  Denote by (R,.) the multiplicative semigroup of an associative algebra R over
an infinite field, and let (R,*) represent R when viewed as a semigroup via the
circle operation x*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,*)
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.
