Rings of continuous functions with values in a topological division ring
type: Item
instance of: scholarly article
title: Rings of continuous functions with values in a topological division ring
zbMATH Open document ID: 0868.46018
DOI: 10.1007/BF02363066
author: E. M. Vechtomov
published in: Journal of Mathematical Sciences (New York)
publication date: 1996-10-30T00:00:00Z
review text: This survey article treats topological and algebraic aspects of the theory of rings of continuous functions \(C(X,F)\) on a space \(X\) with values in a topological division ring \(F\). The results presented in the article are mainly due to the author who answers several problems posed by Henriksen, Skornyakov and some others. To give an idea of the contents of the article, we list the titles of sections: 1. Divisibility in rings of continuous rings of functions. 2. Greatest common divisors and least common multiples. 3. Harmonicity of rings \(C(X,F)\). 4. Rings of continuous functions on compact spaces. Kaplansky problem. 5. The sheaf approach. 6. A generalization of the Gelfand-Kolmogorov theorem. 7. Subspaces of \(F\)-ideals and the Gelfand subring. 8. Dualities and definability. 9. Prime and semiprime ideals. 10. Pure and projective ideals. 11. Decomposable ideals. 12. When is the ring \(C(X,F)\) a Bézout-ring? 13. Ring characterization of topological properties. 14. Rings of continuous functions and injectivity. The bibliography contains 102 references.
Mathematics Subject Classification ID: 46E25
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 54C35
zbMATH DE Number: 907038
zbMATH Keywords: divisibility
zbMATH Keywords: greatest common divisors
zbMATH Keywords: dualities
zbMATH Keywords: prime and semiprime ideals
zbMATH Keywords: pure and projective ideals
zbMATH Keywords: decomposable ideals
zbMATH Keywords: rings of continuous functions
zbMATH Keywords: values in a topological division ring
zbMATH Keywords: least common multiples
zbMATH Keywords: Kaplansky problem
zbMATH Keywords: Gelfand-Kolmogorov theorem
zbMATH Keywords: Gelfand subring
zbMATH Keywords: Bézout-ring
reviewed by: Mikhail Tkachenko
MaRDI profile type: Publication
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Rings of continuous functions with values in a topological division ring (full work available at URL)
full work available at URL: https://doi.org/10.1007/bf02363066

Rings of continuous functions with values in a topological division ring (OpenAlex ID)
OpenAlex ID: W2077321328