Primitive elements in rings of continuous functions

摘要

Let π : X → Y be a surjective continuous map between compact Hausdorff spaces. The map n induces, by composition, an injective morphism C(Y) → C(X) between the corresponding rings of real-valued continuous functions, and this morphism allows us to consider C(Y) as a subring of C(X). This paper deals with algebraic properties of the ring extension C(Y) is contained in C(X) in relation to topological properties of the map π:X→Y. We prove that if the extension C(Y) is contained in C(X) has a primitive element, i.e., C(X) = C(Y)|f|, then it is a finite extension and, consequently, the map π is locally injective. Moreover, for each primitive element f we consider the ideal l_f = (P(t) ∈ C(Y)|t|: P(f) = 0} and prove that, for a connected space Y. I_f is a principal ideal if and only if π : X → Y is a trivial covering.