@ Appl. Gen. Topol. 20, no. 1 (2019), 109-117doi:10.4995/agt.2019.10171
c© AGT, UPV, 2019

A class of ideals in intermediate rings of

continuous functions

Sagarmoy Bag, Sudip Kumar Acharyya and Dhananjoy Mandal

Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata

700019, West Bengal, India (sagarmoy.bag01@gmail.com, sdpacharyya@gmail.com, dmandal.cu@gmail.com)

Communicated by A. Tamariz-Mascarúa

Abstract

For any completely regular Hausdorff topological space X, an inter-

mediate ring A(X) of continuous functions stands for any ring lying
between C

∗(X) and C(X). It is a rather recently established fact that
if A(X) 6= C(X), then there exist non maximal prime ideals in A(X).
We offer an alternative proof of it on using the notion of z

◦
-ideals. It is

realized that a P-space X is discrete if and only if C(X) is identical to
the ring of real valued measurable functions defined on the σ-algebra

β(X) of all Borel sets in X. Interrelation between z-ideals, z◦-ideal and
ZA-ideals in A(X) are examined. It is proved that within the family of
almost P-spaces X, each ZA-ideal in A(X) is a z

◦
-ideal if and only if

each z-ideal in A(X) is a z◦-ideal if and only if A(X) = C(X).

2010 MSC: Primary 54C40; Secondary 46E25.

Keywords: P-space; almost P-space; UMP-space; z-ideal; z◦-ideal; ZA-
ideal.

1. introduction

Let C(X) be the ring of all real valued continuous functions on a completely
regular Hausdorff topological space X. C∗(X) is the subring of C(X) consisting
of those functions which are bounded over X. A ring A(X) lying between
C∗(X) and C(X) is called an intermediate ring. These intermediate rings have
an important common property which says that the structure space of all these
rings are one and the same and is the Stone-Čech compactification βX of X

Received 17 May 2018 – Accepted 10 September 2018

http://dx.doi.org/10.4995/agt.2019.10171


S. Bag, S. K. Acharyya and D. Mandal

(see [7]). The structure space of a commutative ring R with unity stands for
the set of all maximal ideals of R equipped with hull kernel topology. In the
present paper our purpose is to point out a few dissimilarities existing between
the ambient ring C(X) and its proper intermediate subrings. To achieve that
we have chosen three special classes of ideals z-ideals, z◦-ideals and ZA-ideals
in a typical intermediate ring A(X). An ideal I unmodified in a commutative
ring R with unity will also designate a proper ideal of R.

For each a in R, let Ma(Pa) be the intersection of all maximal ideals (minimal
prime ideals) of R containing a. An ideal I in R is called a z-ideal (respectively
a z◦-ideal) if for each a in I, Ma ⊆ I( respectively Pa ⊆ I). It is well known
that if R is a reduced ring meaning that ′0′ is the only nilpotent element of R,
then each z◦-ideal of R is a z-ideal also (see [4]). In particular therefore each
z◦-ideal in an intermediate ring A(X) is a z-ideal. The notion z-ideals and
z◦-ideals in commutative rings are quite well known and are being investigated
since around 1970’s. However the concept of ZA-ideals in an intermediate ring
is rather recent and is initiated in [15]. Given E ⊆ X an f ∈ A(X) is called
E-regular if there exist g ∈ A(X) such that f(x)g(x) = 1 for each x ∈ E.
For any non-invertible f ∈ A(X), ZA(f) = {E ∈ Z[X] : f is X \ E-regular}
and ZA(f) = {E ∈ Z[X] : f is H-regular for each zero set H ⊆ X \ E} are
z-filters on X, here Z[X] stands for the family of all zero sets in X. For any
ideal I in A(X) , ZA[I] = ∪f∈IZA(f) and ZA[I] = ∪f∈IZA(f) are z-filters

on X. For any z-filter F on X, Z−1A [F] = {f ∈ A(X) : ZA(f) ⊆ F} and

Z
−1
A

[F] = {f ∈ A(X) : ZA(f) ⊆ F} are easily seen to be ideals in A(X).

