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 [15].
Mathematics Subject Classification MSC2000: 16S10, 16N40, 16W50, 16U99Key words: free subalgebras, extensions of algebras, nil rings nilpotent matrices 2 [1,4,8,11,13,14,17,18,22]. In the influential paper of Makar-Limanov [12] several interesting open questions have been asked. In particular Makar-Limanov conjectured that if R is a finitely generated infinite dimensional algebraic division algebra then R contains a free subalgebra in two generators.Another question along this line was asked by Anick [1] in mid 1980's: Let R be a finitely presented algebra with exponential growth. Does it follow that R contains a free subalgebra in two generators? In the same paper he shown that finitely presented monomial algebras with exponential growth contain free subalgebras in two generators [1]. In [13] Makar-Limanov proved that the quotient algebra of the Weyl algebra contains a free subalgebra in two generators. He also conjectured that the following holds.
Conjecture 1.1 (Makar-Limanov, [15], [2])If R is an algebra without free subalgebras of rank two and S is an extensions of R obtained by extending the field K then S doesn't contain a free Kalgebra of rank two.Makar-Limanov mentioned that the truth of this conjecture would imply that we need only to consider algebras over uncountable fields in his mentioned above conjecture on the division algebras [12]. Conjecture 1.1 in the case of skew-fields, as stated in [12], attracted a lot of attention and is known to be true in several important cases [3,4,5,6,10,14,17,19]. In 1996 Reichstein showed that Conjecture 1.1 holds for algebras over uncountable fields [16].The purpose of this paper is to show that the situation is completely different for algebras over countable fields, as shown in the next theorem.Theorem 1.1 Over every countable field K there is an algebra A without free noncommutative subalgebras of rank two such that the polynomial ring A[x] in one indeterminate x over A contains a free noncommutative Kalgebra of rank two.Note that if an algebra contains a noncommutative free algebra of rank two then it also contains a noncommutative free algebra of arbitrarily high rank.As an application the following result is obtained. nilpotent matrices