@
Appl. Gen. Topol. 21, no. 2 (2020), 195-200

doi:10.4995/agt.2020.11903

c© AGT, UPV, 2020

Closure formula for ideals in intermediate rings

John Paul Jala Kharbhih and Sanghita Dutta

Department of Mathematics, North Eastern Hill University, Mawkynroh, Umshing, Shillong - 22,

India (jpkharbhih@gmail.com, sanghita22@gmail.com)

Communicated by F. Mynard

Abstract

In this paper, we prove that the closure formula for ideals in C(X)
under m topology holds in intermediate rings also. i.e. for any ideal I
in an intermediate ring with m topology, its closure is the intersection
of all the maximal ideals containing I.

2010 MSC: 46E25; 54C30; 54C35; 54C40.

Keywords: m topology; rings of continuous functions; β-ideals.

1. Introduction

The m topology on C(X) was defined by Hewitt in [9]. Let Cm(X) denote
the ring C(X) equipped with m topology. Cm(X) was shown to be a topological
ring. In any topological ring, the closure of a proper ideal is either a proper ideal
or the whole ring [8, 2M1]. Amongst other results, Hewitt in [9] showed that
every maximal ideal in C(X) under m topology is closed. He conjectured that
every m closed ideal of C(X) is an intersection of maximal ideals of C(X). This
conjecture was settled by Gillman, Henriksen and Jerison [7]. It was also settled
independently by T.Shirota [12]. In [7](also [8, 7Q.3]), it was further shown
that the closed ideals in C∗(X) (under subspace m topology) coincide with the
intersections of maximal ideals in C∗(X) if and only if X is pseudocompact.

Intermediate rings denoted by A(X), are rings of continuous functions which
lie in between C∗(X) and C(X). These rings were studied by Donald Plank
as β- subalgebras in [10]. Subsequently, a number of researchers generated
renewed interests in these intermediate rings as can be seen in [11], [5], [2], [4],
[3] and [1].

Received 27 May 2019 – Accepted 26 April 2020

http://dx.doi.org/10.4995/agt.2020.11903


J. P. J. Kharbhih and S. Dutta

Given a real number " > 0 and g ∈ A(X), let E!(g) [8, 2L] denote the set
{x ∈ X : |g(x)| ≤ "}. Given " > 0, f ∈ A(X), it is not difficult to construct a
function t satisfying ft = 1 on the complement of E!(f). i.e. E!(f) ∈ ZA(f) ∀
" > 0. Given an ideal I in A(X), let I′ denote the intersection of all the
maximal ideals of Am(X) that contain I. Evidently I

′ is closed. Let f ∈ A(X)
and E ∈ Z(X). Then, f is said to be Ec-regular, if ∃ g ∈ A(X) such that
fg|Ec = 1. For each f ∈ A(X), let ZA(f) denote the set {E ∈ Z(X): f is
Ec − regular}. For an ideal I of A(X), ZA[I] denote the set

!
f∈I

ZA(f). The

set of cluster points of a z-filter F is denoted by S[F ]. An ideal I in A(X) is
said to be a β-ideal if ZA(f) ⊂ ZA[I] =⇒ f ∈ I. We shall denote intermediate
rings A(X) with m topology by Am(X). For undefined terms and references,
we refer the reader to [8].

In this paper, we ask if Hewitt’s formula for closure of an ideal holds for
the case of Am(X) also. We answer this question in the affirmative, and as an
outcome we obtain the result that an ideal in an intermediate ring is closed iff
the ideal is a β-ideal.

Theorem 1.1 ([5, Theorem 3.3]). Let M
p
A be the maximal ideal of A(X) cor-

responding to the point p of βX. Then

M
p
A = {f ∈ A(X): p ∈ S[ZA(f)]}.

2. Closure formula in intermediate rings

Let UA(X) denote the set of positive units of A(X). For each f ∈ A(X) and
each u ∈ UA(X), let BA(f, u) denote the collection {g ∈ A(X): |f − g| < u}.
For each f ∈ A(X), the set Bf = {BA(f, u) : u ∈ UA(X)} forms a base for
the neighborhood system at f and the topology so formed is the m topology
in A(X).

Definition 2.1. Let A(X) be an intermediate subring. For an ideal I in A(X),
let ∆A(I) = {p ∈ βX : M

p
A ⊃ I}.

