Filtered Algebraic Algebras

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 A. 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 algebas over infinite fields.