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 | References | Similar Articles | Additional Information
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".
- J. R. Gard and R. D. Johnson, Four dimension equivalences, Canadian J. Math. 20 (1968), 48–50. MR 222863, DOI https://doi.org/10.4153/CJM-1968-006-7
- Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
- Irving Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc. 66 (1949), 464–491. MR 31470, DOI https://doi.org/10.1090/S0002-9947-1949-0031470-3
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Keywords:
Principal ideals,
uniqueness of generators,
dimension of a ring,
rings of continuous functions,
topological dimension
Article copyright:
© Copyright 1970
American Mathematical Society