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On the Set of Zero Divisors of a Topological Ring

Published online by Cambridge University Press:  20 November 2018

Kwangil Koh
Kwangil Koh*
Affiliation:
North Carolina State University, Raleigh, North Carolina
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Let R be a topological (Hausdorff) ring such that for each a ∊ R, aR and Ra are closed subsets of R. We will prove that if the set of non - trivial right (left) zero divisors of R is a non-empty set and the set of all right (left) zero divisors of R is a compact subset of R, then R is a compact ring. This theorem has an interesting corollary. Namely, if R is a discrete ring with a finite number of non - trivial left or right zero divisors then R is a finite ring (Refer [1]).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 10 , Issue 4 , October 1967 , pp. 595 - 596
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Ganesan, N., “Properties of Rings with a finite Number of Zero Divisors II”, Math Annalen 161, 241-246 (1966).Google Scholar
2. Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis. Vol. I, Springer-Verlag, Berlin 1963.Google Scholar