FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2007, VOLUME 13, NUMBER 2, PAGES 3-29

**The Kurosh problem, height theorem, nilpotency of the radical, and
algebraicity identity**

A. Ya. Belov

Abstract

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The paper is devoted to relations between the Kurosh problem and the
Shirshov height theorem.
The central point and main technical tool is the identity of
algebraicity.
The main result of this paper is the following.
Let $A$ be
a finitely generated PI-algebra and $Y$ be a finite subset
of $A$.
For any Noetherian associative and commutative ring $R$É
**F**, let any factor of $R$Ä
A such that all projections of elements
from $Y$ are
algebraic over $$p(R) be
a Noetherian $R$-module.
Then $A$ has
bounded essential height over $Y$.
If, furthermore, $Y$ generates $A$ as an algebra, then
$A$ has bounded
height over $Y$
in the Shirshov sense.

The paper also contains a new proof of the
Razmyslov--Kemer--Braun theorem on radical nilpotence of affine
PI-algebras.
This proof allows one to obtain some constructive estimates.

The main goal of the paper is to develope a "virtual operator
calculus." Virtual operators (pasting, deleting and transfer) depend
not only on an element of the algebra but also on its representation.

Location: http://mech.math.msu.su/~fpm/eng/k07/k072/k07201h.htm

Last modified: May 23, 2007