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Near-rings in which each element is a power of itself

Published online by Cambridge University Press:  17 April 2009

Howard E. Bell
Howard E. Bell
Affiliation:
Brock University, St Catharines, Ontario, Canada.
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Abstract

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Let R denote a near-ring such that for each x ∈ R, there exists an integer n(x) > 1 for which xn(x) = x. We show that the additive group of R is commutative if 0.x; = 0 for all x ∈ R and every non-trivial homomorphic image R¯ of R contains a non-zero idempotent e commuting multiplicatively with all elements of R¯. As the major consequence, we obtain the result that if R is distributively-generated, then R is a ring – a generalization of a recent theorem of Ligh on boolean near-rings.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 2 , Issue 3 , June 1970 , pp. 363 - 368
Copyright
Copyright © Australian Mathematical Society 1970

References

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