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On a class of near-rings

Published online by Cambridge University Press:  09 April 2009

George Szeto
George Szeto
Affiliation:
Mathematics Department, Bradley UniversityPeoria, Illinois 61606, U.S.A.
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In [3], Ligh proved that every distributively generated Boolean near-ring is a ring, and he gave an example to which the above fact can not be extended. That is, let G be an additive group and let the multiplication on G be defined by xy = y for all x, y in G. Ligh called this Boolean near-ring G a general Boolean near-ring. near-ring. Then in [4], Ligh called Ra β-near-ring if for each x in R, x2 = x and xyz = yxz for all x, y, z in R, and he proved that the structure of a β-near-ring is “very close” to that of a usual Boolean ring. We note that general Boolean near-rings and Boolean semirings as defined in [5] are β-near-rings. The purpose of this paper is to generalize the structure theorem on β-near-rings given by Ligh in [4] to a broader class of near-rings.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 1 , July 1972 , pp. 17 - 19
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Birkhoff, G., ‘Subdirect unions in universal algebra’, Bull. Amer. Math. Soc. 50 (1944), 764–768.CrossRefGoogle Scholar
[2]Blackett, D., ‘Simple and semisimple near-rings’, Proc. Amer. Math. Soc. 4 (1953), 772–785.CrossRefGoogle Scholar
[3]Ligh, S., ‘On Boolean near-rings’, Bull. Austral. Math. Soc. 1 (1969), 375–379.CrossRefGoogle Scholar
[4]Ligh, S., ‘The structure of a special class of near-rings’, J. Austral Math. Soc. (to appear).Google Scholar
[5]Subrahmanyam, N., ‘Boolean semirings’, Math. Ann. 148 (1962), 395–401.CrossRefGoogle Scholar