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

What is it about?

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 used for obtaining, for example, a direct combinatorial proof of the theorem for the radical nilpotency and the explicit estimate of its nilpotency index. Let $A$ be a finitely generated PI-algebra and $Y$ be a finite subset of $A$. For any Noetherian associative and commutative ring $R \supset \bold{F}$, 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 a bounded essential height over $Y$. If, furthermore, $Y$ generates $A$ as an algebra, then $A$ has a 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 develop a technique connected with the identity of algebraicity and a special "operational calculus'' for operators with symbolic expressions in PI-algebras (operators of transfer, pasting and deleting). Reviewed by Tsetska Grigorova Rashkova

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