Abstract. Let D be a weakly locally finite division ring and n a positive integer. In this paper, we investigate the problem on the existence of non-cyclic free subgroups in non-central almost subnormal subgroups of the general linear group GLn(D). Further, some applications of this fact are also investigated. In particular, all infinite finitely generated almost subnormal subgroups of GLn(D) are described.

Let D be a division algebra with center F and K a (not necessarily central) subfield of D. An element a ∈ D is called left algebraic (resp. right algebraic) over K, if there exists a non-zero left polynomial a 0 + a 1 x + · · · + a n x n (resp. right polynomial a 0 +xa 1 +· · ·+x n a n ) over K such that a 0 +a 1 a+· · ·+a n a n = 0 (resp. a 0 +aa 1 +· · ·+a n a n ). Bell et al. proved that every division algebra whose elements are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. In this paper we generalize this result and prove that every division algebra whose all multiplicative commutators are left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite provided that the center of division algebra is infinite. Also, we show that every division algebra whose multiplicative group of commutators is left (right) algebraic of bounded degree over a (not necessarily central) subfield must be centrally finite. Among other results we present similar result regarding additive commutators under certain conditions.

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