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A Commutativity Theorem for Near-Rings

Published online by Cambridge University Press:  20 November 2018

Howard E. Bell
Howard E. Bell*
Affiliation:
Department of MathematicsBrock UniversitySt. Catharines, Ontario
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A ring or near-ring R is called periodic if for each xϵR, there exist distinct positive integers n, m for which xn = xm. A well-known theorem of Herstein states that a periodic ring is commutative if its nilpotent elements are central [5], and Ligh [6] has asked whether a similar result holds for distributively-generated (d-g) near-rings. It is the purpose of this note to provide an affirmative answer.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 20 , Issue 1 , 01 March 1977 , pp. 25 - 28
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Bell, H. E., Near-rings in which each element is a power of itself, Bull. Australian Math. Soc. 2 (1970), 363-368.Google Scholar
2. Bell, H. E., Certain near-rings are rings, J. London Math. Soc. (2) 4 (1971), 264-270.Google Scholar
3. Chacron, M., On a theorem of Herstein, Canadian J. Math. 21 (1969), 1348-1353.Google Scholar
4. Fröhlich, A., Distributively-generated near-rings, I, Ideal theory, Proc. London Math. Soc. (3) 8, (1958), 76-94.Google Scholar
5. Herstein, I. N., A note on rings with central nilpotent elements, Proc. Amer. Math. Soc. 5 (1954), 620.Google Scholar
6. Ligh, S., Some commutativity theorems for near-rings, Kyungpook Math. J. 13 (1973), 165-170.Google Scholar