Filtered Algebraic Algebras

TL;DR

This paper investigates conditions under which elements of certain graded algebras are nilpotent, providing new results for associated graded algebras of filtered algebraic algebras and extending previous work on algebraic structures.

Contribution

It proves that in associated graded algebras of filtered algebraic algebras, all homogeneous elements are nilpotent, affirming a specific case of Small and Zelmanov's question.

Findings

Homogeneous elements in associated graded algebras are nilpotent

Extends results to filtered algebraic algebras over uncountable fields

Based on Amitsur's theorems on algebras over infinite fields

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.