FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

The Nagata--Higman theorem for hemirings

I. I. Bogdanov

Abstract

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In this paper the hemirings (in general, with noncommutative addition) with the identity $xn=0$ are studied. The main results are the following ones.

Theorem. If a $n!$-torsionfree general hemiring satisfies the identity $xn=0$, 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 $xn=0$ are equal.

The proof is based on the following lemma.

Lemma. If a general semiring $S$ satisfies the identity $xn=0$, then $Sn$ is a ring.

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Last modified: December 23, 2001