2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222723It 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-LimanovRings and Algebras16S10, 16N40, 16W50text