The Nagata–Higman theorem for hemirings

In this paper the hemirings (in general, with noncommutative addition) with the identity $x^n=0$ are studied. The main results are the following ones.
Theorem. If a $n!$-torsionfree general hemiring satisfies the identity $x^n=0$, then it is nilpotent. The estimates of the nilpotency index are equal for $n!$-torsionless rings and general hemirings.
Theorem. The estimates of the nilpotency index of $l$-generated rings and general hemirings with identity $x^n=0$ are equal.
The proof is based on the following lemma.
Lemma. If a general semiring $S$ satisfies the identity $x^n=0$, then $S^n$ is a ring.