  First some old as well as new results about P.I. algebras, Ore extensions,
and degrees are presented. Then quantized $n\times r$ matrices as well as
quantized factor algebras of $M_q(n)$ are analyzed. The latter are the
quantized function algebra of rank $r$ matrices obtained by working modulo the
ideal generated by all $(r+1)\times (r+1)$ quantum subdeterminants and a
certain localization of this algebra is proved to be isomorphic to a more
manageable one. In all cases, the quantum parameter is a primitive $m$th roots
of unity. The degrees and centers of the algebras are determined when $m$ is a
prime and the general structure is obtained for arbitrary $m$.
