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 Fundam. Prikl. Mat., 2007, Volume 13, Issue 2, Pages 3–29 (Mi fpm16)

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

A. Ya. Belovab

a Moscow Institute of Open Education
b Hebrew University of Jerusalem

Abstract: 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 $R\supset\mathbb F$, let any factor of $R\otimes A$ such that all projections of elements from $Y$ are algebraic over $\pi(R)$ 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.

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

Bibliographic databases:

UDC: 512.552.4+512.554.32+512.664.2

Citation: A. Ya. Belov, “The Kurosh problem, height theorem, nilpotency of the radical, and algebraicity identity”, Fundam. Prikl. Mat., 13:2 (2007), 3–29; J. Math. Sci., 154:2 (2008), 125–142

Citation in format AMSBIB
\Bibitem{Bel07} \by A.~Ya.~Belov \paper The Kurosh problem, height theorem, nilpotency of the radical, and algebraicity identity \jour Fundam. Prikl. Mat. \yr 2007 \vol 13 \issue 2 \pages 3--29 \mathnet{http://mi.mathnet.ru/fpm16} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2322971} \zmath{https://zbmath.org/?q=an:1175.16015} \elib{http://elibrary.ru/item.asp?id=11162639} \transl \jour J. Math. Sci. \yr 2008 \vol 154 \issue 2 \pages 125--142 \crossref{https://doi.org/10.1007/s10958-008-9156-6} \elib{http://elibrary.ru/item.asp?id=13572350} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-54249097337} 

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This publication is cited in the following articles:
1. I. M. Isaev, A. V. Kislitsin, “Identities in vector spaces and examples of finite-dimensional linear algebras having no finite basis of identities”, Algebra and Logic, 52:4 (2013), 290–307
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