beginning with Theorem 3.14, are also valid for
L1(X). The following theorem is the counter part of [10, Theorem 3.10], see
also the comment preceding [10, Theorem 3.10].

Theorem 3.25. Let I be an ideal in Lc(X). Then I and
√
I have the same

largest zl-ideal.

4. LOCALLY COUNTABLE COMPLETELY REGULAR SPACES

Definition 4.1. A Hausdorff space X is called locally countable completely
regular (briefly, lc-completely regular) if whenever F ⊆ X is a closed set and
x ∈ X\F , then there exists f ∈ Lc(X) with f(F) = 0 and f(x) = 1.

We should remind the reader that, in this section, whenever the proof of a
result is very similar to the proof of its counterpart in the literature, the proof
is avoided.

The proof of the following result is evident.

Proposition 4.2. A Hausdorff space X is lc-completely regular if and only if
whenever F ⊆ X is closed and x ∈ X \ F, then x and F have two disjoint
zero-set neighborhoods in Zl(X). Consequently, there exist g, h ∈ Lc(X) with
x ∈ X \ Z(h) ⊆ Z(g) ⊆ X \ F.

Clearly X is a lc-completely regular space if and only if F = {Z(f) : f ∈
Lc(X)} is a base for the closed sets in X or equivalently if and only if B =
{int(Z(f)) : f ∈ Lc(X)} is a base for the open sets in X. The next proposition
is the counterpart of [13, 3.11(a)], [10, Proposition 4.3].

Proposition 4.3. Let X be a lc-completely regular space and A, B be two
disjoint closed sets in X such that A is compact, then there is f ∈ Lc(X) with
f(A) = 0 and f(B) = 1.

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On the locally functionally countable sub algebra of C(X)

Proposition 4.4. Let X be a compact space. Then X is lc-completely regular
if and only if Lc(X) separates points in X.

Since Cc(X) is a subring of Lc(X), the next result is evident, see [10, Propo-
sition 4.4]

Proposition 4.5. If X is a zero-dimensional space, then X is a lc-completely
regular space.

The following fact is also similar to [13, Theorem 3.6], and [10, Corollary
4.5].

Proposition 4.6. Let X be a Hausdorff space. Then X is a lc-completely
regular space if and only if its topology coincides with the weak topology induced
by Lc(X).

We recall that X is a P-space (resp., CP-space) if and only if C(X) (resp.,
Cc(X)) is a regular ring, see [13, 4J] and [10]. In [10], it is shown that if C(X)
is regular, then so too is Cc(X). If X is zero-dimensional, then the regularity
of C(X) and Cc(X) coincide. We have already observed, see Proposition 2.18,
that if X is a P-space, then L1(X) = LF (X) = Lc(X) = C(X). The next
definition is now in order.

Definition 4.7. A space X is called a locally countably P-space (briefly,
LCP-space) if Lc(X) is regular.

By the above comment we have the following result.

Proposition 4.8. Every P-space is LCP-space.

Proposition 4.9. If A is any regular subring of C(X) such that Cc(X) ⊆ A ⊆
C(X), then Cc(X) is regular. In particular, if Lc(X) is regular, then Cc(X) is
regular, too.

Proof. Let A be a regular ring, we must show that for each f ∈ Cc(X), there
exists g ∈ Cc(X) such that f = f2g. Since A is regular, there is h ∈ A with
f = f2h. Consequently, f = f2g, where g = h2f. It is also evident that
Z(f) ⊆ Z(g) and g(x) = 1

f(x)
, whenever x /∈ Z(f). Hence |g(X)| = |f(X)| ,

i.e., g ∈ Cc(X), and we are done. �

Corollary 4.10. Let X be a zero-dimensional space. Then X is P-space if
and only if any of the rings Cc(X), Lc(X) is regular.

Remark 4.11. It is wroth mentioning that if X is a zero-dimensional space,
then the regularity of C(X), Lc(X), Cc(X), and C(X, K) (where C(X, K), is
a subring of C(X) whose elements take values in K, a subfield of R) coincide,
see the above proposition and [10, Remark 7.5].

The following theorem is the counterpart of [10, Theorem 5.5] and its proof
is also the same as the proof of its counterpart. We present a proof for the sake
of completeness.

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O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

Theorem 4.12. A space X is a LCP-space if and only if every zero-set in
Zl(X) is open. Moreover, in this case whenever {fi}∞i=1 is a countable set in
Lc(X), then

∞⋂
i=1

Zl(fi) is an open zero-set in Zl(X).

Proof. Let X be a LCP-space and f ∈ Lc(X), hence f = f2g for some g ∈
Lc(X). It is evident that e = fg is an idempotent in Lc(X) and Z(f) = Z(e) =
X\Z(1 − e) is clopen. Conversely, let Z(f) be open for each f ∈ Lc(X), we
are to show that Lc(X) is regular. Since Z(f) = Z(f

2) for all f ∈ Lc(X),
we infer that f = f2g for some g ∈ Lc(X), by Proposition 3.5. Hence Lc(X)
is regular. Finally, let I be an ideal in Lc(X), generated by {fi}∞i=1, i.e.,
I =

∑∞
i=1 fiLc(X). If I = C(X), then we are done in this case, for

∞⋂
i=1

Zl(fi) =

∅. Hence we assume that I 6= C(X). Since Lc(X) is regular, we infer that
I =

∑∞
i=1 ⊕eiLc(X), where each ei, i ∈ I is an idempotent in Lc(X), and for

each i 6= j, eiej = 0, see [10, Theorem 5.5], or [17, Lemma 2], [8, Proposition
1.4]. If x ∈ X, ej(x) 6= 0, then for each i 6= j, ei(x) = 0, and

