Ideals in rings and intermediate rings of measurable functions
| Autor: | Joshua Sack, Sudip Kumar Acharyya, Sagarmoy Bag |
|---|---|
| Rok vydání: | 2019 |
| Předmět: |
Pure mathematics
Ring (mathematics) Algebra and Number Theory Measurable function Computer Science::Information Retrieval Applied Mathematics General Topology (math.GN) Astrophysics::Instrumentation and Methods for Astrophysics Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Mathematics - Commutative Algebra Commutative Algebra (math.AC) Space (mathematics) Functional Analysis (math.FA) Mathematics - Functional Analysis Set (abstract data type) TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES FOS: Mathematics ComputingMethodologies_DOCUMENTANDTEXTPROCESSING Computer Science::General Literature Ideal (ring theory) 54C40 46E30 ComputingMilieux_MISCELLANEOUS Topology (chemistry) Mathematics - General Topology Mathematics |
| Zdroj: | Journal of Algebra and Its Applications. 19:2050038 |
| ISSN: | 1793-6829 0219-4988 |
| DOI: | 10.1142/s0219498820500383 |
| Popis: | The set of all maximal ideals of the ring $\mathcal{M}(X,\mathcal{A})$ of real valued measurable functions on a measurable space $(X,\mathcal{A})$ equipped with the hull-kernel topology is shown to be homeomorphic to the set $\hat{X}$ of all ultrafilters of measurable sets on $X$ with the Stone-topology. This yields a complete description of the maximal ideals of $\mathcal{M}(X,\mathcal{A})$ in terms of the points of $\hat{X}$. It is further shown that the structure spaces of all the intermediate subrings of $\mathcal{M}(X,\mathcal{A})$ 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 $C(X) = \mathcal{M}(X,\mathcal{A})$ where $\mathcal{A}$ is the $\sigma$-algebra consisting of the zero-sets of $X$. Comment: 15 pages, 0 figures |
| Databáze: | OpenAIRE |
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