@ Appl. Gen. Topol. 22, no. 1 (2021), 47-65doi:10.4995/agt.2021.13165 © AGT, UPV, 2021 Intermediate rings of complex-valued continuous functions Amrita Acharyya a, Sudip Kumar Acharyya b, Sagarmoy Bag b and Joshua Sack c a Department of Mathematics and Statistics, University of Toledo, Main Campus, Toledo, OH 43606-3390. (amrita.acharyya@utoledo.edu) b Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata 700019, West Bengal, India (sdpacharyya@gmail.com, sagarmoy.bag01@gmail.com) c Department of Mathematics and Statistics, California State University Long Beach, 1250 Bellflower Blvd, Long Beach, CA 90840, USA (joshua.sack@csulb.edu) Communicated by A. Tamariz-Mascarúa Abstract For a completely regular Hausdorff topological space X, let C(X, C) be the ring of complex-valued continuous functions on X, let C∗(X, C) be its subring of bounded functions, and let Σ(X, C) denote the collection of all the rings that lie between C∗(X, C) and C(X, C). We show that there is a natural correlation between the absolutely convex ideals/ prime ideals/maximal ideals/z-ideals/z◦-ideals in the rings P(X, C) in Σ(X, C) and in their real-valued counterparts P(X, C) ∩ C(X). These correlations culminate to the fact that the structure space of any such P(X, C) is βX. For any ideal I in C(X, C), we observe that C∗(X, C)+I is a member of Σ(X, C), which is further isomorphic to a ring of the type C(Y, C). Incidentally these are the only C-type intermediate rings in Σ(X, C) if and only if X is pseudocompact. We show that for any maximal ideal M in C(X, C), C(X, C)/M is an algebraically closed field, which is furthermore the algebraic closure of C(X)/M ∩C(X). We give a necessary and sufficient condition for the ideal CP(X, C) of C(X, C), which consists of all those functions whose support lie on an ideal P of closed sets in X, to be a prime ideal, and we examine a few special cases thereafter. At the end of the article, we find estimates for a few standard parameters concerning the zero-divisor graphs of a P(X, C) in Σ(X, C). 2010 MSC: 54C40; 46E25. Keywords: z-ideals; z◦-ideals; algebraically closed field; C-type rings; zero divisor graph. Received 20 February 2020 – Accepted 06 October 2020 http://dx.doi.org/10.4995/agt.2021.13165 A. Acharyya, S. K. Acharyya, S. Bag and J. Sack 1. Introduction In what follows, X stands for a completely regular Hausdorff topological space and C(X,C) denotes the ring of all complex-valued continuous functions on X. C∗(X,C) is the subring of C(X,C) containing those functions which are bounded over X. As usual C(X) designates the ring of all real-valued continuous functions on X and C∗(X) consists of those functions in C(X) which are bounded over X. An intermediate ring of real-valued continuous functions on X is a ring that lies between C∗(X) and C(X). Let Σ(X) be the aggregate of all such rings. Likewise an intermediate ring of complex-valued continuous functions on X is a ring lying between C∗(X,C) and C(X,C). Let Σ(X,C) be the family of all such intermediate rings. It turns out that each member P(X,C) of Σ(X,C) is absolutely convex in the sense that |f| ≤ |g|,g ∈ P(X,C),f ∈ C(X,C) implies f ∈ P(X,C). It follows that each such P(X,C) is conjugate-closed in the sense that if whenever f + ig ∈ P(X,C) where f,g ∈ C(X), then f −ig ∈ P(X,C). It is realised that there is a natural correlation between the prime ideals/ maximal ideals/ z-ideals/ z◦-ideals in the rings P(X,C) and the prime ideals/ maximal ideals/ z-ideals/ z◦-ideals in the ring P(X,C) ∩ C(X). In the second and third sections of this article, we examine these correlations in some detail. Incidentally an interconnection between prime ideals in the two rings C(X,C) and C(X) is already observed in Corollary 1.2[7]. As a follow up of our investigations on the ideals in these two rings, we establish that the structure spaces of the two rings P(X,C) and P(X,C)∩C(X) are homeomorphic. The structure space of a commutative ring R with unity stands for the set of all maximal ideals of R equipped with the well- known hull-kernel topology. It was established in [21] and [22], independently that the structure space of all the intermediate rings of real-valued continuous functions on X are one and the same viz the Stone-Čech compactification βX of X. It follows therefore that the structure space of each intermediate ring of complex-valued continuous functions on X is also βX. This is one of the main technical results in our article. We like to mention in this context that a special case of this result telling that the structure space of C(X,C) is βX is quite well known, see [19]. We call a ring P(X,C) in the family Σ(X,C) a C-type ring if it is isomorphic to a ring of the form C(Y,C) for Tychonoff space Y . We establish that if I is any ideal of C(X,C), then the linear sum C∗(X,C) + I is a C-type ring. This is the complex analogue of the corresponding result in the intermediate rings of real-valued continuous functions on X as proved in [16]. We further realise that these are the only C-type intermediate rings in the family Σ(X,C) when and only when X is pseudocompact i.e. C(X,C) = C∗(X,C). It is well-known that if M is a maximal ideal in C(X), then the residue class field C(X)/M is real closed in the sense that every positive element in this field is a square and each odd degree polynomial over this field has a root in the same field [17, Theorem 13.4]. The complex analogue of this result as we realise © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 48 Intermediate rings of complex-valued continuous functions is that for a maximal ideal M in C(X,C),C(X,C)/M is an algebraically closed field and furthermore this field is the algebraic closure of C(X)/M ∩ C(X). In section 4 of this article, we deal with a few special problems originat- ing from an ideal P of closed sets in X and a certain class of ideals in the ring C(X,C). A family P of closed sets in X is called an ideal of closed sets in X if for any two sets A,B in P,A ∪ B ∈ P and for any closed set C contained in A,C is also a member of P. We let CP(X,C) be the set of all those functions f in C(X,C) whose support clX(X \ Z(f)) is a mem- ber of P; here Z(f) = {x ∈ X : f(x) = 0} is the zero set of f in X. We determine a necessary and sufficient condition for CP(X,C) to become a prime ideal in the ring C(X,C) and examine a few special cases correspond- ing to some specific choices of the ideal P. The ring C∞(X,C) = {f ∈ C(X,C) : f vanishes at infinity in the sense that for each n ∈ N,{x ∈ X : |f(x)| ≥ 1 n } is compact} is an ideal of C∗(X,C) but not necessarily an ideal of C(X,C). On the assumption that X is locally compact, we determine a neces- sary and sufficient condition for C∞(X,C) to become an ideal of C(X,C). The fifth section of this article is devoted to finding out the estimates of a few standard parameters concerning zero divisor graphs of a few rings of complex-valued continuous functions on X. Thus for instance we have checked that if Γ(P(X,C)) is the zero divisor graph of an intermediate ring P(X,C) belonging to the family Σ(X,C), then each cycle of this graph has length 3, 4 or 6 and each edge is an edge of a cycle with length 3 or 4. These are the complex analogues of the corresponding results in the zero divisor graph of C(X) as obtained in [9]. 2. Ideals in intermediate rings Notation: For any subset A(X) of C(X) such that 0 ∈ A(X), we set [A(X)]c = {f + ig : f,g ∈ A(X)} and call it the extension of A(X). Then it is easy to see that [A(X)]c ∩ C(X) = A(X) = [A(X)]c ∩ A(X). From now on, unless otherwise stated, we assume that A(X) is an intermediate ring of real-valued continuous functions on X, i.e. A(X) is a member of the family Σ(X). It follows at once that [A(X)]c is an intermediate ring of complex-valued continuous functions and it is not hard to verify that [A(X)]c is the smallest intermediate ring in Σ(X,C) which contains A(X) and the constant function i. Furthermore [A(X)]c is conjugate-closed meaning that if f +ig ∈ [A(X)]c with f,g ∈ A(X), then f −ig ∈ [A(X)]c. The following result tells that intermediate rings in the family Σ(X,C) are the extensions of intermediate rings in Σ(X). Theorem 2.1. Let P(X,C) be an intermediate ring of C(X,C). Then P(X,C) is absolutely convex. Proof. Let |f| ≤ |g|, f ∈ C(X,C),g ∈ P(X,C). Then f = f 1+g2 (1 + g2) ∈ P(X,C). Hence P(X,C) is absolutely convex. � Theorem 2.2. An intermediate ring P(X,C) of C(X,C) is conjugate closed. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 49 A. Acharyya, S. K. Acharyya, S. Bag and J. Sack Proof. Let f + ig ∈ P(X,C). We have |f| ≤ |f + ig|, |g| ≤ |f + ig| and f + ig ∈ P(X,C). Since P(X,C) is absolutely convex, then f,g ∈ P(X,C). This implies f,ig ∈ P(X,C) as i ∈ P(X,C). Thus f − ig ∈ P(X,C). Hence P(X,C) is conjugate closed. � Theorem 2.3. A ring P(X,C) of complex valued continuous functions on X is a member of Σ(X,C) if and only if there exists a ring A(X) in the family Σ(X) such that P(X,C) = [A(X)]c. Proof. Assume that P(X,C) ∈ Σ(X,C) and let A(X) = P(X,C)∩C(X). Then it is clear that A(X) ∈ Σ(X) and [A(X)]c ⊆ P(X,C). To prove the reverse containment, let f + ig ∈ P(X,C). Here f,g ∈ C(X). Since P(X,C) is conjugate closed, f − ig ∈ P(X,C), and hence 2f and 2ig both belong to P(X,C). Since constant functions are bounded and hence in P(X,C), both the constant functions 1 2 and 1 2i are in P(X,C). It follows that both f and g are in P(X,C) ∩ C(X), and hence in A(X). Consequently, f + ig ∈ [A(X)]c. Thus, P(X,C) ⊆ [A(X)]c. � The following facts involving convex sets will be useful. A subset S of C(X) is called absolutely convex if whenever |f| ≤ |g| with g ∈ S and f ∈ C(X), then f ∈ S. Theorem 2.4. Let A(X) ∈ Σ(X). Then (a) A(X) is an absolutely convex subring of C(X) (in the sense that if |f| ≤ |g| with g ∈ A(X) and f ∈ C(X), then f ∈ A(X)) ([16, Proposition 3.3]). (b) A prime ideal P in A(X) is an absolutely convex subset of A(X) ([13, Theorem 2.5]). The following convenient formula for [A(X)]c with A(X) ∈ Σ(X) will often be helpful to us. Theorem 2.5. For any A(X) ∈ Σ(X), [A(X)]c = {h ∈ C(X,C) : |h| ∈ A(X)}. Proof. First assume that h = f + ig ∈ [A(X)]c with f,g ∈ A(X). Then |h| ≤ |f| + |g|. This implies, in view of Theorem 2.4(a), that h ∈ A(X) and also |h| ∈ A(X). Conversely, let h = f+ig ∈ C(X,C) with f,g ∈ C(X), be such that |h| ∈ A(X). This means that (f2 + g2) 1 2 ∈ A(X). Since |f| ≤ (f2 + g2) 1 2 , this implies in view of Theorem 2.4(a) that f ∈ A(X). Analogously g ∈ A(X). Thus h ∈ [A(X)]c. � Theorem 2.6. If I is an ideal in A(X) ∈ Σ(X), then Ic = {f + ig : f,g ∈ I} is the smallest ideal in [A(X)]c containing I. Furthermore Ic ∩ A(X) = I = Ic ∩ C(X). Proof. It is easy to show that Ic is an ideal in [A(X)]c containing I. Let K be an ideal of [A(X)]c containing I. To show Ic ⊆ K. Let f + ig ∈ K, where f,g ∈ I. Since I ⊆ K, then f,g ∈ K. Now K is an ideal of [A(X)]c, f,g ∈ K © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 50 Intermediate rings of complex-valued continuous functions implies f +ig ∈ K. Therefore Ic ⊆ K. Hence Ic is the smallest ideal of [A(X)]c containing I. Proof of the second part is trivial. � Theorem 2.7. If I and J are ideals in A(X) ∈ Σ(X), then I ⊆ J if and only if Ic ⊆ Jc. Also I ( J when and only when Ic ( Jc. Proof. If I ⊆ J, then clearly Ic ⊆ Jc. Conversely, let Ic ⊆ Jc. Let f ∈ I. Since I ⊂ Ic, we have f ∈ Ic ⊆ Jc. Now f = f + i0 and Jc = {g + ih : g,h ∈ J}. Therefore f ∈ J. Hence I ⊆ J. For the second part we consider I ( J and f ∈ J \ I. Then f ∈ Jc \ Ic. Thus Ic ( Jc. Conversely, let Ic ( Jc and f + ig ∈ Jc \ Ic. Then either f or g is outside I. Let f /∈ I. Then f ∈ J \ I. Hence I ( J. This completes the proof. � We have the following convenient formula for Ic when I is an absolutely convex ideal of A(X). Theorem 2.8. If I is an absolutely convex ideal of A(X) (in particular if I is a prime ideal or a maximal ideal of A(X)), then Ic = {h ∈ [A(X)]c : |h| ∈ I}. Proof. Let h = f + ig ∈ Ic. Then f,g ∈ I. Since |h| ≤ |f| + |g|, the absolute convexity of I implies that |h| ∈ I. Conversely, let h = f +ig ∈ [A(X)]c be such that |h| ∈ I. Here f,g ∈ A(X). Since |f| ≤ (f2 + g2) 1 2 = |h|, it follows from the absolute convexity of I that f ∈ I. Analogously g ∈ I. Hence h ∈ Ic. � The above theorem prompts us to define the notion