Primitive elements in rings of continuous functions

AbstractLet π:X→Y be a surjective continuous map between compact Hausdorff spaces. The map π 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)⊆C(X) in relation to topological properties of the map π:X→Y. We prove that if the extension C(Y)⊆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 If={P(t)∈C(Y)[t]:P(f)=0} and prove that, for a connected space Y, If is a principal ideal if and only if π:X→Y is a trivial covering