The ring of real-valued continuous functions on a completely regular frame L is denoted by RL. As usual, βL denotes the Stone-Cech compactification of ˇ L. In the thesis we study ideals of RL induced by sublocales of βL. We revisit the notion of purity in this ring and use it to characterize basically disconnected frames. The socle of the ring RL is characterized as an ideal induced by the sublocale of βL which is the join of all nowhere dense sublocales of βL. A localic map f : L → M induces a ring homomorphism Rh: RM → RL by composition, where h: M → L is the left adjoint of f. We explore how the sublocale-induced ideals travel along the ring homomorphism Rh, to and fro, via expansion and contraction, respectively. The socle of a ring is the sum of its minimal ideals. In the literature, the socle of RL has been characterized in terms of atoms. Since atoms do not always exist in frames, it is better to express the socle in terms of entities that exist in every frame. In the thesis we characterize the socle as one of the types of ideals induced by sublocales. A classical operator invented by Gillman, Henriksen and Jerison in 1954 is used to create a homomorphism of quantales. The frames in which every cozero element is complemented (they are called P-frames) are characterized in terms of some properties of this quantale homomorphism. Also characterized within the category of quantales are localic analogues of the continuous maps of R.G. Woods that characterize normality in the category of Tychonoff spaces. / Mathematical Sciences / Ph. D. (Mathematics)

Text in English / Submitted in partial fulfillment of a Master's Degree at the University of South Africa / Our main goal is to collate into a single document what is presently known regarding pointfree convergence. This will be done by exposing some well-known results on pointfree convergence in a much more simpler way. We will start to study the convergence and clustering of filters in frames in terms of covers and use this to characterise compact frames and some type of uniform frames. We will extend this study to a more general type of filters. We will then discuss convergence and clustering of filters on a locale, where a filter on a locale L is just a filter in the sublattice of all the sublocales of L. This convergence has many applications like characterising compact locales and also characterising sharp points which will also be studied. Finally, the latter concepts of convergence and clustering will be reconciled with the previous one. / Mathematical Sciences