Denote by $(R,\cdot)$ the multiplicative semigroup of an associative algebra $R$ over an infinite field, and let $(R,\circ)$ represent $R$ when viewed as a semigroup via the circle operation $x\circy=x+y+xy$. 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 $(R,\circ)$ satisfies an identity; the semigroup $(R,\cdot)$ 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.

1991 Mathematics Subject Classification 16R40, 20M07, 20M25