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Variants of P-frames and associated rings

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dc.contributor.advisor Dube, T. A.
dc.contributor.author Nsayi, Jissy Nsonde
dc.date.accessioned 2016-11-18T05:44:59Z
dc.date.available 2016-11-18T05:44:59Z
dc.date.issued 2015-12
dc.identifier.citation Nsayi, Jissy Nsonde (2015) Variants of P-frames and associated rings, University of South Africa, Pretoria, <http://hdl.handle.net/10500/21795> en
dc.identifier.uri http://hdl.handle.net/10500/21795
dc.description.abstract We study variants of P-frames and associated rings, which can be viewed as natural generalizations of the classical variants of P-spaces and associated rings. To be more precise, we de ne quasi m-rings to be those rings in which every prime d-ideal is either maximal or minimal. For a completely regular frame L, if the ring RL of real-valued continuous functions of L is a quasi m-ring, we say L is a quasi cozero complemented frame. These frames are less restricted than the cozero complemented frames. Using these frames we study some properties of what are called quasi m-spaces, and observe that the property of being a quasi m-space is inherited by cozero subspaces, dense z- embedded subspaces, and regular-closed subspaces among normal quasi m-space. M. Henriksen, J. Mart nez and R. G. Woods have de ned a Tychono space X to be a quasi P-space in case every prime z-ideal of C(X) is either minimal or maximal. We call a point I of L a quasi P-point if every prime z-ideal of RL contained in the maximal ideal associated with I is either maximal or minimal. If all points of L are quasi P-points, we say L is a quasi P-frame. This is a conservative de nition in the sense that X is a quasi P-space if and only if the frame OX is a quasi P-frame. We characterize these frames in terms of cozero elements, and, among cozero complemented frames, give a su cient condition for a frame to be a quasi P-frame. A Tychono space X is called a weak almost P-space if for every two zero-sets E and F of X with IntE IntF, there is a nowhere dense zero-set H of X such that E F [H. We present the pointfree version of weakly almost P-spaces. We de ne weakly regular rings by a condition characterizing the rings C(X) for weak almost P-spaces X. We show that a reduced f-ring is weakly regular if and only if every prime z-ideal in it which contains only zero-divisors is a d-ideal. We characterize the frames L for which the ring RL of real-valued continuous functions on L is weakly regular. We introduce the notions of boundary frames and boundary rings, and use them to give another ring-theoretic characterization of boundary spaces. We show that X is a boundary space if and only if C(X) is a boundary ring. A Tychono space whose Stone- Cech compacti cation is a nite union of closed subspaces each of which is an F-space is said to be nitely an F-space. Among normal spaces, S. Larson gave a characterization of these spaces in terms of properties of function rings C(X). By extending this notion to frames, we show that the normality restriction can actually be dropped, even in spaces, and thus we sharpen Larson's result. en
dc.language.iso en en
dc.subject P-frame en
dc.subject Quasi P-frame en
dc.subject Quasi cozero complemented frame en
dc.subject Quasi m-space en
dc.subject Weak almost P-frame en
dc.subject Weakly regular ring en
dc.subject Boundary frame en
dc.subject Boundary ring en
dc.subject Finitely an F-frame en
dc.subject.ddc 514.325
dc.subject.lcsh Quasi-metric spaces
dc.subject.lcsh Quasiparticles (Physics)
dc.subject.lcsh Semirings (Mathematics)
dc.subject.lcsh Rings (Algebra)
dc.title Variants of P-frames and associated rings en
dc.type Thesis en
dc.description.department Mathematics
dc.description.degree D. Phil. (Mathematics)


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