Theorem 2.2. Let I be an ideal in A(X) and p ∈ βX − ∆A(I). Then, ∃ f ∈
I ∩ C∗(X) such that fβ(p) = 1.

Proof. Since p ∕∈ ∆A(I), so M
p
A ∕⊃ I. Therefore, ∃ g ∈ I, such that g ∕∈ M

p
A. So,

∃ a neighborhood U of p (in βX) which does not meet E, for some E ∈ ZA(g).
Now E ∈ ZA(g) =⇒ gl|Ec = 1 for some l ∈ A(X). Let f ∈ C

∗(X) be such

that 0 ≤ f ≤ 1, fβ(p) = 1 and

(2.1) fβ(Uc) = 0.

We define h: X → R by

h(x) =

"
f(x)

(|f(x)|+1)l(x)g(x), if x ∈ clβXU ∩ X
0, if x ∈ (βX − U) ∩ X.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 196



Closure formula for ideals in intermediate rings

Then, h is well-defined and continuous. In fact h ∈ A(X) since h ∈ C∗(X).
Moreover the definition of h shows that f is a multiple of g so that f ∈ I, which
completes the proof. □
Theorem 2.3. Let Ω be an open subset of βX such that Ω ⊃ ∆A(I) for some
ideal I in A(X). Then, given " with 0 < " < 1, ∃ g ∈ I with 0 ≤ g ≤ 1 such
that Ω ∩ X ⊃ E!(g).

Proof. Let p ∈ βX − Ω. Then, p ∕∈ ∆A(I). By theorem 2.2 we see that
∃ gp ∈ I ∩ C∗(X) such that gβp (p) = 1. We choose an " ∈ R with 0 < " < 1.
Let

Σp = {q ∈ βX : gβp (q) >
√
"0}.

Then, Σp is open in βX and non-empty as p ∈ Σp. Now, the collection
{Σp : p ∈ βX − Ω} forms an open cover for the compact set βX − Ω. Let
{Σp1, Σp2, . . . , Σpn} be a finite subcover of this open cover. Let g = g2p1 + g

2
p2

+

. . . + g2pn. For any p ∈ βX − Ω, we then have g
β(p) = (gβp1(p))

2 + (gβp2(p))
2 +

. . . + (gβpn(p))
2 > ". Therefore, if |gβ(p)| ≤ ", then p ∕∈ βX − Ω. i.e. p ∈ Ω.

Hence, E!(g) ⊂ Ω ∩ X. □
Definition 2.4. Let f ∈ A(X). We say that f is ZC-related to I, if ∃ " > 0,
such that Z(f) ⊃ C ⊃ E!(g) for some cozero-set C and some g ∈ I.

Definition 2.5. For an ideal I of A(X), we define

KA(I) = {f ∈ A(X) : f is ZC-related to I}.

Theorem 2.6. For every ideal I of an intermediate subring Am(X), we have
KA(I) ⊂ I and clm(KA(I)) = clm(I).

Proof. Let f ∈ KA(I). Then, ∃ " > 0 such that Z(f) ⊃ C ⊃ E!(g) for some
cozero-set C and some g ∈ I. Let us denote E!(g) by E. Since E ∈ ZA(g),
∃ l ∈ Am(X) such that (gl)|Ec = 1. Now, we define h by

h(x) =

"
0, if x ∈ clXC

f
(|f|+1)lg if x ∕∈ C.

Then, h is a well-defined bounded function. Moreover, h is continuous. i.e.
h ∈ C∗(X) ⊂ A(X). Also, we get f = h(|f| + 1)lg, which shows that f ∈ I.
Thus KA(I) ⊂ I and hence clm(KA(I)) ⊂ clm(I). To prove that clm(I) ⊂
clm(KA(I)), it is enough to prove that I ⊂ clm(KA(I)). So, we take a g ∈ I.
Let π ∈ UA(X). We define f by

f(x) =

#
$%

$&

0, if − π(x)
2

≤ g(x) ≤ π(x)
2

g(x) − π(x)
2

, if g(x) >
π(x)
2

g(x) +
π(x)
2

, if g(x) < −π(x)
2

.

Then, f lies in the π neighborhood of g. We also notice that f ∈ Am(X) since
f may be rewritten as follows :

f(x) =
'(
g(x) −

π(x)

2

)
∨ 0

*
+
'(
g(x) +

π(x)

2

)
∧ 0

*
.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 197



J. P. J. Kharbhih and S. Dutta

We shall now show that f ∈ KA(I). Let C = {x ∈ X : −
π(x)
2

< g(x) <
π(x)
2

}.
Then Z(f) ⊃ C. Moreover, C is the cozero-set of the function h ∈ A(X)
defined by:

h(x) =
(
|g(x)| −

π(x)

2

)
∧ 0.

