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The structure of a special class of near-rings

Published online by Cambridge University Press:  09 April 2009

Steve Ligh
Steve Ligh
Affiliation:
Department of Mathematics University of FloridaGainesville, Florida 32601
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It is well known that a Boolean ring is isomorphic to a subdirect sum of two-element fields. In [3] a near-ring (B, +, ·) is said to be Boolean if there exists a Boolean ring (B, +, Λ, 1) with identity such that · is defined in terms of +, Λ, and 1 and, for any b ∈ B, b · b = b. A Boolean near-ring B is called special if a · b = (a ν x) Λ b, where x is a fixed element of B. It was pointed out that a special Boolean near-ring is a ring if and only if x = 0. Furthermore, a special Boolean near-ring does not have a right identity unless x = 0. It is natural to ask then whether any Boolean near-ring (which is not a ring) can have a right identity. Also, how are the subdirect structures of a special Boolean near-ring compared to those of a Boolean ring. It is the purpose of this paper to give a negative answer to the first question and to show that the subdirect structures of a special Boolean near- ring are very ‘close’ to those of a Boolean ring. In fact, we will investigate a class of near-rings that include the special Boolean near-rings and the Boolean semi- rings as defined in [8].

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 13 , Issue 2 , March 1972 , pp. 141 - 146
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. W., ‘Simple and semisimple near-rings’, Proc. Amer. Math. Soc. 4 (1953), 772–785.Google Scholar
[3]Clay, J. R. and Lawyer, D. A., ‘Boolean near-rings’, Canad. Math. Bull 12 (1969), 265–274.CrossRefGoogle Scholar
[4]Fain, C. G., Some structure theorems for near-rings. Doctoral Dissertation, University of Oklahoma, 1968.Google Scholar
[5]Frohlich, A., ‘Distributively generated near-rings. I. Ideal Theory, Proc. London Math. Soc. 8 (1958), 76–94.CrossRefGoogle Scholar
[6]Ligh, S., ‘On Boolean near-rings’, Bull. Australian Math. Soc. 1 (1969), 375–380.CrossRefGoogle Scholar
[7]Ligh, S., ‘On regular near-rings’, (to appear).Google Scholar
[8]Subrahmanyam, N. V., ‘Boolean semirings’, Math. Ann. 148 (1962), 395–401.CrossRefGoogle Scholar