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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Uniqueness of generators of principal ideals in rings of continuous functions


Author: M. J. Canfell
Journal: Proc. Amer. Math. Soc. 26 (1970), 571-573
MSC: Primary 13.20; Secondary 46.00
MathSciNet review: 0288109
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Abstract: Let $ aR$ denote the principal right ideal generated in a ring $ R$ by an element $ a$. Kaplansky has raised the question: If $ aR = bR$, are $ a$ and $ b$ necessarily right associates? In this note we show that for rings of continuous functions the answer is affirmative if and only if the underlying topological space is zero-dimensional. This gives an algebraic characterization of the topological concept ``zero-dimensional". By extending the notion of uniqueness of generators of principal ideals we are able to give an algebraic characterization of the concept ``$ n$-dimensional".


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0288109-8
PII: S 0002-9939(1970)0288109-8
Keywords: Principal ideals, uniqueness of generators, dimension of a ring, rings of continuous functions, topological dimension
Article copyright: © Copyright 1970 American Mathematical Society