of an absolutely convex ideal in P(X,C) ∈ Σ(X,C) as follows: Definition 2.9. An ideal J in P(X,C) in Σ(X,C) is called absolutely convex if for g,h in C(X,C) with |g| ≤ |h| and h ∈ J, it follows that g ∈ J. The first part of the following proposition is immediate, while the second part follows from Theorem 2.3 and Theorem 2.8. Theorem 2.10. Let P(X,C) ∈ Σ(X,C). (i) If J is an absolutely convex ideal of P(X,C), then J ∩ C(X) is an ab- solutely convex ideal of the intermediate ring P(X,C) ∩C(X) ∈ Σ(X). (ii) An ideal I in P(X,C) ∩ C(X) is absolutely convex in this ring if and only if Ic is an absolutely convex ideal of P(X,C). (iii) If J is an absolutely convex ideal of P(X,C), then J = [J ∩ C(X)]c. Proof. (iii) It is trivial that [J ∩ C(X)]c ⊆ J. To prove the reverse implication relation let h = f + ig ∈ J, with f,g ∈ C(X). The absolute convexity of J implies that |h| ∈ J. Consequently |h| ∈ J∩C(X). But since |f| ≤ (f2+g2) 1 2 = |h|, it follows again due to the absolute convexity of P(X,C) as a subring of C(X,C) that f ∈ P(X,C). We further use absolute convexity of J in P(X,C) to assert that f ∈ J. Analogously g ∈ J. Thus h = f + ig ∈ [J ∩ C(X)]c. Therefore J ⊆ [J ∩ C(X)]c. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 51 A. Acharyya, S. K. Acharyya, S. Bag and J. Sack Remark 2.11. For any P(X,C) ∈ Σ(X,C), the assignment I 7→ Ic provides a one-to-one correspondence between the absolutely convex ideals of P(X,C) ∩ C(X) and those of P(X,C). The following theorem gives a one-to-one correspondence between the prime ideals of P(X,C) and those of P(X,C) ∩ C(X). Theorem 2.12. Let P(X,C) be member of Σ(X,C). An ideal J of P(X,C) is prime if and only if there exists a prime ideal Q in P(X,C) ∩ C(X) such that J = Qc. Proof. Let J be a prime ideal in P(X,C) and let Q = J ∩ C(X) and A(X) = P(X,C) ∩ C(X). Then Q is a prime ideal in the ring A(X). It is easy to see that Qc ⊆ J. To prove the reverse containment, let h = f + ig ∈ J, where f,g ∈ P(X,C). Note that P(X,C) = [A(X)]c by Theorem 2.3. Hence f,g ∈ A(X) and therefore f − ig ∈ P(X,C). As J is an ideal of P(X,C), it follows that (f + ig)(f − ig) ∈ J i.e, f2 + g2 ∈ J ∩ C(X) = Q. Since Q is a prime ideal in A(X), we can apply Theorem 2.4(b), yielding f2 ∈ Q and hence f ∈ Q. Analogously g ∈ Q. Thus h ∈ Qc. Therefore J ⊆ Qc. To prove the converse of this theorem, let Q be a prime ideal in A(X). It follows from Theorem 2.8 that Qc = {h ∈ P(X,C) : |h| ∈ Q} and therefore Qc is a prime ideal in P(X,C). Finally we note that Qc ∩ C(X) = Q. � Remark 2.13. For any P(X,C) ∈ Σ(X,C), the collection of all prime ideals in P(X,C) is precisely {Qc : Q is a prime ideal in P(X,C) ∩ C(X)}. Remark 2.14. The collection of all minimal prime ideals in P(X,C) is precisely {Qc : Q is a minimal prime ideal in P(X, C) ∩ C(X)}. [This follows from Remark 2.13 and Theorem 2.7]. Theorem 2.15. For any P(X,C) ∈ Σ(X,C), the collection of all maximal ideals in P(X,C) is {Mc : M is a maximal ideal of P(X,C) ∩ C(X)}. Proof. Let M be a maximal ideal in P(X,C) ∩ C(X) = A(X). Then by Theo- rem 2.12, Mc is a prime ideal in P(X,C). Suppose that Mc is not a maximal ideal in P(X,C), then there exists a prime ideal T in P(X,C) such that Mc ( T . By remark 2.11, there exists a prime ideal P in A(X) such that J = Pc. So Mc ( Pc. This implies in view of Theorem 2.5 that M ( P , a contradiction to the maximality of M in A(X). Conversely, let J be a maximal ideal of P(X,C). In particular J is a prime ideal in this ring. By Remark 2.13, J = Qc for some prime ideal Q in A(X). We claim that Q is a maximal ideal in A(X). Suppose not; then Q ( K for some proper ideal K in A(X). Then by Theorem 2.7, Qc ( Kc and Kc a proper ideal in P(X,C); this contradicts the maximality of J = Qc. � We next prove analogoues of Remark 2.13 and Theorem 2.15 for two impor- tant classes of ideals viz z-ideals and z◦-ideals in P(X,C) ∈ Σ(X,C). These ideals are defined as follows. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 52 Intermediate rings of complex-valued continuous functions Definition 2.16. Let R be a commutative ring with unity. For each a ∈ R, let Ma (respectively Pa) stand for the intersection of all maximal ideals (respectively all minimal prime ideals) which contain a. An ideal I in R is called a z-ideal (respectively z◦-ideal) if for each a ∈ I,Ma ⊆ I (respectively Pa ⊆ I). This notion of z-ideals is consistent with the notion of z-ideal in C(X) (see [17, 4A5]). Since each prime ideal in an intermediate ring A(X) ∈ Σ(X) is absolutely convex (Theorem 2.4(b)), it follows from Theorem 2.10(ii) and Remark 2.13 that each prime ideal in P(X,C) ∈ Σ(X,C) is absolutely convex. In particular each maximal ideal is absolutely convex. Now if I is a z-ideal in P(X,C) ∈ Σ(X,C) and |f| ≤ |g|,g ∈ I,f ∈ P(X,C), then Mg ⊆ I. Let M be a maximal ideal in P(X,C) containing g. It follows due to the absolute convexity of M that f ∈ M. Therefore f ∈ Mg ⊂ I. Thus each z-ideal in P(X,C) is absolutely convex. Analogously it can be proved that each z◦-ideal in P(X,C) is absolutely convex. The following subsidiary result can be proved using routine arguments. Lemma 2.17. For any family {Iα : α ∈ Λ} of ideals in an intermediate ring A(X) ∈ Σ(X), ( ⋂ α∈Λ Iα)c = ⋂ α∈Λ(Iα)c. Theorem 2.18. An ideal J in a ring P(X,C) ∈ Σ(X,C) is a z-ideal in P(X,C) if and only if there exists a z-ideal I in P(X,C) ∩ C(X) such that J = Ic. Proof. First assume that J is a z-ideal in P(X,C). Let I = J ∩ C(X). Since J is absolutely convex, it follows from Theorem 2.10(iii) that J = Ic. We show that I is a z-ideal in P(X,C) ∩ C(X). Choose f ∈ I. Suppose {Mα : α ∈ Λ} is the set of all maximal ideals in the ring P(X,C) ∩ C(X) which contain f. It follows from Theorem 2.15 that {(Mα)c : α ∈ Λ} is the set of all maximal ideals in P(X,C) containing f. Since f ∈ J and J is a z-ideal in P(X,C), it follows that ⋂ α∈Λ(Mα)c ⊆ J. This implies in the view of Lemma 2.17 that ( ⋂ α∈Λ Mα)c ∩ C(X) ⊆ I, and hence f ∈ ⋂ α∈Λ Mα ⊆ I. Thus it is proved that I is a z-ideal in P(X,C) ∩ C(X). Conversely, let I be a z-ideal in the ring P(X,C)∩C(X). We shall prove that Ic is a z-ideal in P(X,C). We recall from Theorem 2.3 