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Almost nilpotent gamma rings

Published online by Cambridge University Press:  17 April 2009

G.L. Booth  and
N.J. Groenewald
G.L. Booth
Affiliation:
Department of MathematicsUniversity of ZululandPrivate Bag X1001Kwadlangezwa 3886South Africa
N.J. Groenewald
Affiliation:
Department of MathematicsUniversity of Port ElizabethP.O. Box 1600Port Elizabeth 6000South Africa
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Abstract

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In this paper we introduce the concept of almost nilpotence for Γ-rings, similar to the corresponding concept for rings, as defined by Van Leeuwen and Heyman. An almost mlpotent radical property Α0 is introduced for Γ-rings, and shown to be supernilpotent. If M is a Γ-ring with left and right operator rings L and R respectively, then Α(L)+ = Α0(M) = Α(R)*, where Α(−) denotes the almost nilpotent radical of a ring. If M is a Γ-ring and m, n are positive integers, then Α0(Mm, n) is the almost nilpotent radical of the Γn, m-ring Mm, n.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 41 , Issue 2 , April 1990 , pp. 237 - 244
Copyright
Copyright © Australian Mathematical Society 1990

References

[1]Booth, G.L., A contribution to the radical theory of gamma rings (Ph.D. thesis, University of Stellenbosch, 1986).Google Scholar
[2]Booth, G.L., ‘Characterizing the nil radical of a gamma ring’, Quasestiones Math 9 (1986), 55–67.CrossRefGoogle Scholar
[3]Booth, G.L., ‘Operstor rings of a gamma ring’, Math. Japon. 31 (1986), 175–184.Google Scholar
[4]Booth, G.L.On antisimple gamma rings, Quaestiones Math. 11 (1988), 7–15.CrossRefGoogle Scholar
[5]Booth, G.L., ‘Supernilpotent radicals of gamma rings’, Acta. Math. Hungar. (to appear).Google Scholar
[6]Coppage, W.E. and Luh, J., ‘Radicals of gamma rings’, J. Math. Soc. Japan 23 (1971), 40–52.CrossRefGoogle Scholar
[7]Divinsky, N., Rings and Radicals (Univ. of Toronto Press, Toronto, 1965).Google Scholar
[8]Heyman, G.A.P., Jenkins, T.L. and Le Roux, H.J., ‘Variations on almost nilpotent rings, their radicals and partitions’, Acta. Math. Hungar. 39 (1982), 11–15.CrossRefGoogle Scholar
[9]Kyuno, S., ‘Prime ideals in gamma rings’, Pacific J. Math. 98 (1982), 375–379.CrossRefGoogle Scholar
[10]Le Roux, H.J., Lower radicals of Γ-rings, (Preprint).Google Scholar
[11]Tzintzis, G., ‘An upper radical property and an answer to a problem of L.C.A. van Leeuwen and G.A.P. Heyman’, Acta. Math. Hungar 39 (1982), 381–385.CrossRefGoogle Scholar
[12]Van Leeuwen, L.C.A. and Heyman, G.A.P., ‘A class of almost nilpotent rings’, Tomus, Acta. Math. Hungar. 26 (1975), 259–262.CrossRefGoogle Scholar