Topics in point-free rings of continuous functions

Abstract

Let L be a completely regular frame and RL be the ring of real-valued continuous functions on L. In this dissertation, we will study the maximal ideals and prime ideals of the ring RL which were described by Dube in [20] and [25]. We will also study the z-ideals and d-ideals of point-free function rings and how they are related. In the class of f-rings, sufficient conditions for the sum of z-ideals (resp. d-ideals) to be a z-ideal (resp. d-ideal) will be given. In [29], Dube and Ighedo show that, as in C(X), the sum of two z-ideals in RL is a z-ideal. In [64], the authors study algebraic properties of quotient rings in the ring C(X) of real-valued continuous functions on a Tychonoff space X. We show that this result is actually purely ring theoretic and thus deduce its extensions to the f-rings RL of real-valued continuous functions on a frame L. We will show that the maximal essential ideals of RL are precisely the M-ideals indexed with the nowhere dense sublocales of βL (the Stone-ˇCech compactification of L). We will also show that the nowhere dense sublocales of βL are perfect if and only if every prime ideal in RL/MA is essential, where A is a closed sublocale of βL.