that [P(X,C)∩C(X)]c = P(X,C). Choose f from Ic. From Theorem 2.8, it follows that (taking care of the fact that each z-ideal in P(X,C) is absolutely convex) |f| ∈ I. Let {Nβ : β ∈ Λ ∗} be the set of all maximal ideals in P(X,C)∩C(X) which contain the function |f|. The hypothesis that I is a z-ideal in P(X,C)∩C(X) therefore implies that ⋂ β∈Λ∗ Nβ ⊆ I. This further implies in view of Lemma 2.17 that⋂ β∈Λ∗(Nβ)c ⊆ Ic. Again it follows from Theorem 2.8 that, for any maximal ideal M in P(X,C) ∩ C(X) and any g ∈ P(X,C), g ∈ Mc if and only if |g| ∈ M. Thus for any β ∈ Λ∗, |f| ∈ Nβ if and only if f ∈ (Nβ)c. This means that {(Nβ)c}β∈Λ∗ is the collection of maximal ideals in P(X,C) which contain f, and we have already observed that f ∈ ∩β∈Λ∗(Nβ)c ⊆ Ic. Consequently Ic is a z-ideal in P(X,C). � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 53 A. Acharyya, S. K. Acharyya, S. Bag and J. Sack If we use the result embodied in Remark 2.14 and take note of the fact that each minimal prime ideal in P(X,C) is absolutely convex and argue as in the proof of Theorem 2.18, we get the following proposition: Theorem 2.19. An ideal J in a ring P(X,C) ∈ Σ(X,C) is a z◦-ideal in P(X,C) if and only if there exists a z◦-ideal I in P(X,C) ∩ C(X) such that J = Ic. An ideal J in P(X,C) ∈ Σ(X,C) is called fixed if ⋂ f∈J Z(f) 6= ∅. The following proposition is a straightforward consequence of Theorem 2.6. Theorem 2.20. An ideal J in a ring P(X,C) ∈ Σ(X,C) is a fixed ideal in P(X,C) if and only if J ∩ C(X) is a fixed ideal in P(X,C) ∩ C(X). We recall that a space X is called an almost P space if every non-empty Gδ subset of X has non-empty interior. These spaces have been characterized via z-ideals and z◦-ideals in the ring C(X) in [8]. We would like to mention that the same class of spaces have witnessed a very recent characterization in terms of fixed maximal ideals in a given intermediate ring A(X) ∈ Σ(X). We reproduce below these two results to make the paper self-contained. Theorem 2.21 ([8]). X is an almost P space if and only if each maximal ideal in C(X) is a z◦-ideal if and only if each z-ideal in C(X) is a z◦-ideal. Theorem 2.22 ([12]). Let A(X) ∈ Σ(X) be an intermediate ring of real-valued continuous functions on X. Then X is an almost P space if and only if each fixed maximal ideal M p A = {g ∈ A(X) : g(p) = 0} of A(X) is a z◦-ideal. It is further realised in [12] that if X is an almost P space, then the statement of Theorem 2.21 cannot be improved by replacing C(X) by an intermediate ring A(X), different from C(X). Indeed it is shown in [12, Theorem 2.4] that if an intermediate ring A(X) 6= C(X), then there exists a maximal ideal in A(X) (which is incidentally also a z-ideal in A(X)), which is not a z◦-ideal in A(X). We record below the complex analogue of the above results. Theorem 2.23. X is an almost P space if and only if each maximal ideal of C(X,C) is a z◦-ideal if and only if each z-ideal in C(X,C) is a z◦-ideal. Proof. This follows from combining Theorems 2.15, 2.18, 2.19, and 2.21. � Theorem 2.24. Let P(X,C) ∈ Σ(X,C). Then X is almost P if and only if each fixed maximal ideal M p P = {g ∈ P(X,C) : g(p) = 0} of P(X,C) is a z◦-ideal. Proof. This follows from combining Theorems 2.15, 2.20, and 2.22. � Theorem 2.25. Let X be an almost P space and let P(X,C) be a member of Σ(X,C) such that P(X,C) ( C(X,C). Then there exists a maximal ideal in P(X,C), which is not a z◦-ideal in P(X,C). © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 54 Intermediate rings of complex-valued continuous functions Thus, within the class of almost P-spaces X, C(X,C) is characterized amongst all the intermediate rings P(X,C) of Σ(X,C) by the requirement that z-ideals and z◦-ideals (equivalently maximal ideals and z◦-ideals) in P(X,C) are one and the same. Proof. This follows from combining Theorems 2.15, 2.18, and 2.19 of this article together with [12, Theorem 2.4]. � We recall the classical result that X is a P space if and only if C(X) is a von-Neumann regular ring meaning that each prime ideal in C(X) is maximal. Incidentally the following fact was rather recently established: Theorem 2.26 ([3, 20, 12]). If A(X) ∈ Σ(X) is different from C(X), then A(X) is never a regular ring. Theorems 2.12, 2.15, and 2.26 yield in a straight forward manner the follow- ing result: Theorem 2.27. If P(X,C) ∈ Σ(X,C) is a proper subring of C(X,C), then P(X,C) is not a von-Neumann regular ring. It is well-known that if P is a non maximal prime ideal in C(X) and M is the unique maximal ideal containing P , then the set of all prime ideals in C(X) that lie between P and M makes a Dedekind complete chain containing no fewer than 2ℵ1 many members (see [17, Theorem 14.19]). If we use this standard result and combine with Theorems 2.7, 2.12, and 2.15, we obtain the complex-version of this fact: Theorem 2.28. Suppose P is a non maximal prime ideal in the ring C(X,C). Then there exists a unique maximal ideal M containing P in this ring. Fur- thermore, the collection of all prime ideals that are situated between P and M constitutes a Dedekind complete chain containing at least 2α1 many members. Thus for all practical purposes (say for example when X is not a P space), C(X,C) is far from being a Noetherian ring. Incidentally we shall decide the Noetherianness condition of C(X,C) by deducing it from a result in Section 4; in particular, we show that C(X,C) is Noetherian if and only if X is a finite set. 3. Structure spaces of intermediate rings We need to recall a few technicalities associated with the hull-kernel topology on the set of all maximal ideals M(A) of a commutative ring A with unity. If we set for any element a of A, M(A)a = {M ∈ M(A) : a ∈ M}, then the family {M(A)a : a ∈ A} constitutes a base for closed sets of the hull-kernel topology on M(A). We may write Ma for M(A)a when context is clear. The set M(A) equipped with this hull-kernel topology is called the structure space of the ring A. