  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.
