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A construction of supernilpotent radical classes

Published online by Cambridge University Press:  09 April 2009

Dwight M. Olson
Dwight M. Olson
Affiliation:
Department of Mathematics Cameron UniversityLawton, Oklahoma 73505, U.S.A.
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Abstract

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In a recent paper van Leeuwen and Heyman constructed a supernilpotent radical class using the class of almost nilpotent rings. Using a similar construction, for any class C satisfying the following four properties we obtain a superrnlpotent radical class containing C.

(N1) C contains the class Z of all zero rings.

(N2) C is hereditary.

(N3) C is homomorphically closed.

(N4) If A and A/I are elements of C for some ideal I of a ring A, then A ∈ C.

Every supernilpotent radical class P clearly satisfies these conditions. For any such radical class we define the class of almost radical rings and use these to construct a new radical class P2 which contains the given one. Also, we give a characterization for dual supernilpotent radicals.

Keywords

almost nilpotentsupernilpotentweakly specialdual radicals.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 33 , Issue 1 , August 1982 , pp. 86 - 91
Copyright
Copyright © Australian Mathematical Society 1982

References

[1]Andrunakievič, V. A., ‘Radicals in associative rings I’, Mat. Sb. 44 (1958), 179–212. (Russian)Google Scholar
[2]Divinsky, N., Rings and radicals (University of Toronto Press, 1965).Google Scholar
[3]Hoffman, A. E. and Leavitt, W. G., ‘Properties inherited by the lower radical’, Portugal. Math. 27 (1968), 63–68.Google Scholar
[4]Leavitt, W. G., ‘The intersection property of an upper radical’, Arch. Math. (Basel) 24 (1973), 486–492.CrossRefGoogle Scholar
[5]Rjabuhin, Ju. M., ‘On hypernilpotent and special radicals’, Studies in Algebra and Math. Analysis, 65–72 (Izdat., “Karta Modlovenjaska”, Kishinev, 1965).Google Scholar
[6]Sulinski, A., Anderson, T. and Divinsky, N., ‘Lower radical properties for associative and alternative rings’, J. London Math. Soc. 41 (1966), 417–424.CrossRefGoogle Scholar
[7]Van Leeuwen, L. C. A. and Heyman, G. A. P., ‘A class of almost nilpotent rings’, Acta. Math. Sci. Hungar. 26 (1975), 259–262.CrossRefGoogle Scholar