We choose any real number " > 0 and define a function θ by θ(x) =
4!g(x)
π(x)

.

Clearly, θ ∈ I. Moreover |θ(x)| ≤ " ⇐⇒ |g(x)| ≤ π(x)
4

. In otherwords,

x ∈ E!(θ) ⇐⇒ |g(x)| ≤
π(x)
4

. But, |g(x)| ≤ π(x)
4

=⇒ x ∈ Z(f). Hence
Z(f) ⊃ C ⊃ E!(θ) which completes the proof. □
Example 2.7. Now, we will give an example of an ideal I such that KA(I) ⊊ I.
Let X = R and A(X) = C(X). Let I = M0. We will show that KA(I) = O0.
Firstly, if f ∈ O0, then ∃ an open set C such that 0 ∈ C ⊂ Z(f). Now, ∃ " > 0
such that E = [−", "] ⊂ C. Then E = E!(g), where g is the identity map on R.
Moreover, C is a cozero-set as X is a metric space. Hence we have f ∈ KA(I).
Secondly, if f ∈ KA(I), then ∃ g ∈ I, " > 0 such that Z(f) ⊃ C ⊃ E!(g) for
some cozero-set C. Since 0 ∈ E!(g), this gives that Z(f) is a neighborhood of
0 i.e. f ∈ O0.
Theorem 2.8. k ∈ I′ ⇐⇒ S[ZA(k)] ⊃ ∆A(I).
Proof. (⇒) We assume that k ∈ I′. Let p ∈ ∆A[I]. Then, M

p
A ⊃ I and so

k ∈ MpA. By definition of M
p
A, p ∈ S[ZA(k)].

(⇐) Let MpA be a maximal ideal which contains I. So, p ∈ ∆A(I) and thus,
p ∈ S[ZA(k)]. Therefore, k ∈ M

p
A and hence k ∈ I

′. □
We now prove the main result.

Theorem 2.9. The m closure of any ideal I in Am(X) is the intersection of
all the maximal ideals containing I.

Proof. We have clm(I) ⊂ I′ as I′ is closed. To prove I′ ⊂ clm(I), it is sufficient
to prove that KA(I

′) ⊂ KA(I). Then, by theorem 2.6, we will get I′ ⊂ clmI.
Let f ∈ KA(I′). Then, ∃ a cozero-set C, a real number " > 0 and θ ∈ I′

such that

Z(f) ⊃ C ⊃ E!(θ) = E(say).(2.2)
Let Z = X − C. Then, Z and E are completely separated being disjoint zero-
sets. Therefore, ∃ h ∈ C∗(X), 0 ≤ h ≤ 1 such that h(E) = 0 and h(Z) = 1.

Let Ω = {p ∈ βX : hβ(p) < 1}. We observe that X = C ∪ Z, so βX =
clβXC ∪ clβXZ. If p ∈ Ω, i.e. hβ(p) < 1, then p ∕∈ clβXZ as hβ(clβXZ) = 1. So
p ∈ clβXC.

i.e. clβXC ⊃ Ω.(2.3)
Since E ∈ ZA(θ), therefore Ω ⊃ S[ZA(θ)] because p ∈ S[ZA(θ)] gives

hβ(p) = 0. Hence by theorem 2.8, we see that Ω ⊃ ∆A(I). Theorem 2.3 now
gives a g ∈ I with 0 ≤ g ≤ 1 and some " with 0 < " < 1 such that

Ω ∩ X ⊃ E!(g).(2.4)

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 198



Closure formula for ideals in intermediate rings

From (2.2) and (2.3), we get,

clβXZ(f) ⊃ clβXC ⊃ Ω.
Then clβXZ(f) ∩ X ⊃ Ω ∩ X. Thus Z(f) ⊃ Ω ∩ X. Therefore, by (2.4)
Z(f) ⊃ Ω ∩ X ⊃ E!(g). Finally, we have Ω ∩ X is a co-zero-set as Ω ∩ X =
{p ∈ X : h(p) < 1}. □
Corollary 2.10. Every closed ideal is a β-ideal.

