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Associative algebras satisfying asemigroup identity

Published online by Cambridge University Press:  01 October 1999

David M. Riley  and
Mark C. Wilson
David M. Riley
Affiliation:
Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487-0350, USA. E-mail:driley@gp.as.ua.edu
Mark C. Wilson
Affiliation:
Department of Mathematics, University of Auckland, Private Bag 92019 Auckland, New Zealand. E-mail:wilson@math.auckland.ac.nz
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Abstract

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Denoteby $(R,\cdot)$ the multiplicative semigroup of an associative algebra$R$ over an infinite field, and let $(R,\circ)$ represent $R$ whenviewed as a semigroup via the circle operation $x\circy=x+y+xy$. In thispaper we characterize the existence of an identity in these semigroupsin terms of the Lie structure of $R$. Namely, we prove that thefollowing conditions on $R$ are equivalent: the semigroup $(R,\circ)$satisfies an identity; the semigroup $(R,\cdot)$ satisfies a reducedidentity; and, the associated Lie algebra of $R$ satisfies the Engelcondition. When $R$ is finitely generated these conditions are eachequivalent to $R$ being upper Lienilpotent.

1991 Mathematics Subject Classification 16R40, 20M07, 20M25

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 41 , Issue 3 , October 1999 , pp. 453 - 462
Copyright
1999 Glasgow Mathematical Journal Trust