It is easily verified that for any ideal I in A(X), Z−1A ZA[I] ⊇ I. I is called

ZA-ideal if Z
−1
A ZA[I] = I. It is plain that ZC-ideals and z-ideals in C(X)

are same. For an arbitrary A(X), we check that every ZA-ideal is a z-ideal
(Theorem 3.4). We establish a partial converse of this theorem that, within
the class of P spaces X if every z-ideal is ZA-ideal in A(X) then A(X) = C(X)
(Theorem 3.5). A ZA-ideal in A(X) need not be a z

◦-ideal. Indeed a ZC-ideal
in C(X) is not necessarily a z◦-ideal, a fact which is not hard to realize on
choosing X = R. We prove that such things do not happen if and only if X
is an almost P-space (Theorem 3.6). This further yields that if X is almost
P and A(X) 6= C(X), then there does exist a ZA ideal in A(X) which is not
a z◦-ideal (Theorem 3.7). Relations between maximal ideals and z◦-ideals in
A(X) have also been investigated by the present authors, indeed an improved
version of such interrelations have already been pondered upon in a recently
communicated paper [5] However to make the present article self contained and
also for the benefit of the readers we reproduce a few of these relevant facts
from this paper in the technical section §2.

It is a rather recently established fact that if an intermediate ring A(X)
is different from C(X), then there exist non maximal prime ideals in A(X)
[1], [11]. We give an alternative proof of it in the concluding section §4 of
this article. In this last section we find out two conditions each necessary and
sufficient for a P-space X to be discrete (Theorem 4.3). On using certain

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A class of ideals in intermediate rings of continuous functions

facts about z◦-ideals we prove a second special result which tells that certain
important subspaces of an UMP-space are also UMP-spaces. A space X is
called an UMP-space if every maximal ideal in C(X) is a union of minimal
prime ideals contained in it (see [6] for results about these spaces).

2. z◦-ideals in intermediate rings versus almost Pspaces

We start with the following characterization of minimal prime ideals in a
commutative ring R with unity ([12]), which is also recorded in ([10], Lemma
1.1).

Theorem 2.1. A prime ideal P in R is a minimal prime ideal if and only if
given a ∈ P, there is a b ∈ R \ P such that a.b is a nilpotent element of R, in
particular ab = 0, if R is assumed to be a reduced ring.

It follows from this theorem that each non zero element of a minimal prime
ideal in a reduced ring R is a divisor of zero in R. Therefore non zero elements
of a z◦-ideal in such a ring R are all divisors of zero. We shall use this singularly
important fact in the proof of several theorems that follow. Incidentally it is
not hard to prove on using Theorem 3.1 that if Ann(a) is the annihilator of an
element a in a reduced ring R, then Pa = {b ∈ R : Ann(a) ⊆ Ann(b)}. This
formula together with the fact that each bounded function in C(X) belongs to
each intermediate ring A(X) yields the following theorem:

Theorem 2.2. For f ∈ A(X), Pf = {g ∈ A(X) : intXZ(f) ⊆ intXZ(g)}, here
Z(f) stands for the zero set of f in X.

As recorded in [9], P-spaces are fairly rare. Interesting examples of non
discrete P-spaces are rather pathological (see Examples 7.3,7.4,7.5,7.6 in [9]).
A larger family of spaces, the so-called almost P-spaces X viz those for which
non empty zero sets in X have non empty interior (equivalently non empty Gδ
sets in X have non empty interior) have been introduced in [13]. It turns out
that almost P-spaces are far more abundant than P-spaces. Those spaces have
already been characterized viz z◦-ideals and z-ideals in C(X) in [3]. We have
offered a some what improved characterization of those spaces via z◦-ideals in
intermediate rings. The following proposition attests to this fact:

Theorem 2.3. Suppose A(X) is an intermediate ring. Then X is an almost
P-space if and only if each fixed maximal ideal M

p
A = {f ∈ A(X) : f(p) =

0}, p ∈ X of A(X) is a z◦-ideal.

Proof. Let X be almost P-space and p ∈ X. Choose f ∈ M
p
A

and g ∈ Pf ≡
the intersection of all minimal prime ideals of A(X) which contain f. Then
from Theorem 3.2, intXZ(f) ⊆ intXZ(g). Therefore the hypothesis that X
is almost P implies that Z(f) = clXintXZ(f) ⊆ clXintXZ(g) = Z(g) and
therefore g ∈ M

p
A. Thus Pf ⊆ M

p
A and hence M

p
A is a z

◦-ideal of A(X). To
prove the other containment le X be not almost P . So there exists f ∈ C∗(X)
such that Z(f) 6= φ but intXZ(f) = φ. Hence f ∈ M

p
A yet f is not a divisor of

zero in A(X). As members of z◦-ideals are necessarily divisors of zero in the
ambient ring, if follows that M

p
A
is not a z◦-ideal in A(X). �

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S. Bag, S. K. Acharyya and D. Mandal

We would like to mention in this context that in the paper [3] a space X
was realized as almost P when and only when each maximal ideal of C(X) is
a z◦ideal. We show in the next result that, the last characterization can not
be improved to say that X is almost P if and only if each maximal ideal of an
intermediate ring A(X) is a z◦-ideal.

