Closed ideals in the uniform topology on the ring of real-valued continuous functions on a frame

Abstract

For a completely regular frame $L$, the ring $\mathcal{R}L$ of real-valued continuous functions on $L$ is equipped with the uniform topology. The closed ideals of $\mathcal{R}L$ in this topology are studied, and a new, merely algebraic characterization of these ideals is given. This result is used to describe the real ideals of $\mathcal{R}L$, and to characterize pseudocompact frames and Lindelöf frames. It is shown that a frame $L$ is finite if and only if every ideal of $\mathcal{R}L$ is closed. Finally, we prove that every closed ideal in $\mathcal{R}L$ is an intersection of maximal ideals.