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A note on the Levitzki radical of a near-ring

Published online by Cambridge University Press:  09 April 2009

N. J. Groenewald  and
P. C. Potgieter
N. J. Groenewald
Affiliation:
Department of Mathematics, University of Port Elizabeth6000 Port Elizabeth, South Africa
P. C. Potgieter
Affiliation:
Department of Mathematics, University of Port Elizabeth6000 Port Elizabeth, South Africa
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Abstract

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It is known that in a near-ring N the Levitzki radical L(N), that is, the sum of all locally nilpotent ideals, is the intersection of all the prime ideals P in N such that N/P has zero Levitzki radical. The purpose of this note is to prove that L(N) is the intersection of a certain class of prime ideals, called l-prime ideals. Every l-prime ideal P is such that N/P has zero Levitzki radical. We also introduce an l-semi-prime ideal and show that P is an l-semi-prime ideal if and only if N/P has zero Levitzki radical. We get another characterization of the Levitzki radical of the near-ring as the intersection of all the l-semi-prime ideals.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 3 , June 1984 , pp. 416 - 420
Copyright
Copyright © Australian Mathematical Society 1984

References

[1]Bhandari, M. C. and Saxena, P. K., ‘A note on Levitzki radical of a near-ring,’ Kyungpook Math. J. 20 (1980), 183–188.Google Scholar
[2]Roux, H. J. Le, Contribution to the theory of radicals in associative rings (Doctoral Thesis, University of the Orange Free State, R.S.A., 1977 (in Afrikaans)).Google Scholar
[3]Pilz, G., Near-rings (North-Holland, Amsterdam and New York, 1977).Google Scholar
[4]Van der Walt, A. P. J., ‘On the Levitzki nil radical,’ Arch. Math. 16 (1965), 22–24.Google Scholar
[5]Van der Walt, A. P. J., ‘Prime ideals and nil radicals in near-rings,’ Arch. Math. 15 (1964), 408–414.CrossRefGoogle Scholar