⋂
Z[I] =

∞⋂

i=1

Z(fi) =

∞⋂

i=1

Z(ei)

Now, we may define g =
∑∞

i=1
ei

pi(1+ei)
, where p ≥ 2 is a real number. Clearly

g ∈ C(X), and Z(g) =
⋂∞

i=1 Z(ei). On the other hand for each x ∈ X,
there exists at most a unique i ≥ 1 such that ei(x) 6= 0. Therefore g(x) =

ei(x)
pi(1+ei(x))

= 1
2pi

. Hence g(X) ⊆ {0, 1
2p
, 1
2p2

, . . .} i.e., g ∈ Cc(X), therefore
g ∈ Lc(X). �

Remark 4.13. In view of the previous proof we may record an interesting fact,
which follows. Let X be a LCP-space and {fi}i∈I be an infinite countable
set of elements in C(X), then

⋂
i∈I Z(fi) = Z(g), where g ∈ Cc(X) ⊆ Lc(X)

can be chosen with the property that g(X) is an infinite subset of an arbitrary
subfield of R.

It is well known that X is a P-space if and only if every Gδ-set is open, see
[13, 4J(3)]. The following theorem is the counterpart of this result, see also
[10, Corollary 5.7].

Corollary 4.14. Let X be a lc-completely regular LCP-space. Then every
Gδ-set A containing a compact set S contains a zero-set in Zl(X) containing
S. In particular, every lc-completely regular LCP-space is a P-space.

If Mlp = Mp ∩Lc(X) and Olp = Op ∩Lc(X), where p ∈ X and Op is the ideal
of C(X) consisting of all f in C(X) for which Z(f) is a neighborhood of p. It
goes without saying that Mlp is a maximal ideal in Lc(X) and O

l
p is a zl-ideal

in Lc(X). The following theorem is the counterpart of [13, 4J], [10, Theorem
5.8].

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 196



On the locally functionally countable sub algebra of C(X)

Theorem 4.15. Let X be a topological space. Then the following statements
are equivalent.

(1) X is a LCP-space.
(2) Lc(X) is a regular ring.
(3) Each ideal in Lc(X) is a zl-ideal.
(4) Each prime ideal in Lc(X) is a maximal ideal.
(5) For each p ∈ X, Mlp = Olp.
(6) Every zero-set in Zl(X) is open.
(7) Each ideal in Lc(X) is an intersection of maximal ideals.
(8) For all f, g ∈ Lc(X), (f, g) = (f2 + g2).
(9) For every f ∈ Lc(X), Zl(f) (X \ Zl(f)) is C-embedded.

(10) If {fi : i ∈ N} ⊆ Lc(X), then
⋂∞

i=1 Zl(fi) is an open zero-set in Zl(X).

The following results are the counterparts of [11, Proposition 2.5] and [11,
Corollary 2.6].

Proposition 4.16. Lc(X) is regular if and only if every pseudoprime ideal in
Lc(X) is prime.

Corollary 4.17. Let X be a lc-completely regular. Then every pseudoprime
ideal in C(X) is prime if and only every pseudoprime ideal in Lc(X) is prime.

The following theorem is similar to [13, Theorem 4.11], [11, Theorem 3.8].

Theorem 4.18. Let X be a lc-completely regular space, then the following
statements are equivalent.

(1) X is compact.
(2) Every ideal of Lc(X) is fixed.
(3) Every maximal ideal of Lc(X) is fixed.
(4) Every prime ideal of Lc(X) is fixed.

If X is any topological space and x ∈ X, Mlx = Mx ∩ Lc(X), then as we
pointed out earlier Mlx is a maximal ideal of Lc(X) and in fact

Lc(X)
Mlx

∼= R.
Consequently, the Jacobson radical of Lc(X) is zero.

Definition 4.19. A maximal ideal M in Lc(X) is called a real maximal ideal

of Lc(X) if
Lc(X)

M
∼= R. A topological space X is called locally countably

realcompact space (briefly, lc-realcompact) if every real maximal ideal M of
Lc(X) is of the form M = M

l
x for some x ∈ X.

The following results are the counterparts of [13, 10.5(c)] and [11, Theorem
3.11].

Theorem 4.20. X is a lc-realcompact space if and only if each nonzero ho-
momorphism from Lc(X) into R is a valuation map.

If X is a compact zero-dimensional space, the corresponding x → Mlx is
one-one from X onto the set of maximal ideals of Lc(X), say Max(Lc(X)), and
hence the space X is homeomorphic to Max(Lc(X)) with the Stone topology

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O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

(note, the proof is similar to [13, 4.9(a)], see also the comment above [10,
Theorem 3.9] and [19]). The proof of the following result which is similar to
its counterpart in [13, Theorem 8.3], is omitted.

Proposition 4.21. Two zero-dimensional lc-realcompact spaces X and Y are
homeomorphic if and only if Lc(X) ∼= Lc(Y ).

We recall that if X, Y are compact zero-dimensional spaces, then C(X) ∼=
C(Y ) if and only if Cc(X) ∼= Cc(Y ). In what follows we show that this result
also holds if we replace Cc(X) by Lc(X), but the proof is not as evident.

