Primitive elements in rings of continuous functions

Abstract

Let 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 I 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.