Theorem 2.4. Let X be almost P. Then for an intermediate ring A(X), each
maximal ideal is a z◦-ideal in A(X) if and only if each z-ideal in A(X) is a
z◦-ideal if and only if A(X) = C(X).

Proof. If A(X) = C(X), then it follows from [3], that each z ideal of A(X), in
particular each maximal ideal of A(X) is a z◦-ideal. To prove the converse let
A(X) 6= C(X). Then there exists f ∈ C(X) such that f /∈ A(X). Since A(X)
is an absolutely convex subring of C(X) (see [7]), it follows that |f| /∈ A(X). It
follows that g = 1

1+|f|
is an element of A(X) which is not invertible in this ring.

Accordingly there exists a maximal ideal M of A(X) (which is incidentally a
z ideal of A(X) also) such that g ∈ M. As Z(g) = φ, g is not a divisor of zero,
hence M can not be a z◦-ideal of A(X). �

Remark 2.5. Since the converse part of Theorem 2.4 does not use the almost
P hypothesis on X, we can say that for any space X (not necessarily almost
P), if A(X) is an intermediate subring of C(X) properly contained in C(X),
then there exists a free maximal ideal of A(X), which is not a z◦-ideal.

3. ZA-ideals in A(X) versus P-spaces/ almost P-spaces X

For any z-filter F on X, the hull hF of F is the set of all z-ultrafilters
containing F and for every set U of z-ultrafilters on X, the kernel kU of U is
the intersection of all z-ultrafilters belonging to U. We reproduce the following
theorem established in [16].

Theorem 3.1. For any f in A(X), ZA(f) = khZA(f).

This further yields the following result:

Theorem 3.2. For f, g ∈ A(X), hZA(f) ⊆ hZA(g) if and only if ZA(g) ⊆
ZA(f).

Proof. Let hZA(f) ⊆ hZA(g). then khZA(g) ⊆ khZA(f) and hence from The-
orem 3.1, ZA(g) ⊆ ZA(f). To prove the other containment let ZA(g) ⊆ ZA(f).
Choose p from the set hZA(f), then ZA(f) ⊆ U

p (here we are identifying
the point p in βX with the z-ultrafilter Up on X associated with p). This
means that Up ∈ hZA(f) and therefore ZA(g) ⊆ khZA(g) = ZA(g) ⊆ ZA(f) =
khZA(f) ⊆ U

p.
Consequently, Up ∈ hZA(g). Thus hZA(f) ⊆ hZA(g). �

The next proposition furnishes us with a convenient formula for the inter-
section of all maximal ideals containing a function f in A(X).

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A class of ideals in intermediate rings of continuous functions

Theorem 3.3. Let f ∈ A(X). Then the intersection of all maximal ideals of
A(X) containing f is given by: Mf = {g ∈ A(X) : hZA(f) ⊆ hZA(g)} = {g ∈
A(X) : ZA(g) ⊆ ZA(f)}.

Proof. It follows from Theorem 3.3 of [7] that the maximal ideal in A(X)
corresponding to a point p ∈ βX is given by: M

p
A

= {f ∈ A(X) : ZA(f) ⊆

Up} ≡ Z−1A [U
p] = {f ∈ A(X) : p ∈ hZA(f)}. The desired result therefore

follows in view of Theorem 3.2. �

The following result comes out as a consequence of the last one:

Theorem 3.4. Every ZA ideal in A(X) is a z-ideal.

Proof. Let I be a ZA-ideal in A(X) and f ∈ I. Let g ∈ Mf . Then from
Theorem 3.3, we have ZA(g) ⊆ ZA(f). As f ∈ I, it is plain that ZA(f) ⊆ ZA[I]
and hence ZA(g) ⊆ ZA[I]. Since I is a ZA-ideal in A(X), it follows that g ∈ I.
thus Mf ⊆ I and hence I is a z-ideal in A(X). �

It is recently established in [11], Theorem 3.7 that if X is a P-space and
each ideal in A(X) is a ZA ideal then A(X) = C(X).

