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