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 55 A. Acharyya, S. K. Acharyya, S. Bag and J. Sack For any subset M◦ of M(A), its closure M◦ in this topology is given by: M◦ = {M ∈ M(A) : M ⊇ ⋂ M◦}. For further information on this topology, see [17, 7M]. Following the terminology of [14], by a (Hausdorff) compactification of a Tychonoff space X we mean a pair (α,αX), where αX is a compact Hausdorff space and α : X → αX a topological embedding with α(X) dense in αX. For simplicity, we often designate such a pair by the notation αX. Two compact- ifications αX and γX of X are called topologically equivalent if there exists a homeomorphism ψ : αX → γX with the property ψ ◦ α = γ. A compactifi- cation αX of X is said to possess the extension property if given a compact Hausdorff space Y and a continuous map f : X → Y , there exists a continu- ous map fα : αX → Y with the property fα ◦ α = f. It is well known that the Stone-Čech compactification βX of X or more formally the pair (e,βX), where e is the evaluation map on X induced by C∗(X) defined by the formula: e(x) = (f(x) : f ∈ C∗(X)) such that e : X 7→ RC ∗(X) , enjoys the extension property. Furthermore this extension property characterizes βX amongst all the compactifications of X in the sense that whenever a compactification αX of X has extension property, it is topologically equivalent to βX. For more information on these topic, see [14, Chapter 1]. The structure space M(A(X)) of an arbitrary intermediate ring A(X) ∈ Σ(X) has been proved to be homeomorphic to βX, independently by the au- thors in [21] and [22]. Nevertheless we offer yet another independent technique to establish a modified version of the same fact by using the above terminology of [14]. Theorem 3.1. Let ηA : X → M(A(X)) be the map defined by ηA(x) = MxA = {g ∈ A(X) : g(x) = 0} (a fixed maximal ideal in A(X)). Then the pair (ηA,M(A(X))) is a (Hausdorff) compactification of X, which further sat- isfies the extension property. Hence the pair (ηA,M(A(X))) is topologically equivalent to the Stone-Čech compactification βX of X. Proof. Since X is Tychonoff, ηA is one-to-one. Also clM(A(X))ηA(X) = {M ∈ M(A(X)) : M ⊇ ⋂ x∈X M x A} = {M ∈ M(A(X)) : M ⊇ {0}} = M(A(X)). It follows from a result proved in Theorem 3.3 and Theorem 3.4 [23] that M(A(X)) is a compact Hausdorff space and ηA is an embedding. Thus (ηA,M(A(X))) is a compactification of X. To prove that this compactifi- cation of X possesses the extension property we take a compact Hausdorff space Y and a continuous map f : X → Y . It suffices to define a continuous map fβA : M(A(X)) → Y with the property that fβA ◦ ηA = f. Let M be any member of M(A(X)) i.e. M is a maximal ideal of the ring A(X). Define M̂ = {g ∈ C(Y ) : g ◦ f ∈ M}. Note that if g ∈ C(Y ) then g ◦ f ∈ C(X). Further note that since Y is compact and g ∈ C(Y ), g is bounded i.e. g(Y ) is a bounded subset of R. It follows that (g ◦ f)(X) is a bounded subset of R and hence g ◦ f ∈ C∗(X). Consequently g ◦ f ∈ A(X). Thus the definition of M̂ is without any ambiguity. It is easy to see that M̂ is an ideal of C(Y ). It follows, since M is a maximal ideal and therefore a prime ideal of A(X), that © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 56 Intermediate rings of complex-valued continuous functions M̂ is a prime ideal of C(Y ). Since C(Y ) is a Gelfand ring, M̂ can be extended to a unique maximal ideal N in C(Y ). Since Y is compact, N is fixed (see [17, Theorem 4.11]). Thus we can write: N = Ny = {g ∈ C(Y ) : g(y) = 0} for some y ∈ Y . We observe that y ∈ ⋂ g∈M̂ Z(g). Indeed ⋂ g∈M̂ Z(g) = {y} for if y1,y2 ∈ ⋂ g∈M̂ Z(g), for y1 6= y2, then M̂ ⊆ Ny1 and M̂ ⊆ Ny2 which is impos- sible as Ny1 6= Ny2 and C(Y ) is a Gelfand ring. We then set f βA(M) = y. Note that {fβA(M)} = ⋂ g∈M̂ Z(g). Thus f βA : M(A(X)) → Y is a well defined map. Now choose x ∈ X and then g ∈ M̂x A ; then g ◦ f ∈ MxA, which implies that (g ◦ f)(x) = 0. Consequently f(x) ∈ Z(g) for each g ∈ M̂xA. On the other hand {fβA(MxA)} = ⋂ g∈M̂x A Z(g). This implies that fβA(MxA) = f(x); in other words (fβA ◦ ηA)(x) = f(x) and this relation is true for each x ∈ X. Hence fβA ◦ ηA = f. Now towards the proof of the continuity of the map fβA, choose M ∈ M(A(X)) and a neighbourhood W of fβA(M) in the space Y . In a Ty- chonoff space every neighbourhood of a point x contains a zero set neigh- bourhood of x, which contains, a co-zero set neighbourhood of x. So there exist some g1,g2 ∈ C(Y ), such that f βA(M) ∈ Y \ Z(g1) ⊆ Z(g2) ⊆ W . It follows that g1g2 = 0 as Z(g1) ∪ Z(g2) = Y which means that Z(g1g2) = Y . Furthermore fβA(M) /∈ Z(g1). Since {f βA(M)} = ⋂ g∈M̂ Z(g), as observed earlier, we then have g1 /∈ M̂. This means that g1 ◦ f /∈ M. In other words M ∈ M(A(X)) \ Mg1◦f, which is an open neighbourhood of M in M(A(X)). We shall check that fβA(M(A(X)) \ Mg1◦f) ⊆ W and that set- tles the continuity of fβA at M. Towards that end, choose a maximal ideal N ∈ M(A(X)) \ Mg1◦f. This means that N /∈ Mg1◦f, i.e. g1 ◦ f /∈ N. Thus g1 /∈ N̂. But as g1g2 = 0 and N̂ is prime ideal in C(Y ), it must be that g2 ∈ N̂. Since {fβA(N)} = ⋂ g∈N̂ Z(g), it follows that fβA(N) ∈ Z(g2) ⊆ W . � To achieve the complex analogue of the above mentioned theorem, we need to prove the following proposition, which is by itself a result of independent interest. Theorem 3.2. Let A(X) ∈ Σ(X). Then the map ψA : M([A(X)]c) → M(A(X)) mapping M → M ∩ A(X) is a homeomorphism from the structure space of [A(X)]c onto the structure space of A(X). Proof. That the above map ψA is a bijection between the structure spaces of [A(X)]c and A(X) follows from Theorems 2.3, 2.6, 2.7, and 2.15. Recall (same notation as before) that M([A(X)]c)f is the set of maximal ideals in the ring [A(X)]c containing the function f ∈ [A(X)]c. A typical basic closed set in the structure space M([A(X)]c) is given by M([A(X)]c)h where h ∈ [A(X)]c. Note that M([A(X)]c)h = {J ∈ M([A(X)]c) : h ∈ J}. So for h ∈ [A(X)]c, J ∈ M([A(X)]c)h if and only if h ∈ J, and this is true in view of Theorem 2.8 and the absolute convexity of maximal ideals (see Theorem 2.4(b) of the present article) if and only if |h| ∈ J ∩ A(X), and this holds when and only when J ∩ A(X) ∈ M(A(X))|h|, which is a basic closed set in the structure space © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 57 A. Acharyya, S. K. Acharyya, S. Bag