Ring ideals and the Stone-Čech compactification in pointfree topology

This paper shows that the compact completely regular coreflection in the category of frames is given by the frame of Jacobson radical ideals of the ring RL of real-valued continuous functions on L, as an alternative to its familiar representations in terms of (i) the l-ideals of RL as lattice-ordered ring or (ii) the ideals of the bounded part of RL which are closed in the usual uniform topology. Further, in analogy with this, the compact zero-dimensional coreflection will also be described in terms of ring ideals, this time of the ring ZL of integer-valued continuous functions on L.