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Right alternative algebras with commutators in a nucleus

Published online by Cambridge University Press:  17 April 2009

Erwin Kleinfeld  and
Harry F. Smith
Erwin Kleinfeld
Affiliation:
Division of Mathematical Sciences, University of Iowa Iowa City, IA 52242, United States of America
Harry F. Smith
Affiliation:
Department of Mathematics Statistics and Computing, Science University of New England, Armidale NSW 2351
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Abstract

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Let A be a right alternative algebra, and [A, A] be the linear span of all commutators in A. If [A, A] is contained in the left nucleus of A, then left nilpotence implies nilpotence. If [A, A] is contained in the right nucleus, then over a commutative-associative ring with 1/2, right nilpotence implies nilpotence. If [A, A] is contained in the alternative nucleus, then the following structure results hold: (1) If A is prime with characteristic ≠ 2, then A is either alternative or strongly (–1, 1). (2) If A is a finite-dimensional nil algebra, over a field of characteristic ≠ 2, then A is nilpotent. (3) Let the algebra A be finite-dimensional over a field of characteristic ≠ 2, 3. If A/K is separable, where K is the nil radical of A, then A has a Wedderburn decomposition

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 1 , August 1992 , pp. 81 - 90
Copyright
Copyright © Australian Mathematical Society 1992

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