On free subsemigroups of associative algebras

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 k -algebras R R , where k k is a field not algebraic over a finite subfield. We show that R R contains a free noncyclic subsemigroup in the following cases: (1) R R satisfies a polynomial identity and is noncommutative modulo its prime radical. (2) R R has at least one nonartinian primitive subquotient. (3) k k is uncountable and R 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.