Engel conditions and symmetric tensors

TL;DR

This paper simplifies the proof of Engel conditions in Lie rings and extends the results to tensor algebra, broadening their applicability in algebraic structures.

Contribution

It provides a more straightforward proof of Engel conditions and generalizes these conditions within tensor algebra frameworks.

Findings

Simplified proof of Engel conditions in Lie rings

Generalization of Engel conditions to tensor algebra

Broader applicability of Engel conditions in algebraic contexts

Abstract

In a recent study of Engel Lie rings, Serena Cicalo` and Willem de Graaf have given a practical set of conditions for an additively finitely generated Lie ring L to satisfy an Engel condition. We present a simpler and more direct proof of this fact. Then we generalize it to a result in the language of tensor algebra, which can be applied to other contexts.