Quantized Rank R Matrices

First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized n × r matrices as well as certain quantized factor algebras M q (n) of Mq(n) are analyzed. For r = 1, . . . , n − 1, M q (n) is the quantized function algebra of rank r matrices obtained by working modulo the ideal generated by all (r+1)×(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 mth 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.