TY - JOUR

T1 - The Kurosh problem, the height theorem, the nilpotency of the radical, and the algebraicity identity

AU - Belov, A.

N1 - English version:
Journal of Mathematical Sciences (New York), 2008, 154:2, 125–142

PY - 2008

Y1 - 2008

N2 - 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 AA be a finitely generated PI-algebra and YY be a finite subset of AA. For any Noetherian associative and commutative ring R⊃FR⊃F, let any factor of R⊗AR⊗A such that all projections of elements from YY are algebraic over π(R)π(R) be a Noetherian RR-module. Then AA has bounded essential height over YY. If, furthermore, YY generates AA as an algebra, then AA has bounded height over YY 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.

AB - 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 AA be a finitely generated PI-algebra and YY be a finite subset of AA. For any Noetherian associative and commutative ring R⊃FR⊃F, let any factor of R⊗AR⊗A such that all projections of elements from YY are algebraic over π(R)π(R) be a Noetherian RR-module. Then AA has bounded essential height over YY. If, furthermore, YY generates AA as an algebra, then AA has bounded height over YY 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.

UR - http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=fpm&paperid=16&option_lang=eng

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VL - 13

SP - 3

EP - 29

JO - Fundamentalnaya i Prikladnaya Matematika (Moscow)

JF - Fundamentalnaya i Prikladnaya Matematika (Moscow)

IS - 2

ER -