Makar-Limanov's conjecture on free subalgebras

TL;DR

The paper demonstrates the existence of specific nil algebras over countable fields that, after field extension, contain noncommutative free subalgebras of arbitrary or specific ranks, addressing a question posed by Makar-Limanov.

Contribution

It constructs examples of nil algebras over countable fields with controlled free subalgebra properties, advancing understanding of Makar-Limanov's conjecture.

Findings

Existence of nil algebras with arbitrarily high rank free subalgebras after extension

Existence of nil algebras without rank two free subalgebras, yet their extensions contain such subalgebras

Answers to Makar-Limanov's question about free subalgebras in extended algebras

Abstract

It is proved that over every countable field K there is a nil algebra R such that the algebra obtained from R by extending the field K contains noncommutative free subalgebras of arbitrarily high rank. It is also shown that over every countable field K there is an algebra R without noncommutative free subalgebras of rank two such that the algebra obtained from R by extending the field K contains a noncommutative free subalgebra of rank two. This answers a question of Makar-Limanov