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Associative rings satisfying the Engel condition

Authors: D. M. Riley and Mark C. Wilson
Journal: Proc. Amer. Math. Soc. 127 (1999), 973-976
MSC (1991): Primary 16R40; Secondary 16W10, 17B60, 16U60
MathSciNet review: 1473677
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Abstract: Let $C$ be a commutative ring, and let $R$ be an associative $C$-algebra generated by elements $\{x_1,\ldots,x_d\}$. We show that if $R$ satisfies the Engel condition of degree $n$, then $R$ is upper Lie nilpotent of class bounded by a function that depends only on $d$ and $n$. We deduce that the Engel condition in an arbitrary associative ring is inherited by its group of units, and implies a semigroup identity.

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Additional Information

D. M. Riley
Affiliation: Department of Mathematics, University of Alabama, Tuscaloosa, Alabama 35487-0350

Mark C. Wilson
Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019 Auckland, New Zealand
Address at time of publication: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607-7045

Keywords: Engel identity, Lie nilpotent, strongly Lie nilpotent, upper Lie nilpotent, nonmatrix
Received by editor(s): March 20, 1997
Received by editor(s) in revised form: April 15, 1997, and July 29, 1997
Additional Notes: The first author received support from NSF-EPSCoR in Alabama and the University of Alabama Research Advisory Committee.
The second author was supported by a NZST Postdoctoral Fellowship.
Communicated by: Ken Goodearl
Article copyright: © Copyright 1999 American Mathematical Society