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 Mr + 1q(n) of Mq(n) are analyzed. For r = 1,…, n − 1, Mr + 1q(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 almost all cases, the quantum parameter is a primitive mth root 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.