The following theorem is a somewhat improved version of this fact.

Theorem 3.5. Let X be a P-space. Then A(X) = C(X) if and only if every
z-ideal in A(X) is a ZA-ideal.

Proof. If A(X) = C(X), then z-ideals and ZC-ideals in C(X) are the same. To
prove the converse let A(X) ( C(X). Then by Theorem 3.10 in [11], it follows
that there is a point p ∈ βX for which O

p
A
= {f ∈ A(X) : p ∈ intβXhZA(f)} (

M
p
A = {f ∈ A(X) : p ∈ hZA(f)}. It is not hard to verify that O

p
A is a z-ideal

in A(X) indeed let f ∈ O
p
A and g ∈ Mf . Then from Theorem 3.3, we have

hZA(f) ⊆ hZA(g). Hence p ∈ intβXhZA(f) ⊆ intβXhZA(g), this implies
that g ∈ O

p
A. We assert that O

p
A is not a ZA ideal in A(X). We argue by

contradiction and let O
p
A

be a ZA-ideal. Since X is a P-space it follows from
Corollary 2.4 of [11] that for any ideal I in A(X), we have ZA[I] = ZA[I].
Again from [7], Theorem 4.1 it follows that ZA[O

p
A] = ZA[M

p
A]. Thus we can

write ZA[O
p
A
] = ZA[M

p
A
], which implies in view of the assumption that O

p
A
is a

ZA ideal in A(X) that M
p
A = O

p
A, a contradiction. �

A ZA-ideal in A(X) need not be a z
◦-ideal, indeed a maximal ideal in C(X)

and therefore a ZC-ideal in C(X) is not necessarily a z
◦-ideal. An easy example

is produced by M0 = {f ∈ C(R) : f(0) = 0}, we only note that the function
i ∈ C(R) defined by i(r) = r, r ∈ R is a member of M0 without being a divisor
of zero in the ring C(R). The following theorem settles the exact class of spaces
X for which ZC-ideals in C(X) are z

◦-ideals.

Theorem 3.6. X is an almost P-space if and only if every ZC-ideal in C(X)
is a z◦-ideal.

Proof. Let X be almost P and I a ZC-ideal of C(X). Then from Theorem
3.4, I is a z ideal of C(X). The hypothesis, X is almost P implies in view of
Theorem 2.14 in [3] that I is a z◦-ideal of C(X).

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S. Bag, S. K. Acharyya and D. Mandal

To prove the converse let every ZC-ideal in C(X) be a z
◦-ideal. Since maxi-

mal ideals in C(X) are always ZC-ideals (see [15]), it follows from the Theorem
2.3 that, X is almost P-space. �

The next proposition shows that, the last result characterizes C(X) amongst
all the intermediate rings within the class of almost P-spaces.

Theorem 3.7. Let X be an almost P-space. then A(X) = C(X) if and only
if each ZA ideal in A(X) is a z

◦-ideal.

Proof. Let A(X) = C(X). then from Theorem 3.6, it follows that every ZA-
ideal in A(X) is a z◦-ideal. To prove the converse let A(X) ( C(X). then as
in the proof of the converse part of Theorem 2.4, we can ensure the existence
of a maximal ideal M of A(X), which is not a z◦-ideal. surely M is a ZA-ideal
in A(X). �

4. Two special results

It has been established recently by the authors in [1] and [11], independently
that if A(X) is an intermediate ring, properly contained in C(X), then A(X)
is never regular in the sense of Von-Neumann, which means that there exist
non maximal prime ideals in A(X). We offer yet another proof of the above
mentioned fact by using the notion of z◦-ideals. We will need the following
general result for commutative rings.

Theorem 4.1. Let R be a commutative reduced regular ring with unity. Then
each ideal in R is a z◦-ideal.

Proof. Let I be an ideal in R. Let a ∈ I and b ∈ Pa, then Ann(a) ⊆ Ann(b).
Since R is regular there exists x ∈ R such that a = a2x. Therefore a(1−ax) = 0
and hence 1 − ax ∈ Ann(a) ⊆ Ann(b). This implies that (1 − ax)b = 0, hence
b = abx and therefore b ∈ I as a ∈ I. Thus Pa ⊆ I. Hence I is a z

◦-ideal in
R. �

Theorem 4.2. Let A(X) 6= C(X). then A(X) is not Von-Neumann regular,
equivalently if A(X) is a regular ring, then A(X) = C(X).

