Associative algebras satisfying a semigroup identity

D. M. Riley and Mark C. Wilson, Glasgow Mathematical Journal 41 (1999), 453-462.

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Abstract: Denote by $latex (R,cdot)$ the multiplicative semigroup of an associative algebra $latex R$ over an infinite field, and let $latex (R,circ)$ represent $latex R$ when viewed as a semigroup via the circle operation $latex xcirc y=x+y+xy$. In this paper we characterize the existence of an identity in these semigroups in terms of the Lie structure of $latex R$. Namely, we prove that the following conditions on $latex R$ are equivalent: the semigroup $latex (R,circ)$ satisfies an identity; the semigroup $latex (R,cdot)$ satisfies a reduced identity; and, the associated Lie algebra of $latex R$ satisfies the Engel condition. When $latex R$ is finitely generated these conditions are each equivalent to $latex R$ being upper Lie nilpotent.