Theorem 4.22. Let X and Y be two lc-completely regular compact spaces
(e.g., zero-dimensional compact spaces). Then X and Y are homeomorphic
if and only if Lc(X) ∼= Lc(Y ). In particular, if X, Y are compact zero-
dimensional spaces, then Lc(X) ∼= Lc(Y ) if and only if Cc(X) ∼= Cc(Y ) if
and only if CF (X) ∼= CF (Y ) if and only if C(X) ∼= C(Y ).
Proof. Clearly if Lc(X) ∼= Lc(Y ), then Max(Lc(X)) and Max(Lc(Y )) are
homeomorphic (with the Stone topology), i.e., X, Y are homeomorphic, see the
comment preceding Proposition 4.21. Conversely, let ϕ : X → Y be a home-
omorphism from X onto Y . If f ∈ Lc(Y ), then we claim that foϕ ∈ Lc(X).
To see this, since f ∈ Lc(Y ), we infer that Y = Cf =

⋃
i∈I Vi, where for each

i ∈ I, Vi is open in Y and |f(Vi)| ≤ ℵ0. Let us put Ui = ϕ−1(Vi), where i ∈ I.
Clearly Ui is open in X and |foϕ(Ui)| = |foϕ(ϕ−1(Vi))| = |f(Vi)| ≤ ℵ0, hence
Cfoϕ ⊇

⋃
i∈I Ui. Since ϕ is open (note, ϕ

−1 is continuous), we infer that

X = ϕ−1(Y ) = ϕ−1(
⋃

i∈I

Vi) ⊆ ϕ−1(
⋃

i∈I

Vi) =
⋃

i∈I

ϕ−1(Vi) =
⋃

i∈I

Ui

Therefore Cfoϕ = X, i.e., foϕ ∈ Lc(X). Now we define σ : Lc(Y ) → Lc(X)
with σ(f) = foϕ. It is evident that σ is an isomorphism from Lc(Y ) onto
Lc(X). The last part is evident. �

Remark 4.23. The above result shows that if X, Y are compact zero-dimensional
spaces, such that C(X) ∼= C(Y ), then Lc(X) ∼= Lc(Y ). In the comment fol-
lowing [11, Corollary 9.5], it is observed that whenever X, Y are two arbi-
trary spaces (not necessary compact zero-dimensional) and C(X) ∼= C(Y ),
then Cc(X) ∼= Cc(Y ) and CF (X) ∼= CF (Y ) (i.e., Cc(X) and CF (X) are al-
gebraic objects). This naturally raises the question that whether Lc(X) is
also an algebraic object, too (i.e., if C(X) ∼= C(Y ), then is Lc(X) ∼= Lc(Y ))?
Clearly, if X, Y are strongly zero-dimensional spaces with C(X) ∼= C(Y ), then
Lc(βX) ∼= Lc(βY ).

Let us recall that a commutative ring R is selfinjective (resp., ℵ0-selfinjective),
if every homomorphism f : I → R, where I is an ideal (resp., countably gen-
erated ideal) in R, can be extended to f̂ : R → R. We recall that a subset
S of a commutative ring R is said to be orthogonal, provided xy = 0 for all
x, y ∈ S with x 6= y. In the following result we show that [10, Theorem 6.10] is
also true for Lc(X). In contrast to the proofs of some of the previous results,

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On the locally functionally countable sub algebra of C(X)

we should emphasize that the next proof can not be easily obtained from the
proof of its counterpart (i.e., [10, Theorem 6.10]). It is well known that the
ℵ0-selfinjectivity of a ring is not a consequence of its regularity, in general, see
[14, Examples 14.7, 14.9]. But, the following worthwhile fact shows that Lc(X)
as well as C(X) and Cc(X) have this rare property, see [8], [10]. We should
remind the reader that CF (X) does not satisfy this property in general, see
[10, Remark 6.11, Example 7.1] (note, CF (X) is always regular, see [11, the
comment preceding Proposition 4.2].

Theorem 4.24. Let X be a topological space. Then Lc(X) is regular if and
only if Lc(X) is ℵ0-selfinjective.

Proof. If Lc(X) is ℵ0-selfinjective, then Lc(X) is regular by [18, Proposition
1.2], or [10, Lemmas 6.7, 6.8, Remark 6.9]. Conversely, by [18, Lemma 1.9] and
[10, Lemma 6.8, Remark 6.9], it suffices to show that if S is an orthogonal subset
in Lc(X), then there exists f ∈ Lc(X) such that for each g ∈ S, fg = g2. Let
S = {fi}∞i=1, where fi 6= 0, for each i ∈ I. Since Lc(X) is regular,

⋂∞
i=1 Z(fi) =

Z(h) is an open zero-set in Lc(X), by Theorem 4.12. Put Gi = X\Z(fi), for
each i ≥ 1. Since fifj = 0, hence Gi ∩ Gj = ∅, for each i 6= j, and Gi’s are
clopen for each i ≥ 1. Let us put G =

⋃∞
i=1 Gi, hence X =

⋃∞
i=1 Gi ∪ (X\G).

We may define f : X → R by f(x) =
{

fi(x) , x ∈ Gi
0 , x /∈ G i.e., f|Gi = fi for all

i ≥ 1 and f(x) = 0 for all x ∈ X\G. Hence f is continuous by [13, 1A(2)] and
we must show that f ∈ Lc(X). Let V ⊆ X be an arbitrary open set, then we
are to show that there exists an open set U in X such that |f(U)| ≤ ℵ0 and
U ∩ V 6= ∅. Now we consider two cases. First let V ⊆ X\G, then f(V ) = 0,
hence V ⊆ Cf . Otherwise V ∩ G 6= ∅, hence there exists a nonempty open
subset Gi such that V ∩ Gi 6= ∅. Since fi ∈ Lc(X) i.e., Cfi = X, hence there
exists an open set H ⊆ Cfi such that |fi(H)| ≤ ℵ0 and ∅ 6= H ∩ (V ∩ Gi) = U.
Now clearly, |f(U)| = |fi(U)| ≤ |fi(H)| ≤ ℵ0 i.e., we are done. Finally, we
claim that ffi = f

