# On free subsemigroups of associative algebras

## TL;DR

This paper investigates conditions under which associative algebras contain free subsemigroups, confirming Klein's conjecture in several classes of noncommutative domains over countable fields.

## Contribution

It identifies specific algebraic conditions ensuring the existence of free subsemigroups, extending Klein's conjecture to new classes of associative algebras.

## Key findings

- Free subsemigroups exist in algebras satisfying polynomial identities and noncommutative modulo prime radical.
- Presence of nonartinian primitive subquotients guarantees free subsemigroups.
- Uncountable base fields and noncommutativity modulo Jacobson radical also imply free subsemigroups.

## Abstract

In 1992, following earlier conjectures of Lichtman and Makar-Limanov, Klein conjectured that a noncommutative domain must contain a free, multiplicative, noncyclic subsemigroup. He verified the conjecture when the center is uncountable. In this note we consider the existence (or not) of free subsemigroups in associative $k$-algebras $R$, where $k$ is a field not algebraic over a finite subfield. We show that $R$ contains a free noncyclic subsemigroup in the following cases: (1) $R$ satisfies a polynomial identity and is noncommutative modulo its prime radical. (2) $R$ has at least one nonartinian primitive subquotient. (3) $k$ is uncountable and $R$ is noncommutative modulo its Jacobson radical. In particular, (1) and (2) verify Klein's conjecture for numerous well known classes of domains, over countable fields, not covered in the prior literature.