Concerning ideals of pointfree function rings

Abstract

We study ideals of pointfree function rings. In particular, we study the lattices of z-ideals and d-ideals of the ring RL of continuous real-valued functions on a completely regular frame L. We show that the lattice of z-ideals is a coherently normal Yosida frame; and the lattice of d-ideals is a coherently normal frame. The lattice of z-ideals is demonstrated to be atly projectable if and only if the ring RL is feebly Baer. On the other hand, the frame of d-ideals is projectable precisely when the frame is cozero-complemented. These ideals give rise to two functors as follows: Sending a frame to the lattice of these ideals is a functorial assignment. We construct a natural transformation between the functors that arise from these assignments. We show that, for a certain collection of frame maps, the functor associated with z-ideals preserves and re ects the property of having a left adjoint. A ring is called a UMP-ring if every maximal ideal in it is the union of the minimal prime ideals it contains. In the penultimate chapter we give several characterisations for the ring RL to be a UMP-ring. We observe, in passing, that if a UMP ring is a Q-algebra, then each of its ideals when viewed as a ring in its own right is a UMP-ring. An example is provided to show that the converse fails. Finally, piggybacking on results in classical rings of continuous functions, we show that, exactly as in C(X), nth roots exist in RL. This is a consequence of an earlier proposition that every reduced f-ring with bounded inversion is the ring of fractions of its bounded part relative to those elements in the bounded part which are units in the bigger ring. We close with a result showing that the frame of open sets of the structure space of RL is isomorphic to L.