Filtered algebraic algebras

Regev, Alon

doi:10.1090/S0002-9939-10-10227-5

Filtered algebraic algebras
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Proc. Amer. Math. Soc. 138 (2010), 1941-1947 Request permission

Abstract:

Small and Zelmanov posed the question whether every element of a graded algebra over an uncountable field must be nilpotent, provided that the homogeneous elements are nilpotent. This question has recently been answered in the negative by Smoktunowicz. In this paper we prove that the answer is affirmative for associated graded algebras of filtered algebraic algebras. Our result is based on Amitsur’s theorems on algebras over infinite fields.

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