Maximal ideals in subalgebras of 𝐶(𝑋)

Redlin, Lothar; Watson, Saleem

doi:10.1090/s0002-9939-1987-0894451-2

Cited by 8 publications

(9 citation statements)

References 8 publications

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“…By an intermediate ring of measurable functions we mean a subring N (X, A) of M(X, A) which contains M * (X, A) of all the bounded measurable functions on X. The main technical tool in this section, which we borrow from the articles [13], [14], [15], is that of local invertibility of measurable functions on measurable sets in the given intermediate ring. With each maximal ideal M in N (X, A), we associate an A-ultrafilter Z N [M ] on X which leads to a bijection between the set of all maximal ideals of N (X, A) and the family of all A-ultrafilters on X (Theorems 4.6 and 4.7).…”

Section: Introductionmentioning

confidence: 99%

Ideals in rings and intermediate rings of measurable functions

2019

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The set of all maximal ideals of the ring M(X, A) of real valued measurable functions on a measurable space (X, A) equipped with the hullkernel topology is shown to be homeomorphic to the setX of all ultrafilters of measurable sets on X with the Stone-topology. This yields a complete description of the maximal ideals of M(X, A) in terms of the points ofX. It is further shown that the structure spaces of all the intermediate subrings of M(X, A) containing the bounded measurable functions are one and the same and are compact Hausdorff zero-dimensional spaces. It is observed that when X is a P -space, then C(X) = M(X, A) where A is the σ-algebra consisting of the zero-sets of X.2010 Mathematics Subject Classification. Primary 54C40; Secondary 46E30. Key words and phrases. Rings of Measurable functions, intermediate rings of measurable functions, A-filter on X, A-ultrafilter on X, A-ideal, absolutely convex ideals; hull-kernel topology; Stone-topology; conditionally complete lattice; P -space.The second author thanks the NBHM, Mumbai-400 001, India, for financial support.

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Section: Introductionmentioning

confidence: 99%

Ideals in rings and intermediate rings of measurable functions

2019

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“…This gives an answer to the question raised by Redlin and Watson [5]. It has further been shown that a minimal algebra thus obtained need not be minimal with respect to set inclusion, however it still remains open whether such a minimal algebra exists with respect to set inclusion.…”

Section: Redlin and Watsonmentioning

confidence: 62%

“…Redlin and Watson [5] raised the following question: Given a realcompact space X, does there exist in some sense a minimal algebra A m over R for which X is A m -compact? In this section we give an answer to this question.…”

Section: On a Question Raised By Redlin And Watsonmentioning

confidence: 99%

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On a class of subalgebras of $C(X)$ and the intersection of their free maximal ideals

Acharyya

Chattopadhyay

Ghosh³

1997

Proc. Amer. Math. Soc.

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Abstract. Let X be a Tychonoff space and A a subalgebra of C(X) containing C * (X). Suppose that C K (X) is the set of all functions in C(X) with compact support. Kohls has shown that C K (X) is precisely the intersection of all the free ideals in C(X) or in C * (X). In this paper we have proved the validity of this result for the algebra A. Gillman and Jerison have proved that for a realcompact space X, C K (X) is the intersection of all the free maximal ideals in C(X). In this paper we have proved that this result does not hold for the algebra A, in general. However we have furnished a characterisation of the elements that belong to all the free maximal ideals in A. The paper terminates by showing that for any realcompact space X, there exists in some sense a minimal algebra Am for which X becomes Am-compact. This answers a question raised by Redlin and Watson in 1987. But it is still unsettled whether such a minimal algebra exists with respect to set inclusion.

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“…The structure space of a commutative ring R with unity stands for the set of all maximal ideals of R equipped with the wellknown 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.…”

Section: Introductionmentioning

confidence: 98%

Intermediate rings of complex-valued continuous functions

Acharyya

Bag

et al. 2021

Appl. Gen. Topol.

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<p>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).</p>

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Maximal ideals in subalgebras of 𝐶(𝑋)

Cited by 8 publications

References 8 publications

Ideals in rings and intermediate rings of measurable functions

Ideals in rings and intermediate rings of measurable functions

On a class of subalgebras of $C(X)$ and the intersection of their free maximal ideals

Intermediate rings of complex-valued continuous functions

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