Associative rings satisfying the Engel condition

D.M. Riley and Mark C. Wilson, Proceedings of the American Mathematical Society 127 (1999), 973-976.

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Abstract: Let $latex C$ be a commutative ring, and let $latex R$ be an associative $latex C$-algebra generated by elements $latex {x_1,ldots,x_d}$. We show that if $latex R$ satisfies the Engel condition of degree $latex n$ then $latex R$ is nilpotent as a Lie algebra of class bounded by a function that depends only on $latex d$ and $latex n$. We deduce that the Engel condition in an arbitrary associative ring is inherited by its group of units, and implies a semigroup identity.