Quantized rank R matrices

First some old as well as new results about P.I. algebras, Ore extensions, and degrees are presented. Then quantized $n\times r$ 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 $m$.Comment: 18 pages with 3 eps figures. Some proofs in Section 5 have been changed and a remark has been remove

Remarks on a paper by Skornjakov concerning rings for which every module is a direct sum of left ideals

Jøndrup S, Ringel CM. Remarks on a paper by Skornjakov concerning rings for which every module is a direct sum of left ideals. Archiv der Mathematik. 1978;31(1):329-331

Automorphism and derivations of upper triangular matrix rings

AbstractKezlan proved that for a commutative ring C, every C-automorphism of the ring of upper triangular matrices over C is inner. We generalize this result to rings in which all idempotents are central; moreover we show that for a semiprime ring A and central subring C, every C-automorphism of the ring of upper triangular matrices over C is the composite of an inner automorphism and an automorphism induced from a C-automorphism of A. By the method of proof we re-prove results of S. P. Coelho and C. P. