@article{cd7a7fcbbdbc4155b4e54e0ab61e6bea, title = "Makar-Limanov's conjecture on free subalgebras", 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 [Lenny Makar-Limanov, private communication, Beijing, June 2007].", keywords = "Free subalgebras, Extensions of algebras, Nil rings, DIVISION RINGS, FREE SUBGROUPS, FREE SUBSEMIGROUPS, POLYNOMIAL-RINGS, ALGEBRAS, FRACTIONS", author = "Agata Smoktunowicz", year = "2009", month = "12", day = "20", doi = "10.1016/j.aim.2009.07.010", language = "English", volume = "222", pages = "2107--2116", journal = "Advances in Mathematics", issn = "0001-8708", publisher = "Academic Press Inc.", number = "6", }