FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2001, VOLUME 7, NUMBER 3, PAGES 651-658

I. I. Bogdanov

Abstract

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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.

\textbf{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.

\textbf{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.

\textbf{Lemma.}
If a general semiring $S$ satisfies the identity $x^n=0$ , then
$S^n$ is a ring.

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Last modified: December 23, 2001