Proof. First we claim that an arbitrary intersection of β-ideals is also a β-ideal.
Let {Iα : α ∈ Λ} be a collection of β-ideals. Let ZA(f) ⊂ ZA[

+
α∈Λ

Iα]. Since

each Iα is a β-ideal, it is enough to prove that ZA(f) ⊂ ZA[Iα] ∀ α ∈ Λ,
for this would imply that f ∈ Iα ∀ α ∈ Λ. So take E ∈ ZA(f). Therefore
E ∈ ZA(g) for some g ∈

+
α∈Λ

Iα. This then gives E ∈ ZA[Iα] ∀ α ∈ Λ. Now,

let I be a closed ideal in Am(X). Therefore, I is an intersection of maximal
ideals. But, as every maximal ideal is a β-ideal, therefore I is an intersection
of β-ideals and hence a β-ideal. □
Remark 2.11. In [6, Theorem 3.13], it was shown that the β-ideals of an inter-
mediate ring are just the intersections of maximal ideals of the ring. This says
that β-ideals are closed, since maximal ideals are closed. Hence the class of
β-ideals and the class of closed ideals in intermediate rings coincide. This co-
incidence also occurs in the case of the subring C∗(X) with m topology. Here,
the class of e-ideals is the same as the class of closed ideals [8, 2M]. However,
this coincidence does not extend to z-ideals in Cm(X) since the ideal O

p is a
z-ideal which is not closed.

Remark 2.12. In [1], it was proven that if an intermediate ring A(X) is different
from C(X), then there exists at least one non-maximal prime ideal P in A(X).
Thus, P is not closed in Am(X). On the other hand if A(X) = C(X) and X is
a P space then each ideal in Am(X) is closed [8, 7Q4]. Thus within the class
of P spaces X, for an intermediate ring A(X), each ideal in Am(X) is closed
⇐⇒ A(X) = C(X) - this is a special property of C(X) which distinguishes
C(X) amongst all the intermediate rings (in the category of P spaces X).

Acknowledgements. The authors would like to thank the referee for the
valuable comments and suggestions towards the improvement of the paper. In
particular, Remark 2.12 is due to the referee.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 199



J. P. J. Kharbhih and S. Dutta

References

[1] S. K. Acharyya and B. Bose, A correspondence between ideals and z-filters for certain
rings of continuous functions - some remarks, Topology and its Applications 160, no. 13
(2013), 1603–1605.

[2] S. K. Acharyya, K. C. Chattopadhyay and D. P. Ghosh, A class of subalgebras of C(X)
and the associated compactness, Kyungpook Math. J. 41, no. 2 (2001), 323–324.

[3] S. K. Acharyya and D. De, An interesting class of ideals in subalgebras of C(X) con-
taining C∗(X), Comment. Math. Univ. Carolin. 48, no. 2 (2007), 273–280.

[4] S. K. Acharyya and D. De, Characterization of function rings between C∗(X) and C(X),
Kyungpook Math. J. 46 (2006), 503–507.

[5] H. L. Byun and S. Watson, Prime and maximal ideals in subrings of C(X), Topology
and its Applications 40 (1991), 45–62.

[6] J. M. Domı́nguez and J.-Gómez Pérez, Intersections of maximal ideals in algebras be-
tween C∗(X) and C(X), Topology and its Applications 98 (1999), 149–165.

[7] L. Gillman, M. Henriksen and M. Jerison, On a theorem of Gelfand and Kolmogoroff
concerning maximal ideals in rings of continuous functions, Proc. Amer. Math. Soc. 5
(1954), 447–455.

[8] L. Gillman and M. Jerison, Rings of continuous functions, Univ. Ser. Higher Math, D.
Van Nostrand Company, Inc., Princeton, N. J., 1960.

[9] E. Hewitt, Rings of real-valued continuous functions I, Trans. Amer. Math. Soc. 64, no.
1 (1948), 45–99.

[10] D. Plank, On a class of subalgebras of C(X) with applications to βX \ X, Fund. Math.
64 (1969), 41–54.

[11] L. Redlin and S. Watson, Maximal ideals in subalgebras of C(X), Proc. Amer. Math.
Soc. 100, no. 4 (1987), 763–766.

[12] T. Shirota, On ideals in rings of continuous functions, Proc. Japan Acad. 30, no. 2
(1954), 85–89.

c© AGT, UPV, 2020 Appl. Gen. Topol. 21, no. 2 200