A note on the existence of non-cyclic free subgroups in division rings

Abstract

A division ring D is said to be weakly locally finite if for every finite subset \({S \subset D}\), the division subring of D generated by S is centrally finite. It is known that the class of weakly locally finite division rings strictly contains the class of locally finite division rings. In this note we prove that every non-central subnormal subgroup of the multiplicative group of a weakly locally finite division ring contains a non-cyclic free subgroup. This generalizes the previous result by Goncalves for centrally finite division rings.