2
i , for each fi ∈ S and this complete the proof, by [18,

Lemma 1.9]. To this end, we note that if f(x) = 0, then x /∈ G, hence x /∈ Gi
for all i ≥ 1, i.e., x ∈ Z(fi), for all i ≥ 1. Thus ffi = f2i , on Z(f) for each
fi ∈ S. Since f(x) = fi(x), for each x ∈ Gi = X\Z(fi) and Z(f) ⊆ Z(fi) for
each i ≥ 1, we infer that ffi = f2i , for each fi ∈ S, hence we are done. �

Remark 4.25. Let X be an uncountable discrete space, then C(X) = Lc(X)
is selfinjective but Cc(X) is not selfinjective, see [10, Example 7.1, Remark
7.5]. More generally, if C(X) is ℵ0-selfinjective, then by [8, Theorem 1], X
is a P-space. Hence in view of Proposition 2.18, we have L1(X) = LF (X) =
Lc(X) = C(X). Moreover in view of Theorem 4.22 and Remark 4.11, we note
that the ℵ0-selfinjectivity of C(X), Lc(X), Cc(X), and C(X, K) coincide if X
is a zero-dimensional space.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 199



O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

5. The socle of Lc(X)

We recall that the socle of any commutative ring R, Soc(R), is the sum of its
nonzero minimal ideals (in fact, it can be written as the direct sum of some of
its nonzero minimal ideals). We recall that CF (X) = {f ∈ C(X) : |X\Z(f)| <
∞}, the socle of C(X), is a z-ideal. Clearly, CF (X) ⊆ LF (X) ⊆ Lc(X).
We show that CF (X) ⊆ Soc

(
Lc(X)

)
, i.e., CF (X) is a sum of minimal ideals of

Lc(X). In [20, Proposition 3.1], it is shown that if I is a minimal ideal of C(X),
then I = eC(X), where e is an idempotent such that e(x) = 1 and e(X\{x}) =
0, where x is an isolated point of X. Clearly C(X) = eC(X) ⊕ (1 − e)C(X),
eC(X) = eLc(X) = eCc(X), and Lc(X) = eLc(X) ⊕ (1 − e)Lc(X). We also
note that (1 − e)Lc(X) = (1 − e)C(X) ∩ Lc(X) is a maximal ideal in Lc(X),
therefore eLc(X) = eC(X) is a minimal ideal in Lc(X). Hence every minimal
ideal of C(X) is a minimal ideal in Lc(X), too. Therefore CF (X) is an ideal
in Lc(X), and CF (X) ⊆ Soc

(
Lc(X)

)
. We should also emphasize that since

Soc
(
Lc(X)

)
is a semisimple Lc(X)-module, hence CF (X) is a direct summand

of Soc
(
Lc(X)

)
as a Lc(X)-module. In the proof of Theorem 5.4, we shall

briefly observe that Soc
(
Lc(X)

)
⊆ Soc

(
Cc(X)

)
. Let us also recall that CF (X)

is the subring of C(X) whose elements have finite image. Hence, we have
CF (X) ⊆ CF (X) ⊆ Cc(X) ⊆ Lc(X) ⊆ C(X) and Soc

(
Cc(X)

)
= Soc

(
CF (X)

)
,

see [10], [11]. The following lemma which is similar to [11, Lemma 5.1], some-
how determines the minimal ideals of Lc(X). Let us first remind the reader that
if I is a nonzero minimal ideal in a reduced commutative ring R, then I = eR,
where e ∈ R is an idempotent (note, I = (a) = (a2), for every 0 6= a ∈ I, and
a = a2r, for some r ∈ R, now put e = ar).

In [11, Lemma 5.1], it is shown that if 0 6= e is an idempotent, then eCc(X)
is a minimal ideal in Cc(X) if and only if Z(1 − e) is connected. In the next
lemma the minimal ideals in Lc(X) are characterized, too.

Lemma 5.1. Let I be a nonzero minimal ideal in Lc(X), then I = eLc(X)
where e is an idempotent in Lc(X) such that Z(1−e) is connected. Conversely,
if I = eLc(X) where e 6= 0 is an idempotent in Lc(X) such that Z(1 − e) is a
constant subset of X with respect to Lc(X), then I is a minimal ideal in Lc(X).

Proof. Let I be a nonzero minimal ideal in Lc(X). Since Lc(X) is reduced,
I = eLc(X), where e is an idempotent in Lc(X). If Z(1 − e) is not connected,
there exists a nonempty clopen subset A ( Z(1 − e) (note, A is clopen in
X, too). Now define the idempotent e1 ∈ Lc(X) such that A = Z(1 − e1).
Clearly Z(1 − e1) ( Z(1 − e). Consequently, e1 = ee1 but e 6= e1e. Hence
e1Lc(X) ( eLc(X) = I and this contradicts the minimality of I. Conversely,
let I = eLc(X), where e ∈ Lc(X) such that Y = Z(1 − e) ⊆ X is a constant
subset of X with respect to Lc(X) (i.e., Lc(X) ⊆ C1(Y ), see Definition 3.16).
We are to show that I is minimal in Lc(X). It suffices to show that (1−e)Lc(X)
is a maximal ideal in Lc(X). Now we define ϕ : Lc(X) → R by ϕ(f) = f(Y ).

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 200



On the locally functionally countable sub algebra of C(X)

Clearly, kerϕ = (1 − e)Lc(X) and Lc(X)(1−e)Lc(X)
∼= R, hence (1 − e)Lc(X) is

maximal in Lc(X). �

In [11, Proposition 5.3], the socle of Cc(X) is characterized and in [11,
Remark 5.2, and the introduction of Section 5], it’s observed that CF (X) ⊆
Soc

(
Cc(X)

)
= Soc

(
CF (X)

)
. Next, we topologically characterize the socle of

Lc(X). The next proof is similar to the proof of [11, Proposition 5.3], but it’s
given for the sake of the reader.