and J. Sack M(A(X)) of the ring A(X). Thus (3.1) ψA[M([A(X)]c)h] = M(A(X))|h| Therefore ψA carries a basic closed set in the domain space onto a basic closed set in the range space. Now for a maximal ideal N in A(X) and a func- tion g ∈ A(X),g belongs to N if and only if |g| ∈ N, because of the ab- solutely convexity of a maximal ideal in an intermediate ring. Consequently M(A(X))g = M(A(X))|g| for any g ∈ A(X). Hence from relation (3.1), we get: ψA[M([A(X)]c)g] = M(A(X))g which implies that ψ −1 A [M(A(X))g] = M([A(X)]c)g. Thus ψ −1 A carries a basic closed set in the structure space M(A(X)) onto a basic closed in the structure space M([A(X)]c). Altogether ψA becomes a homeomorphism. � For any x ∈ X and A(X) ∈ Σ(X), set Mx A[C] = {h ∈ [A(X)]c : h(x) = 0}. It is easy to check by using standard arguments, such as those employed to prove the textbook theorem [17, Theorem 4.1], that Mx A[C] is a fixed maximal in [A(X)]c and M x A[C] ∩ A(X) = MxA = {g ∈ A(X) : g(x) = 0}. Let ζA : X 7→ M([A(X)]c) be the map defined by: ζA(x) = M x A[C] . Then we have the following results. Theorem 3.3. (ζA,M([A(X)]c)) is a Hausdorff compactification of X. Fur- thermore (ψA ◦ ζA)(x) = ηA(x) for all x in X. Hence (ζA,M([A(X)]c)) is topologically equivalent to the Hausdorff compactification (ηA,M(A(X))) as considered in Theorem 3.1. Consequently (ζA,M([A(X)]c)) turns out to be topologically equivalent to the Stone-Čech compactification βX of X. Proof. Since M(A(X)) is Hausdorff [23], it follows from Theorem 3.2 that M([A(X)]c) is a Hausdorff space. Now by following closely the arguments made at the very beginning of the proof of Theorem 3.1, one can easily see that (ζA,M([A(X)]c)) is a Hausdorff compactification of X. The second part of the theorem is already realised in Theorem 3.2. The third part of the present theorem also follows from Theorem 3.2. � Definition 3.4. An intermediate ring A(X) ∈ Σ(X) is called C-type in [16], if it is isomorphic to C(Y ) for some Tychonoff space Y . In [16], the authors have shown that if I is an ideal of the ring C(X), then the linear sum C∗(X)+I is a C-type ring and of course C∗(X)+I ∈ Σ(X). Recently the authors in [1] have realised that these are the only C-type intermediate rings of real-valued continuous functions on X if and only if X is pseudocompact. We now show that the complex analogues of all these results are also true. We reproduce the following result established in [15], which will be needed for this purpose. Theorem 3.5. A ring A(X) ∈ Σ(X) is C-type if and only if A(X) is iso- morphic to the ring C(υAX), where υAX = {p ∈ βX : f ∗(p) ∈ R for each f ∈ A(X)} and f∗ : βX 7→ R ∪ {∞} is the Stone extension of the function f. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 58 Intermediate rings of complex-valued continuous functions We extend the notion of C-type ring to rings of complex-valued continuous functions: a ring P(X,C) ∈ Σ(X,C) is a C-type ring if it is isomorphic to a ring C(Y,C) for some Tychonoff space Y . Theorem 3.6. Suppose A(X) ∈ Σ(X) is a C-type intermediate ring of real- valued continuous functions on X. Then [A(X)]c is a C-type intermediate ring of complex-valued continuous functions on X. Proof. Since A(X) is a C-type intermediate ring by Theorem 3.5, there exists an isomorphism ψ : A(X) 7→ C(υAX). Let ψ̂ : [A(X)]c 7→ C(υAX,C) be defined as follows: ψ̂(f +ig) = ψ(f) +iψ(g), where f,g ∈ A(X). It is not hard to check that ψ̂ is an isomorphism from [A(X)]c onto C(υAX,C). � Theorem 3.7. Let I be a z-ideal in C(X,C). Then C∗(X,C) + I is a C-type intermediate ring of complex-valued continuous functions on X. Furthermore these are the only C-type rings lying between C∗(X,C) and C(X,C) if and only if X is pseudocompact. Proof. As mentioned above, it is proved in [16] that for any ideal J in C(X), C∗(X) + J is a C-type intermediate ring of real-valued continuous functions on X. In light of this and Theorem 3.6, it is sufficient to prove for the first part of this theorem that C∗(X,C) + I = [C∗(X) + I ∩ C(X)]c. Towards proving that, let f,g ∈ C∗(X) + I ∩ C(X). We can write g = g1 + g2 where g1 ∈ C ∗(X) and g2 ∈ I ∩ C(X). It follows that ig1 ∈ C ∗(X,C) and ig2 ∈ I and this implies that i(g1 + g2) ∈ C ∗(X,C) + I. Thus f + ig ∈ C∗(X) + I. Hence [C∗(X) + I ∩ C(X)]c ⊆ C ∗(X,C) + I. To prove the reverse inclusion relation, let h1 + h2 ∈ C ∗(X,C) + I, where h1 ∈ C ∗(X,C) and h2 ∈ I. We can write h1 = f1 + ig1,h2 = f2 + ig2, where f1,f2,g1,g2 ∈ C(X). Since h1 ∈ C ∗(X,C), it follows that f1,g1 ∈ C ∗(X). Thus |f2| ≤ |h2| and h2 ∈ I. This implies, because of the absolute convexity of the z-ideal I in C(X,C), that f2 ∈ I. Analogously g2 ∈ I. It is now clear that f1 + f2 ∈ C ∗(X) + I ∩ C(X) and g1 + g2 ∈ C ∗(X) + I ∩ C(X). Thus h1 + h2 = (f1 + f2) + i(g1 + g2) ∈ [C∗(X) + I ∩ C(X)]c. Hence C ∗(X,C) + I ⊆ [C∗(X) + I ∩ C(X)]c. To prove the second part of the theorem, we first observe that if X is pseudocompact, then there is practically nothing to prove. Assume therefore that X is not pseudocompact. Hence by [1], there exists an A(X) ∈ Σ(X) such that A(X) is a C-type ring but A(X) 6= C∗(X) + J for any ideal J in C(X). It follows from Theorem 3.6 that [A(X)]c is a C-type intermediate ring of complex-valued continuous functions belonging to the family Σ(X,C). We assert that there does not exist any z-ideal I in C(X,C) with the rela- tion: C∗(X,C) + I = [A(X)]C and that finishes the present theorem. Sup- pose towards a contradiction, there exists a z-ideal I in C(X,C) such that C∗(X,C) +I = [A(X)]C. Now from the proof of the first part of this theorem, we have already settled that C∗(X,C)+I = [C∗(X)+I∩C(X)]C. Consequently [C∗(X) + I ∩ C(X)]C = [A(X)]C which yields [C ∗(X) + I ∩ C(X)]C ∩ C(X) = [A(X)]C ∩ C(X), and hence C ∗(X) + I ∩ C(X) = A(X), a contradiction. � © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 59 A. Acharyya, S. K. Acharyya, S. Bag and J. Sack We shall conclude this section after incorporating a purely algebraic result pertaining to the residue class field of C(X,C) modulo a maximal ideal in the same field. For each a = (a1,a2, . . . ,an) ∈ C n if P1a,P2a,. . . ,Pna are the zeroes of the