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On the radical theory of Andrunakievich varieties

Published online by Cambridge University Press:  17 April 2009

P.N. Ánh ,
N.V. Loi  and
R. Wiegandt
P.N. Ánh
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest, PO Box 127, H-1364, Hungary.
N.V. Loi
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest, PO Box 127, H-1364, Hungary.
R. Wiegandt
Affiliation:
Mathematical Institute of the Hungarian Academy of Sciences, Budapest, PO Box 127, H-1364, Hungary.
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Abstract

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In 1978 Anderson and Gardner investigated semisimple classes and recently Buys and Gerber developed the theory of special radicals in Andrunakievich varieties. In this note we continue the study of radical theory in Andrunakievich varieties. Sharpening some of the results of Anderson and Gardner we prove versions of Sands' Theorem characterizing semisimple classes by regularity, coinductivity and being closed under extensions. In the proof we follow a new method which avoids calculations with defining identities of the variety. We generalize van Leeuwen's Theorem characterizing semisimple classes of hereditary radicals as classes being regular and closed under essential extensions and subdirect sums.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 31 , Issue 2 , April 1985 , pp. 257 - 269
Copyright
Copyright © Australian Mathematical Society 1985

References

[1]Anderson, T. and Gardner, B.J., “Semi-simple classes in a variety satisfying an Andrunakievich lemma”, Bull. Austral. Math. Soc. 18 (1978), 187–200.CrossRefGoogle Scholar
[2]Anderson, T. and Wiegandt, R., “Semisimple classes of alternative rings”, Proc. Edinburgh Math. Soc. 25 (1982), 21–26.CrossRefGoogle Scholar
[3]Ánh, P.N., “On semisimple classes of topological rings”, Ann. Univ. Sci. Budapest 20 (1977), 59–70.Google Scholar
[4]Ánh, P.N. and Wiegandt, R., “Semisimple classes of Jordan algebras”, preprint.Google Scholar
[5]Buys, A. and Gerber, G.K., “Special classes in Ω-groups”, Ann. Univ. Sci. Budapest (to appear).Google Scholar
[6]Gardner, B.J. and Wiegandt, R., “Characterizing and constructing special radicals”, Acta Math. Acad. Sci. Hungar. 40 (1982), 73–83.CrossRefGoogle Scholar
[7]van Leeuwen, L.C.A., “Properties of semisimple classes”, J. Natur. Sci. Math. 15 (1975), 59–67.Google Scholar
[8]van Leeuwen, L.C.A., Roos, C. and Wiegandt, R., “Characterizations of semisimple classes”, J. Austral. Math. Soc. Ser. A 23 (1977), 172–182.CrossRefGoogle Scholar
[9]Puczytowski, E.R., “On semisimple classes of associative and alternative rings”, Proc. Edinburgh Math. Soc. 27 (1984), 1–5.CrossRefGoogle Scholar
[10]Rjabuhin, Ju.M. and Wiegandt, R., “On special radicals, supernilpotent radicals and weakly homomorphically closed classes”, J. Austral. Math. Soc. Ser. A 31 (1981), 152–162.CrossRefGoogle Scholar
[11]Sands, A.D., “Strong upper radicals”, Quart. J. Math. Oxford Ser. (2) 27 (1976), 21–24.CrossRefGoogle Scholar
[12]Sands, A.D., “A characterization of semisimple classes”, Proc. Edinburgh Math. Soc. 24 (1981), 5–7.CrossRefGoogle Scholar
[13]Terlikowska-Oslowska, B., “Category with self-dual set of axioms”, Bull. Acad. Polon. Sci. 25 (1977), 1207–1214.Google Scholar
[14]Terlikowska-Oslowska, B., “Radical and semisimple classes of objects in categories with a self-dual set of axioms”,Bull. Acad. Polon. Sci. 26 (1978), 7–13.Google Scholar
[15]Wiegandt, R., Radical and semisimple classes of rings (Queen's Papers in Pure and Applied Mathematics, 37. Queen's University, Kingston, Ontario, 1974).Google Scholar
[16]Zhevlakov, K.A., Slin'ko, A.M., Shestakov, I.P. and Shirshov, A.I., Rings that are nearly associative (Acadesic Press, New York, London, 1982).Google Scholar