Proposition 5.2. Let f ∈ Lc(X) be a nonunit element. If f ∈ Soc
(
Lc(X)

)
,

then X\Z(f) ⊆
⋃n

i=1 Ai, where n ∈ N and {A1, A2, . . . , An} is a set of mutually
disjoint clopen connected subsets of X. Conversely, if X\Z(f) ⊆

⋃n
i=1 Ai,

where n ∈ N and {A1, A2, . . . An} is a set of mutually disjoint clopen constant
subsets of X with respect to Lc(X), then f ∈ Soc

(
Lc(X)

)
. In Particular,

Soc
(
Lc(X)

)
is a zl-ideal in Lc(X).

Proof. We put Soc
(
Lc(X)

)
=

∑
i∈I ⊕eiLc(X), where each ei is an idempotent

in Lc(X), and eiLc(X) is a nonzero minimal ideal in Lc(X). Let f = ei1f1 +
ei2f2 + . . .+ einfn be an element in Soc

(
Lc(X)

)
, where fk ∈ Lc(X) and ik ∈ I,

k = 1, 2, . . . , n. We put Aik = Z(1 − eik), for each ik ∈ I, k = 1, 2, . . . , n.
Clearly, Aik , k = 1, 2, . . . , n, are clopen and connected, by Lemma 5.1. Since
the idempotent elements {ei : i ∈ I} are mutually orthogonal, we infer that
{Ai : i ∈ I, Ai = Z(1 − ei)} is a set of mutually disjoint clopen connected
subsets of X. If x /∈

⋃n
i=1 Ai, then eik (x) = 0, k = 1, 2, . . . , n, hence x ∈ Z(f).

Therefore X\Z(f) ⊆
⋃n

i=1 Ai. Conversely, let X\Z(f) ⊆
⋃n

i=1 Ai, where {Ai :
i ∈ I} is a set of mutually disjoint clopen constant subsets in X with respect
to Lc(X), we show that f ∈ Soc

(
Lc(X)

)
. Since each Ai is a clopen set, there

exists an idempotent ei, such that Ai = Z(1 − ei), where i = 1, 2, . . . , n. We
also note that each Ai is constant with respect to Lc(X), hence there is a set
of idempotents in Lc(X), {e1, . . . , en} say, which are mutually orthogonal and
each eiLc(X) is a minimal ideal in Lc(X), by Lemma 5.1. Clearly, f = e1f +
e2f + . . . + enf ∈ Lc(X) which belongs to Soc

(
Lc(X)

)
=

∑
i∈I ⊕eiLc(X). �

Remark 5.3. One can easily observe that if in the previous two results we trade
off Lc(X) with any R-subalgebra of Lc(X), A say, which contains Cc(X), then
the two results are also valid for A.

The next result determines spaces X such that the socles of Lc(X), Cc(X)
and hence of CF (X) coincide.

Theorem 5.4. Soc
(
Lc(X)

)
= Soc

(
Cc(X)

)
if and only if the clopen connected

subsets of X coincide with the clopen constant subsets of X with respect to
Lc(X).

Proof. Soc
(
Lc(X)

)
⊆ Soc

(
Cc(X)

)
, for if I is a minimal ideal in Soc

(
Lc(X)

)
,

then I = eLc(X) where e is an idempotent such that Z(1 − e) is connected, by
Lemma 5.1. Hence I is a minimal ideal in Cc(X), by [11, Lemma 5.1]. Now, let
I be a nonzero minimal ideal in Soc

(
Cc(X)

)
, so I = eCc(X), where e 6= 0, 1 is

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 201



O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

an idempotent and Z(1 − e) is a clopen connected subset in X, by [11, Lemma
5.1]. Hence by our hypothesis Z(1 − e) is constant with respect to Lc(X).
Therefore I = eLc(X) = eCc(X) is a minimal ideal of Lc(X), by Lemma 5.1,
hence it is in Soc

(
Lc(X)

)
. Conversely, let Soc

(
Lc(X)

)
= Soc

(
Cc(X)

)
, and

∅ 6= Y ⊆ X be a clopen constant subspace of X with respect to Lc(X) we
are to show that Y is connected. Clearly, there exists e ∈ Lc(X) such that
e(Y ) = 1, e(X\Y ) = 0. But Y = Z(1 − e), hence by Lemma 5.1, we infer
that e ∈ Soc

(
Lc(X)

)
= Soc

(
Cc(X)

)
. Consequently, Y = Z(1 − e) must be

connected, by [11, Proposition 5.3]. �

We remind the reader in [11, Theorem 6.6], it is proved that Soc
(
Cc(X)

)
=

CF (X) if and only if each clopen connected subsets of X consists of a single
isolated point. Motivated by this fact and Theorem 5.4, we present the next
result.

Theorem 5.5. If every proper nonempty clopen connected subset of X is sin-
gleton, (e.g., any totally disconnected space), then Soc

(
Lc(X)

)
= CF (X). Con-

versely, if Soc
(
Lc(X)

)
= CF (X), then every proper nonempty clopen constant

subspace of X with respect to Lc(X) is singleton.

Proof. Let every proper nonempty clopen connected subset of X be single-
ton, we are to show that Soc

(
Lc(X)

)
= CF (X). It is evident that CF (X) ⊆

Soc
(
Lc(X)

)
. Let I be a nonzero minimal ideal in Soc

(
Lc(X)

)
, so by Lemma

5.1, I = eLc(X), where e 6= 0, 1 is an idempotent and Z(1 − e) is a clopen
connected subset in X. Hence by our hypothesis Z(1 − e) is singleton. There-
fore I = eLc(X) = eC(X) is a minimal ideal in CF (X), by [20, Proposition
3.3]. Conversely, let CF (X) = Soc

(
Lc(X)

)
, and ∅ 6= Y ⊆ X be a clopen con-

stant subspace of X with respect to Lc(X). There exists e ∈ Lc(X) such that
e(Y ) = 1, e(X\Y ) = 0. Clearly, by Lemma 5.1, e ∈ Soc

(
Lc(X)

)
= CF (X),

hence eC(X) is a minimal ideal in CF (X), therefore Y = Z(1 − e) is single-
ton. �

The following remark is now immediate.

