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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
DOI: https://doi.org/10.1090/S0002-9939-1970-0288109-8
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".


References [Enhancements On Off] (What's this?)

  • [1] J. R. Gard and R. D. Johnson, Four dimension equivalences, Canad. J. Math. 20 (1968), 48-50. MR 36 #5913. MR 0222863 (36:5913)
  • [2] L. Gillman and M. Jerison, Rings of continuous functions, The University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #6994. MR 0116199 (22:6994)
  • [3] I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464-491. MR 11, 155. MR 0031470 (11:155b)

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Additional Information

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

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