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Certain conditions under which near-rings are rings

Published online by Cambridge University Press:  17 April 2009

Murtaza A. Quadri
Affiliation:
Department of Mathematics Aligarh, Muslim University Aligarh, 202002, India
M. Ashraf
Affiliation:
Department of Mathematics Aligarh, Muslim University Aligarh, 202002, India
Asma Ali
Affiliation:
Department of Mathematics Aligarh, Muslim University Aligarh, 202002, India
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Abstract

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In 1969, Ligh proved that distributively generated (d-g) Boolean near-rings are rings, and hinted that some of the more complicated polynomial identities implying commutativity in rings may turn d-g near-rings into rings. In the present paper we investigate the following conditions: (1) xy = (xy)n(x, y); (2) xy = (yz)n (xy); (3) xy = ym (x, y)xn (x, y); (4) xy = xy n(x, y)x; (5) xy = xn(x, y)ym (x, y); finally prove that under appropriate additional hypotheses a d-g near-ring must be a commutative ring. Indeed the theorem proved here is a wide generalisation of many recently established results.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Ali, Asma, Ashraf, M. and Quadri, M.A., ‘Some elementary commutativity conditions for near-rings’, Math. Student 56 (1988), 181183.Google Scholar
[2]Bell, H.E., ‘A commutativity condition for rings’, Canad. J. Math. 28 (1976), 986991.CrossRefGoogle Scholar
[3]Bell, H.E., ‘A commutativity study for periodic rings’, Pacific J. Math. 70 (1977), 2936.CrossRefGoogle Scholar
[4]Bell, H.E., ‘A commutativity theorem for near-rings’, Canad. Math. Bull 20 (1977), 2528.CrossRefGoogle Scholar
[5]Bell, H.E., ‘Certain near-rings are rings II’, Internat. J. Math. Math. Sci. 9 (1986), 267272.CrossRefGoogle Scholar
[6]Frohlich, A., ‘Distributively generated near-rings’, (I. Ideal theory), Proc. London Math. Soc. 8 (1958), 7694.CrossRefGoogle Scholar
[7]Herstein, I.N., ‘A note on rings with central nilpotent elements’, Proc. Amer. Math. Soc. 16 (1969), 239243.Google Scholar
[8]Ligh, S., ‘On Boolean near-rings’, Bull. Austral. Math. Soc. 1 (1969), 375379.CrossRefGoogle Scholar
[9]Ligh, S., ‘Some commutativity theorems for near-rings’, Kyungpook Math. J. 13 (1973), 165170.Google Scholar
[10]Searcoid, M.O. and MacHale, D., ‘Two elementary generalizations for Boolean rings’, Amer. Math. Monthly 93 (1986), 121122.CrossRefGoogle Scholar