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Semi-simple classes in a variety satisfying an Andrunakievich Lemma

Published online by Cambridge University Press:  17 April 2009

Tim Anderson  and
B.J. Gardner
Tim Anderson
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, Canada, Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary;
B.J. Gardner
Affiliation:
Department of Mathematics, University of Tasmania, Hobart, Tasmania, Department of Mathematics, Dalhousie University, Halifax, Canada.
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Abstract

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It is shown that in a variety of (not necessarily associative) algebras which satisfies a variant of Andrunakievich's Lemma, a class C containing no solvable algebras is the semi-simple class corresponding to some supernilpotent radical class if and only ifCis hereditary and is closed under extensions and sub-direct products. Semi-simple classes in general are not characterized by these properties. If the variety satisfies the further condition that some proper power of every ideal is an ideal, then analogous results hold for the semi-simple classes corresponding to radical classes containing no solvable algebras. In particular, for algebras over a field in the latter situation, all semi-simple classes are characterized by the three closure properties mentioned.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 18 , Issue 2 , April 1978 , pp. 187 - 200
Copyright
Copyright © Australian Mathematical Society 1978

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