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. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/37206 |
| Date | January 1966 |
| Creators | Su, Li Pi |
| Publisher | University of British Columbia |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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