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 RL of real-valued continuous functions on L is equipped with the uniform topology. The closed ideals of RL 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 RL, and to characterize pseudocompact frames and Lindelöf frames. It is shown that a frame L is finite if and only if every ideal of RL is closed. Finally, we prove that every closed ideal in RL is an intersection of maximal ideals. Mathematics Subject Classification (2010). Primary: 06D22; Secondary: 54C40, 54C50, 16D25, 13J20.