Proof. Assume that A(X) is a regular ring and choose f ∈ C(X). To show
that f lies in A(X) it is sufficient to show in view of the absolute convexity of
A(X) that g = 1

1+|f|
is a multiplicative unit of the ring A(X). If possible let

g be not a unit in A(X). Then there exists a maximal ideal M in A(X) such
that g ∈ M. Surely g is not a divisor of zero in A(X) and therefore M cannot
be a z◦-ideal in A(X). On the other hand it follows from Theorem 4.1 that
each ideal of A(X) is a z◦-ideal, a contradiction. �

Before using the above theorem, we let B(X) be the set of all Borel sets in
the space X. Thus B(X) is the smallest σ-algebra on X containing all the open
sets in X. We call a function f : X 7→ R, B measurable if for any open set V in
R, f−1(V ) is a member of B(X). It is quite well known that the family B(X)
of all B measurable functions on X constitutes a commutative lattice ordered

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A class of ideals in intermediate rings of continuous functions

ring with unity if the relevant operations are defined point wise on X and of
course C(X) ⊆ B(X) (see [2]).

The following theorem gives three conditions involving C(X) and B(X) for
a P-space X to become a discrete one.

Theorem 4.3. Let X be a P-space. then the following three statements are
equivalent:

(1) X is discrete.
(2) Z[X] = B(X), we recall that Z[X] is the family of all zero sets in x.
(3) C(X) = B(X).

Proof. It is trivial that the truth of the statement (1) implies the truth of each
of the statements (2), (3). (2) ⇒ (1) : Let X be not discrete, then there exists
x ∈ X such that {x} is not open in X. But {x} is closed in X implies that
{x} ∈ B(X). On the other hand the fact that each zero set in a P-space X is
open implies that {x} /∈ Z(X). Thus Z[X] 6= B(X). Hence the statements (1)
and (2) are equivalent. We make the further observation that the characteristic
function χ{x} : X 7→ R defined by

χ{x}(y) =

{

1, if y = x

0, if y 6= x

is not continuous as {x} is not open in X. But χ{x} ∈ B(X) because {x} is a
Borel set in X. Thus C(X) 6= B(X). So (3) ⇒ (1) is also proved. �

We shall now prove the last principal theorem of this paper.

Theorem 4.4. Every dense C∗-embedded subspace Y of an UMP-space X is
an UMP-space.

Proof. Define a map φ : C(X) 7→ C(Y ) as follows φ(f) = f|Y . Then φ is
an injective homomorphism. Since Y is C∗-embedded in X, it follows that
φ(C∗(X)) = C∗(Y ). Consequently φ(C(X)) becomes an intermediate subring
of C(Y ), say φ(C(X)) = A(Y ). The hypothesis X is an UMP-space, therefore
ensures that A(Y ) is an UMP-ring meaning that each maximal ideal is union of
minimal prime ideals contained in it. In particular each maximal ideal of A(X)
consists of divisor of zero. But as we have observed in the proof of the second
part of Theorem 2.4 and also in Remark 2.5 that if A(Y ) ( C(Y ), then there
exist a maximal ideal M of A(Y ) and g ∈ M such that g is not a divisor of
zero. Hence we should necessarily have A(Y ) = C(Y ). thus φ(C(X)) = C(Y ).
Hence Y is an UMP-space. �

Corollary 4.5. If X is UMP-space, then every subspace of υX containing X
is UMP-space.

In this context we record the following result proved in [6], Corollary 1.11.

Theorem 4.6. No dense C∗-embedded proper realcompact subspace of a com-
pact UMP-space is a UMP-space.

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S. Bag, S. K. Acharyya and D. Mandal

This above two theorems 4.4 and 4.6 shows that a compact UMP-space does
not contains any proper dense C∗-embedded realcompact subspace.

We conclude this article after raising the following open questions:

Question 4.7. Does an isomorphism between the rings C(X) and B(X) of a
P-space X imply that C(X) = B(X)?

Question 4.8. Is O
p
A
, p ∈ βX necessarily a z◦-ideal of A(X)?

Question 4.9. If A(X) ( C(X), then what is the least cardinal number of the
set of all free maximal ideals of A(X), which are not z◦-ideal?

Acknowledgements. The authors would like to thank the referee for suggest-
ing a few revisions and also pointing out some errors appearing in the original

version of this paper.

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