Associative algebras satisfying a semigroup identity

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