On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ring $R$ we can assign the adjoint Lie ring $R^{(-)}$ (with the operation $(a,b)=ab-ba$) and two semigroups, the multiplicative semigroup $M(R)$ and the associated semigroup $A(R)$ (with the operation $a\circ b=ab+a+b$). It is clear that a Lie ring $R^{(-)}$ is commutative if and only if the semigroup $M(R)$ (or $A(R)$) is commutative. In the present paper we try to generalize this observation to the case in which $R^{(-)}$ is a nilpotent Lie ring. It is proved that if $R$ is an associative algebra with identity element over an infinite field $F$, then the algebra $R^{(-)}$ is nilpotent of length $c$ if and only if the semigroup $M(R)$ (or $A(R)$) is nilpotent of length $c$ (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in which $R$ is an algebra without identity element over $F$, this assertion remains valid for $A(R)$, but fails for $M(R)$. Another similar results are obtained.