Primitive elements in rings of continuous functions

Let pi : X -> Y be a surjective continuous map between compact Hausdorff spaces. The map pi 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) subset of C(X) in relation to topological properties of the map pi : X -> Y. We prove that if the extension C(Y) subset of C(X) has a primitive element, i.e., C(X) = C(Y)[f], then it is a finite extension and, consequently, the map pi is locally injective. Moreover, for each primitive element f we consider the ideal I(f) = {P(t) is an element of C(Y)[t]: P(f) = 0} and prove that, for a connected space Y, I(f) is a principal ideal if and only if pi : X -> Y is a trivial covering. (C) 2009 Elsevier B.V. All rights reserved.