Remark 5.6. CF (X) = Soc
(
Lc(X)

)
= Soc

(
Cc(X)

)
if and only if each clopen

connected subset of X consists of a single isolated point. Consequently, if X
is zero-dimensional or totally disconnected, we have CF (X) = Soc

(
Lc(X)

)
=

Soc
(
Cc(X)

)
.

Let us recall that an ideal in a commutative ring R is essential if it intersects
every nonzero ideal of R nontrivially. It is well known and easy to show that a
nonzero ideal I in a reduced ring R (i.e., no nonzero element in R is nilpotent)
is essential if and only if Ann(I) = 0, see [3, Background and preliminary
results]. The proof of the following corollary is similar to [11, Corollary 5.4],
but we include the proof for the sake of the reader.

Corollary 5.7. Let X be a lc-completely regular space, and Soc
(
Lc(X)

)
=∑

i∈I ⊕eiLc(X), where eiLc(X) is a nonzero minimal ideal of Lc(X), and ei

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 202



On the locally functionally countable sub algebra of C(X)

is an idempotent for each i ∈ I. Put Y =
⋃

i∈I Z(1 − ei), then Soc
(
Lc(X)

)
is

essential in Lc(X) if and only if Y is dense in X.

Proof. Let Y =
⋃

i∈I Z(1−ei) be dense in X, we are to show that Soc
(
Lc(X)

)
is

essential in Lc(X). Since Lc(X) is reduced, in order to prove that Soc
(
Lc(X)

)

is essential in Lc(X) it suffices to show that Ann(Soc
(
Lc(X)

)
) = (0). We note

that f ∈ Ann(Soc
(
Lc(X)

)
) if and only if fei = 0 for each i ∈ I. Now, if fei = 0,

then f(Z(1 − ei)) = 0, hence f(Y ) = {0}. Since Y is dense in X we infer that
f = 0, and we are done. Conversely, let Soc

(
Lc(X)

)
be essential in Lc(X),

hence Ann(Soc
(
Lc(X)

)
) = (0) in Lc(X). Let us now take x ∈ X\Y and obtain

a contradiction. By lc-complete regularity of X, there exists 0 6= f ∈ Lc(X)
with f(Y ) = f(Y ) = 0. Therefore f(Z(1 − ei)) = 0, hence fei = 0 for all i ∈ I.
Thus 0 6= f ∈ Ann(Soc

(
Lc(X)

)
) = (0), which is a contradiction. �

We recall that CF (X) is never a prime ideal of C(X), see [8, Proposition
1.2], or [3, Remark 2.4]. The following result characterizes spaces X such that
CF (X) 6= 0 is a prime ideal in Lc(X) (note, CF (X) 6= 0 if and only if X has
isolated points).

Proposition 5.8. Let |I(X)| < ∞, where I(X) is the set of isolated points in
X. If 0 6= CF (X) is a prime ideal in Lc(X), then X\I(X) is connected in X.
Conversely, if X\I(X) is constant with respect to Lc(X), then 0 6= CF (X) is
prime in Lc(X).

Proof. Let Y = X\I(X) = A ∪ B, where A, B are two nonempty infinite
disjoint clopen subsets of Y and seek a contradiction. Since Y is clopen in X
we infer that A, B are also clopen in X. Clearly, X = I(X) ∪ A ∪ B. Now
define f, g ∈ Lc(X) such that f(A∪I(X)) = 1, f(B) = 0 and g(A∪I(X)) = 0,
g(B) = 1. Clearly fg = 0 ∈ CF (X), but by [20, Proposition 3.3], we infer
that f, g /∈ CF (X), which is a contradiction. Conversely, let Y = X\I(X) be
constant with respect to Lc(X) and take f, g ∈ Lc(X) such that fg ∈ CF (X).
Clearly X = Y ∪ I(X), so X\Z(fg) ⊆ I(X) and fg(Y ) = 0. Since f and g are
constant on Y , we infer that either f(Y ) = 0 or g(Y ) = 0, i.e., X\Z(f) ⊆ I(X)
or X\Z(g) ⊆ I(X), therefore f ∈ CF (X) or g ∈ CF (X), by [20, Proposition
3.3], and we are done. �

In the following corollary, we consider spaces X, such that CF (X) is not a
prime ideal in Lc(X).

Corollary 5.9. If I(X) is an infinite set or Y = X\I(X) is disconnected, then
CF (X) is never a prime ideal in Lc(X).

Proof. Let I(X) be an infinite set and take A = {xn : n ∈ N}, B = {yn :
n ∈ N} to be two disjoint countably infinite subsets of I(X). We now define
f(x) =

{
1
n

, x = xn ∈ A
0 , x /∈ A and g(x) =

{
1
n

, x = yn ∈ B
0 , x /∈ B . Let ǫ > 0

be given, then there exists k ∈ N such that 1
n

< ǫ, for all n ≥ k. Now,
for the clopen subsets G = X\{x1, x2, . . . , xk}, H = X\{y1, y2, . . . , yk} and

