Associative rings satisfying the Engel condition

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