polynomial Pa(λ) = λ n + a1λ n−1 + · · · + an, ordered so that |P1a| ≤ |P2a| ≤ · · · ≤ |Pna|, then by following closely the arguments of [17, 13.3(a)], the following result can be obtained. Theorem 3.8. For each k, the function Pk : C n 7→ C, described above, is continuous. By employing the main argument of [17, Theorem 13.4], we obtain the fol- lowing proposition as a consequence of Theorem 3.8. Theorem 3.9. For any maximal ideal N in C(X,C), the residue class field C(X,C)/N is algebraically closed. We recall from Theorem 2.15 that the assignment M 7→ Mc establishes a one-to-one correspondence between maximal ideals in C(X) and those in C(X,C). Let φ : C(X)/M 7→ C(X,C)/Mc be the induced assignment between the corresponding residue class fields, explicitly φ(f + M) = f + Mc for each f ∈ C(X). It is easy to check that φ is a ring homomorphism and is one-to-one because if f + Mc = g + Mc with f,g ∈ C(X), then f − g ∈ Mc ∩ C(X) = M and hence f + M = g + M. Furthermore, if we choose an element f + ig + Mc from C(X,C)/Mc, with f,g ∈ C(X), then one can verify easily that it is a root of the polynomial λ2 − 2(f +Mc)λ+ (f 2 +g2 +Mc) over the field φ(C(X)/M). Identifying C(X)/M with φ(C(X)/M), and taking note of Theorem 3.9 we get the following result. Theorem 3.10. For any maximal ideal M in C(X), the residue class field C(X,C)/Mc is the algebraic closure of C(X)/M. 4. Ideals of the form CP(X,C) and C P ∞(X,C) Let P be an ideal of closed sets in X. We set CP(X,C) = {f ∈ C(X,C) : clX(X \ Z(f)) ∈ P} and C P ∞(X,C) = {f ∈ C(X,C) : for each ǫ > 0 in R,{x ∈ X : |f(x)| ≥ ǫ} ∈ P}. These are the complex analogues of the rings, CP(X) = {f ∈ C(X) : clX(X\Z(f)) ∈ P} and C P ∞(X) = {f ∈ C(X) : for each ǫ > 0,{x ∈ X : |f(x)| ≥ ǫ} ∈ P} already introduced in [4] and investigated subsequently in [5], [12]. As in the real case, it is easy to check that CP(X,C) is a z-ideal in C(X,C) with CP∞(X,C) just a subring of C(X,C). Plainly we have: CP(X,C) ∩ C(X) = CP(X) and C P ∞(X,C) ∩ C(X) = C P ∞(X). The following results need only routine verifications. Theorem 4.1. For any ideal P of closed sets in X, [CP(X)]c = {f +ig : f,g ∈ CP(X)} = CP(X,C) and [C P ∞(X)]c = C P ∞(X,C). Theorem 4.2. a) If I is an ideal of the ring CP(X), then Ic = {f + ig : f,g ∈ I} is an ideal of CP(X,C) and Ic ∩ CP(X) = I. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 60 Intermediate rings of complex-valued continuous functions b) If I is an ideal of the ring CP∞(X), then Ic is an ideal of C P ∞(X,C) and Ic ∩ C P ∞(X) = I. We record below the following consequence of the above theorem. Theorem 4.3. If I1 ( I2 ( ... is a strictly ascending sequence of ideals in CP(X)(respectively C P ∞(X)), then I1c ( I2c ( · · · becomes a strictly ascending sequence of ideals in CP(X,C)(respectively C P ∞(X,C)). The analogous results for a strictly descending sequence of ideals in both the rings CP(X) and C P ∞(X) are also valid. Definition 4.4. A space X is called locally P if each point of X has an open neighbourhood W such that clXW ∈ P. Observe that if P is the ideal of all compact sets in X, then X is locally P if and only if X is locally compact. Towards finding a condition for which CP(X,C) and C P ∞(X,C) are Noe- therian ring/Artinian rings, we reproduce a special version of a fact proved in [6]: Theorem 4.5 (from [6, Theorem 1.1]). Let P be an ideal of closed sets in X and suppose X is locally P. Then the following statements are equivalent: 1) CP(X) is a Noetherian ring. 2) CP(X) is an Artinian ring. 3) CP∞(X) is a Noetherian ring. 4) CP∞(X) is an Artinian ring. 5) X is finite set. We also note the following standard result of Algebra. Theorem 4.6. Let {R1,R2, ...,Rn} be a finite family of commutative rings with identity. The ideals of the direct product R1 × R2 × · · · × Rn are exactly of the form I1 × I2 × · · · × In, where for k = 1,2, . . . ,n, Ik is an ideal of Rk. Now if X is a finite set, with say n elements, then as it is Tychonoff, it is discrete space. Furthermore if X is locally P, then clearly P is the power set of X. Consequently CP(X,C) = C P ∞(X,C) = C(X,C) = C n, which is equal to the direct product of C with itself ‘n’ times. Since C is a field, it has just 2 ideals, hence by Theorem 4.6 there are exactly 2n many ideals in the ring Cn. Hence CP(X,C) and C P ∞(X,C) are both Noetherian rings and Artinian rings. On the other hand if X is an infinite space and is locally P space then it follows from the Theorem 4.3 and Theorem 4.5 that neither of the two rings CP(X,C) and C P ∞(X,C) is either Noetherian or Artinian. This leads to the following proposition as the complex analogue of Theorem 4.5. Theorem 4.7. Let P be an ideal of closed sets in X and suppose X is locally P. Then the following statements are equivalent: 1) CP(X,C) is a Noetherian ring. 2) CP(X,C) is an Artinian ring. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 61 A. Acharyya, S. K. Acharyya, S. Bag and J. Sack 3) CP∞(X,C) is a Noetherian ring. 4) CP∞(X,C) is an Artinian ring. 5) X is finite set. A special case of this theorem, choosing P to be the ideal of all closed sets in X reads: C(X,C) is a Noetherian ring if and only if X is finite set. The following gives a necessary and sufficient condition for the ideal CP(X,C) in C(X,C) to be prime. Theorem 4.8. Let P be an ideal of closed sets in X and suppose X is locally P. Then the following statements are equivalent: (1) CP(X,C) is a prime ideal in C(X,C). (2) CP(X) is a prime ideal in C(X). (3) X /∈ P and for any two disjoint co-zero sets in X, one has its closure lying in P. Proof. The equivalence of (1) and (2) follows from Theorem 2.12 and Theo- rem 4.1. Towards the equivalence (2) and (3), assume that CP(X) is a prime ideal in C(X). If X ∈ P, then for each f ∈ C(X), clX(X \Z(f)) ∈ P meaning that f ∈ CP(X) and hence CP(X) = C(X), a contradiction to the assumption that CP(X) is a prime ideal and in particular a proper ideal of C(X). Thus X /∈ P. Now consider two disjoint co-zero sets X \ Z(f) and X \ Z(g) in X, with f,g ∈ C(X). It follows that Z(f) ∪ Z(g) = X, i.e. fg = 0. Since CP(X) is prime, this implies that f ∈ CP(X) or g ∈ CP(X), i.e. clX(X \ Z(f)) ∈ P or clX(X \ Z(g)) ∈ P. Conversely let the statement (3) be true. Since a z-ideal I in C(X) is prime if and only if for each f,g ∈ C(X), fg = 0 implies f ∈ I or g ∈ I (see [17, Theorem 2.9]) and since CP(X) is a z-ideal in C(X), it is sufficient to show that for each f,g ∈ C(X), if fg = 0 then f ∈ CP(X) or g ∈ CP(X). Indeed fg = 0 implies that X\Z(f) and X\Z(g) are disjoint co-zero sets in X. Hence by supposition (3), either clX(X \ Z(f))P or clX(X \ Z(g)) ∈ P meaning that f ∈ CP(X) or g ∈ CP(X). � A special case of Theorem 4.8, with P equal to the ideal of all compact sets in X, is proved in [10]. We examine a second special case of Theorem 4.8. A subset Y of X is called a bounded subset of X if each f ∈ C(X) is bounded on Y . Let β denote the family of all closed bounded subsets of X. Then β is an ideal of closed sets in X. It is plain that a pseudocompact subset of X is bounded but a bounded subset of X may not be pseudocompact. Here is a counterexample: the open interval (0,1) in R is a bounded subset of R without being a pseudocompact subset of R. However for a certain class of subsets of X, the two notions of boundedness and pseudocompactness coincide. The following well-known proposition substantiates this fact: Theorem 4.9 (Mandelkar [18]). A support of X, i.e. a subset of X of the form clX(X \ Z(f)) for some f ∈ C(X), is a bounded subset of X if and only if it is a pseudocompact subset of X. © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 62 Intermediate rings of complex-valued continuous functions It is clear that the conclusion of Theorem 4.9 remains unchanged if we replace C(X) by C(X,C). Let Cψ(X) = {f ∈ C(X) : f has pseudocompact support} and recall that Cβ(X) = {f ∈ C(X) : f has bounded support}. We would like to mention here that the closed pseudocompact subsets of a pseudocompact space X might not constitute an ideal of closed sets in X. Indeed a closed subset of a pseu- docompact space may not be pseucdocompact. The celebrated example of a Tychonoff plank in [17, 8.20]: [0,ω1] × [0,ω] \ {(ω1,ω)}, where ω1 is the 1st uncountable ordinal and ω is the first infinite ordinal, demonstrates this fact. Nevertheless Cψ(X) is an ideal of the ring C(X). Indeed it follows directly from Theorem 4.9 that Cψ(X) = Cβ(X). A Tychonoff space X is called locally pseudocompact if each point on X has an open neighbourhood with its closure pseudocompact. On the other hand, X is called locally bounded (or locally β) if each point in X has an open neighbourhood with its closure bounded. Since each open neighbourhooad of a point x in a Tychonoff space X contains a co-zero set neighbourhood of x, it follows from Theorem 4.9 that X is locally bounded if and only if X is locally pseudocompact. This combined with Theorem 2.12 leads to the following special case of Theorem 4.8. Theorem 4.10. Let X be locally pseudocompact. Then the following state- ments are equivalent: (1) Cψ(X) is a prime ideal of C(X). (2) Cψ(X,C) = {f ∈ C(X,C) : f has pseudocompact support} is a prime ideal of C(X,C). (3) X is not pseudocompact and for any two disjoint co-zero sets in X, the closure of one of them is pseudocompact. Since for f ∈ C(X,C), f ∈ C∞(X,C) if and only if |f| ∈ C∞(X), it follows that C∞(X,C) is an ideal of C(X,C) if and only if C∞(X) is an ideal of C(X). In general however C∞(X) need not be an ideal of C(X). If X is assumed to be locally compact, then it is proved in [2] and [11] that C∞(X) is an ideal of C(X) when and only when X is pseudocompact. Therefore the following theorem holds. Theorem 4.11. Let X be locally compact. Then the following three statements are equivalent: 1) C∞(X,C) is an ideal of C(X,C). 2) C∞(X) is an ideal of C(X). 3) X is pseudocompact. 5. Zero divisor graphs of rings in the family Σ(X,C) We fix any intermediate ring P(X,C) in the family Σ(X,C). Suppose G = G(P(X,C)) designates the graph whose vertices are zero divisors of P(X,C) and there is an edge between vertices f and g if and only if fg = 0. For any two vertices f,g in G, let d(f,g) be the length of the shortest path between f and © AGT, UPV, 2021 Appl. Gen. Topol. 22, no. 1 63 A. Acharyya, S. K. Acharyya, S. Bag and J. Sack g and Diam G = sup{d(f,g) : f,g ∈ G}. Suppose Gr G designates the length of the shortest cycle in G, often called the girth of G. It is easy to check that a vertex f in G is a divisor of zero in P(X,C) if and only if IntXZ(f) 6= ∅. This parallels the statement that a vertex f in the zero-divisor graph ΓC(X) of C(X) considered in [9] is a divisor of zero in C(X) if and only if IntXZ(f) 6= ∅. We would like to point out in this connection that a close scrutiny into the proof of various results in [9] reveal that several facts related to the nature of the vertices and the length of the cycles related to ΓC(X) have been established in [9] by employing skillfully the last mentioned simple characterization of divisors of zero in C(X). It is expected that the anlogous facts pertaining to the various parameters of the graph G(P(X,C)) = G should also hold. We therefore just record the following results related to the graph G, without any proof. Theorem 5.1. Let f,g be vertices of the graph G. Then d(f,g) = 1 if and only if Z(f) ∪ Z(g) = X; d(f,g) = 2 if and only if Z(f) ∪ Z(g) ( X and IntXZ(f) ∩ IntXZ(g) 6= φ; d(f,g) = 3 if and only if Z(f) ∪ Z(g) ( X and IntXZ(f)∩IntXZ(g) = ∅. Consequently on assuming that X contains at least 3 points, Diam G and Gr G are both equal to 3 (compare with [9, Corollary 1.3]). Theorem 5.2. Each cycle in G has length 3,4 or 6. Furthermore every edge of G is an edge of a cycle with length 3 or 4 (compare with [9, Corollary 2.3]). Theorem 5.3. Suppose X contains at least 2 points. Then a) Each vertex of G is a 4 cycle vertex. b) G is a triangulated graph meaning that each vertex of G is a vertex of a triangle if and only if X is devoid of any isolated point. c) G is a hypertriangulated graph in the sense that each edge of G is edge of a triangle if and only if X is a connected middle P space (compare with the analogous facts in [9, Proposition 2.1]). Acknowledgements. 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