Abstract: | Let D be a division ring with center F and K a division subring of D. In this paper, we show that a non-central normal subgroup N of the multiplicative group
$$D^*$$
is left algebraic over K if and only if so is D provided F is uncountable and contained in K. Also, if K is a field and the n-th derived subgroup
$$D^{(n)}$$
of
$$D^{*}$$
is left algebraic of bounded degree d over K, then
$$\dim _FD\le d^2$$
. |