Rings of continuous functions with values in a topological division ring (Q1918663)

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.