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 203



O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

for each x ∈ G, y ∈ H, we have |f(x)| < ǫ, |g(y)| < ǫ, hence f, g ∈ C(X).
Clearly, f, g ∈ Cc(X). Therefore f, g ∈ Lc(X) and 0 = fg ∈ CF (X), but
f, g /∈ CF (X), by [20, Proposition 3.3]. Consequently, in this case CF (X) is
not prime in Cc(X), a fortiori, in Lc(X)). Finally let |I(X)| < ∞ and X\I(X)
be disconnected, hence by Proposition 5.8, we are done. �

In the next result, which is our main theorem in this section, we consider the
maximality of CF (X) in Lc(X). First, let us recall that if ϕ : C(X) → C(Y )
is a ring homomorphism with ϕ(1) = 1, then ϕ(Cc(X)) ⊆ Cc(Y ). This is an
easy consequence of the fact that whenever f ∈ Cc(X), then Im(ϕ(f)) ⊆ Im(f)
(note, let r ∈ Im(ϕ(f)), then ϕ(f) − r is non-unit, but ϕ(f − r) = ϕ(f) − r,
hence f − r is non-unit too, i.e., Z(f − r) 6= ∅, and we are done), see also[11,
the comment following Corollary 3.5].

Theorem 5.10. Let CF (X) be a maximal ideal in Lc(X). Then C
F (X) =

Cc(X) = Lc(X) and Lc(X) is isomorphic to a finite direct product of fields,
each of which, is isomorphic to R and X has a unique infinite clopen connected
subset. Conversely, let X have a unique infinite clopen connected subset, and

assume that every element of Lc(X) is constant on it, and Lc(X) ∼=
n∏

i=1

Fi,

where each Fi is a field. Then C
F (X) = Cc(X) = Lc(X), and CF (X) is

maximal in Lc(X).

Proof. Let CF (X) be a maximal ideal in Lc(X). Let us first take care of the
case, when CF (X) = 0. Clearly in this case Lc(X) = R (note, in this case X is
connected and Cc(X) = C

F (X) = R), and we are done. Hence, we may assume
that that CF (X) 6= 0. In view of the previous corollary we infer that I(X),
the set of isolated points of X must be finite. Let us assume that |I(X)| = n,
where n is a positive integer (note, CF (X) 6= 0 if and only if I(X) 6= ∅, see [20,
Proposition 3.3]). Hence CF (X) is a finitely generated ideal in C(X) (note, by
[20, Proposition 3.1], there is a one-one correspondence between I(X) and the

set of nonzero minimal ideals in C(X)). Consequently, CF (X) =
n∑

i=1

⊕eiC(X),
where each ei is an idempotent and eiC(X) = eiLc(X) is a minimal ideal in
C(X) as well as in Lc(X), see the comment preceding Lemma 5.1. Clearly,
CF (X) = eCF (X) = eC(X) = eLc(X), where e = e1 + e2 + · · · + en (note,
eiej = 0 for i 6= j). Since CF (X) is maximal in Lc(X), we infer that e 6= 1,
which implies that (1−e)Lc(X) is a nonzero minimal ideal in Lc(X). Inasmuch
as CF (X) is maximal in Lc(X) and CF (X) ⊆ Soc

(
Lc(X)

)
, we infer that either

CF (X) = Soc
(
Lc(X)

)
or Lc(X) = Soc

(
Lc(X)

)
. We claim that CF (X) =

Soc
(
Lc(X)

)
leads us to a contradiction. To see this, we note that (1−e)Lc(X)

is a nonzero minimal ideal in Lc(X). Hence if the latter equality holds, we infer
that (1 − e)Lc(X) must be in CF (X). But CF (X) ∩ (1 − e)Lc(X) = 0, which
is absurd. Consequently, we must have Lc(X) = Soc

(
Lc(X)

)
= CF (X) ⊕ (1 −

e)Lc(X). Now, for each ei we can easily show that eiLc(X) ∼= R ∼= (1−e)Lc(X).
To see this, let x ∈ Z(1 − ei) and define ϕ : Lc(X) → R by ϕ(f) = f(x) for

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 204



On the locally functionally countable sub algebra of C(X)

all f ∈ Lc(X). Hence, (1 − ei)Lc(X) ⊆ kerϕ. Since (1 − ei)Lc(X) is maximal
in Lc(X), we infer that (1 − ei)Lc(X) = kerϕ, hence eiLc(X) ∼= Lc(X)(1−ei)Lc(X)

∼=
R. Similarly (1 − e)Lc(X) ∼= Lc(X)eLc(X)

∼= R. Consequently, we have already

shown that Lc(X) ∼=
n+1∏
i=1

Ri, where Ri = R. In view of Lemma 5.1 , and

by the fact that (1 − e)Lc(X) is minimal in Lc(X), we infer that Z(e) is
connected. Consequently, by the comment preceding Lemma 5.1, (1 − e)Cc(X)
is a minimal ideal in Cc(X). Hence Cc(X) = CF (X) ⊕ (1 − e)Cc(X), which
is equal to Soc

(
Cc(X)

)
= Soc

(
CF (X)

)
⊆ CF (X), see the comment preceding

Proposition 5.2. Thus CF (X) = Cc(X). But, Cc(X) = CF (X) ⊕ (1 − e)Cc(X)
is the direct sum of n + 1 minimal ideals in Cc(X), hence by the above proof

for Lc(X), we can also show that Cc(X) ∼=
n+1∏
i=1

Ri, where Ri = R. Let us

consider the natural isomorphism ϕ : Cc(X) → Lc(X) ⊆ C(X). Now in
view of the comment preceding the theorem we have Lc(X) ⊆ Cc(X), hence
Lc(X) = Cc(X) = C

F (X). Finally, in view of [20, Proposition 3.3], it is clear
that the connected clopen set Z(e) is infinite (in fact Z(e) = X \ I(X)). It
is also manifest that every non-singleton connected subset of X must be a
subset of Z(e), hence Z(e) is the only clopen connected subset of X which is

infinite, and we are done. Conversely, since Lc(X) ∼=
n∏

i=1

Fi, where each Fi

is a field, we infer that Lc(X) =
n∑

i=1

⊕uiLc(X) = Soc
(
Lc(X)

