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Algebraic properties of certain rings of continuous functions Su, Li Pi
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 zfilters. Let [fomula omitted] be the ring of all mtimes differentiable functions on a [formula omitted] differentiable nmanifold X , [formula omitted] the ring of all Lcfunctions 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. Lchomeomorphic) 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. Lchomeomorphism) 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.
Item Metadata
Title 
Algebraic properties of certain rings of continuous functions

Creator  
Publisher 
University of British Columbia

Date Issued 
1966

Description 
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 zfilters. Let [fomula omitted] be the ring of all mtimes differentiable functions on a [formula omitted] differentiable nmanifold X , [formula omitted] the ring of all Lcfunctions 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. Lchomeomorphic) 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. Lchomeomorphism) 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.

Genre  
Type  
Language 
eng

Date Available 
20110909

Provider 
Vancouver : University of British Columbia Library

Rights 
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.

DOI 
10.14288/1.0080603

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Campus  
Scholarly Level 
Graduate

Aggregated Source Repository 
DSpace

Item Media
Item Citations and Data
Rights
For noncommercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.