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 Mat. Zametki, 2009, Volume 86, Issue 5, Pages 681–685 (Mi mz6625)

A Finiteness Criterion and Asymptotics for Codimensions of Generalized Identities

A. S. Gordienko

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: Let $A$ be an associative algebra over a field of characteristic zero. Then either all codimensions $\operatorname{gc}_n(A)$ of its generalized polynomial identities are infinite or $A$ is the sum of ideals $I$ and $J$ such that $\dim_FI<\infty$ and $J$ is nilpotent. In the latter case, there exist numbers $n_0\in\mathbb N$, $C\in\mathbb Q_+$, and $t\in\mathbb Z_+$ for which $\operatorname{gc}_n(A)<+\infty$ if $n\ge n_0$ and $\operatorname{gc}_n(A)\sim Cn^td^n$ as $n\to\infty$, where $d=\mathrm{PI}\exp(A)\in\mathbb Z_+$. Thus, in the latter case, conjectures of Amitsur and Regev on generalized codimensions hold.

Keywords: generalized polynomial identity, generalized polylineal polynomial, PI-algebra, PI-exponent, associative algebra, nilpotent ideal, division ring, semi-simple algebra

DOI: https://doi.org/10.4213/mzm6625

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English version:
Mathematical Notes, 2009, 86:5, 645–649

Bibliographic databases:

UDC: 512.552.4

Citation: A. S. Gordienko, “A Finiteness Criterion and Asymptotics for Codimensions of Generalized Identities”, Mat. Zametki, 86:5 (2009), 681–685; Math. Notes, 86:5 (2009), 645–649

Citation in format AMSBIB
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