  In this note, we introduce a new concept of a {\it generalized algebraic
rational identity} to investigate the structure of division rings. The main
theorem asserts that if a non-central subnormal subgroup $N$ of the
multiplicative group $D^*$ of a division ring $D$ with center $F$ satisfies a
non-trivial generalized algebraic rational identity of bounded degree, then $D$
is a finite dimensional vector space over $F$. This generalizes some previous
results.