)
, where each

uiLc(X) is a nonzero minimal ideal in Lc(X), and each ui is idempotent with
1 = u1 +u2 · · · un. Now let 1 6= u ∈ Lc(X) be an idempotent such that Z(1−u)
is the unique infinite clopen subset of X, on which, every element of Lc(X) is
constant. Consequently, uLc(X) is a minimal ideal in Lc(X), by Lemma 5.1.
Multiplying, 1 = u1 + u2 · · · un by u, we get u = uu1 + uu2 + · · · uun. Clearly,
u 6= 0, hence uui 6= 0 for some i. We now claim that there is a unique i, with
1 ≤ i ≤ n such that uui 6= 0. To see this, let uui 6= 0 6= uuj for some i 6= j
and obtain a contradiction. But uui 6= 0 implies that uLc(X)uiLc(X) 6= 0,
hence uLc(X)uiLc(X) = uLc(X) = uiLc(X) and similarly uLc(X) = ujLc(X),
which is a contradiction. Consequently, we may assume that uui = 0 for
1 ≤ i ≤ n − 1 and uun 6= 0. This means that uLc(X) = unLc(X). In view
of [20, Proposition 3.3], and the fact that Z(1 − u) is infinite, we infer that
u /∈ CF (X), i.e, un /∈ CF (X). By Lemma 5.1, and the fact that each uiLc(X)
for 1 ≤ i ≤ n − 1 is minimal, we infer that each Z(1 − ui) is connected,
which by our assumption is not an infinite set, hence it must be a singleton.
Consequently, in view of [20, Proposition 3.3], ui ∈ CF (X) for 1 ≤ i ≤ n − 1.
Inasmuch as Lc(X) =

n∑
i=1

⊕uiLc(X) = Soc
(
Lc(X)

)
, we infer that CF (X) =

n−1∑
i=1

⊕uiLc(X) ⊕ unLc(X)
⋂
CF (X). Since unLc(X) is minimal, we infer that

either unLc(X)
⋂
CF (X) = 0 or unLc(X)

⋂
CF (X) = unLc(X). The latter

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 205



O. A. S. Karamzadeh, M. Namdari and S. Soltanpour

equality is impossible, for by what we have already observed above un /∈ CF (X).
Consequently, CF (X) =

n−1∑
i=1

⊕uiLc(X) and Lc(X) = CF (X)⊕unLc(X), hence
CF (X) is maximal in Lc(X). Now by the proof of the first part we also have
CF (X) = Cc(X) = Lc(X), hence we are done. �

The following theorem shows that for the spaces X in which there exist
certain constant subsets with respect to Lc(X), Lc(X) can not be isomorphic
to any C(Y ).

Theorem 5.11. Let |I(X)| < ∞ and X\I(X) be constant with respect to
Lc(X) (note, in this case Lc(X) = C

F (X)). Then there is no space Y with
Lc(X) ∼= C(Y ).

Proof. Let |I(X)| < ∞ and X\I(X) be constant with respect to Lc(X), then
CF (X) is a prime ideal in Lc(X) by Proposition 5.8. If there exists a space Y
such that Lc(X) ∼= C(Y ), then Soc

(
Lc(X)

) ∼= CF (Y ). Now, since Soc
(
Lc(X)

)

is a zl-ideal containing a prime ideal CF (X), Soc
(
Lc(X)

)
is a prime ideal in

Lc(X), by Theorem 3.14. Hence CF (Y ) is a prime ideal in C(Y ), which is a
contradiction, see the comment preceding Proposition 5.11. �

Remark 5.12. If we replace Lc(X) by LF (X) or by L1(X) in this section, then
some of the results of this section remain valid for these two rings, too.

Remark 5.13. Let X = W ∪{x1, x2, . . . , xn}, where W is constant with respect
to L1(X) (e.g., if we take W as in Remark 2.10) and x1, x2, . . . , xn are the
only isolated points of X (note, W is connected and has no isolated point)
i.e., |I(X)| < ∞ and X\I(X) = W is a constant subset of X with respect to
L1(X). Hence, by Theorem 5.11, Remark 5.12, L1(X) can not be isomorphic
to any C(Y ), in general. But in some special cases, namely, L1(W) and L1(X)
we have L1(W) = R and L1(X) ∼=

∏n
i=1 Ri, where Ri = R, for i = 1, 2, . . . , n.

That is to say L1(W) = C(Y ), where Y is a singleton, and L1(X) = C(Z),
where |Z| < ∞. But, we should remind the reader that we are interested in
infinite spaces.

Remark 5.14. Let K be a subring of R, then Lc(X, K) is a subring of Lc(X)
whose elements take values in K. We denote Lc(X, Z), Lc(X, Q) by Li(X)
and Lr(X), respectively. Clearly, Li(X) = C(X, Z) = Ci(X) and Lr(X) =
C(X, Q) = Cr(X), see also [10, the comment following Definition 2.1]. It is
manifest that Li(X) ⊆ Lr(X) ⊆ C(X, F) ⊆ Cc(X) ⊆ Lc(X), where F is a
countable subfield of R and Li(X) ⊆ Lr(X) ⊆ Lc(X, K) ⊆ Lc(X), where K
is a proper subfield of R. But unfortunately, apart from Cc(X) and Lc(X),
these are not R-subalgebras of C(X), see [10, Remark 7.5], and are not of our
interest, in general.

c© AGT, UPV, 2015 Appl. Gen. Topol. 16, no. 2 206



On the locally functionally countable sub algebra of C(X)

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