Free subgroups and free subsemigroups of division rings (Q1922897)

There are several open questions about free sub-objects of a multiplicative group of a (non-commutative) skew field. Namely, does it contain a non-commutative free subsemigroup (L. Makar-Limanov, 1984) or even a non-commutative free group (A. Lichtman, 1977)? More recently (1992) A. Klein asked whether a multiplicative semigroup of a non-commutative domain contains a non-commutative free subsemigroup. The author answers these questions under some restrictions and improves previously known results. As a sample let me quote one of his results: If \(D\) is a non-commutative division ring with uncountable center \(K\) then \(D^*\) contains a non-commutative free subgroup of the same cardinality as \(K\). The reviewer would also like to mention a similar additive conjecture which contains a question on subsemigroups and has not attracted enough attention: Conjecture. If \(D\) is a finitely generated non-commutative division ring which is infinite dimensional over its center then it contains a non-commutative free subalgebra. (Also submitted to MR).