Geodesic
A Finiteness Criterion and Asymptotics for Codimensions of Generalized Identities
Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 681-685.

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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. 
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A. S. Gordienko. A Finiteness Criterion and Asymptotics for Codimensions of Generalized Identities. Matematičeskie zametki, Tome 86 (2009) no. 5, pp. 681-685. http://geodesic.mathdoc.fr/item/MZM_2009_86_5_a4/

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