On the nilpotency of subrings of skew group rings

The main aim of the present paper is to prove the following theorem.
Theorem. Let $A$ be either a left Goldie ring or a ring satisfying the ascending chain conditions both for left and for right annihilators, $G$ be a free commutative group and $\sigma\colon\,G\to\operatorname{Aut}(A)$ be a group homomorphism. Then any homogeneous nilsubsemigroup of the multiplicative semigroup of the skew group ring $A_{\sigma}[G]$ is nilpotent.
This theorem can be considered as a skew analogue of a well-known classical result in the ring theory, Shock–Fisher theorem.