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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published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k01/k013/k01302h.htm

Last modified: December 23, 2001