  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 $\dim_FD\le d^2$.
