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Algebraic properties of certain rings of continuous functions

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Title: Algebraic properties of certain rings of continuous functions
Author: Su, Li Pi
Degree: Doctor of Philosophy - PhD
Program: Mathematics
Copyright Date: 1966
Subject Keywords Functions, Continuous;Rings (Algebra)
Issue Date: 2011-09-09
Publisher University of British Columbia
Series/Report no. UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/]
Abstract: We study the relations between algebraic properties of certain rings of functions and topological properties of the spaces on which the functions are defined. We begin by considering the relation between ideals of rings of functions and z-filters. Let [fomula omitted] be the ring of all m-times differentiable functions on a [formula omitted] differentiable n-manifold X , [formula omitted] the ring of all Lc-functions on a metric space X , and [formula omitted] the ring of all analytic functions on a subset X of the complex plane. It is proved that two m-(resp. Lc-) realcompact spaces X and Y are [formula omitted] diffeomorphic (resp. Lc-homeomorphic) iff [formula omitted] are ring isomorphic. Again if X and Y are m-(resp. Lc-) realcompact spaces, then X can be [formula omitted] (resp.Lc-) embedded as an open [resp. closed] subset in Y iff [formula omitted] homomorphic image of [formula omitted]. The subrings of [formula omitted] which determine the [formula omitted] diffeomorphism (resp. Lc-homeomorphism) of the spaces are studied. We also establish a representation for a transformation, more general than homomorphism, from a ring of [formula omitted] differentiable functions to another ring of [formula omitted] differentiable functions. Finally, we show that, for arbitrary subsets X and Y of the complex plane, if there is a ring isomorphism from [formula omitted] which is the identity on the constant functions, then X and Y are conformally equivalent.
Affiliation: Science, Faculty of
URI: http://hdl.handle.net/2429/37206
Scholarly Level: Graduate

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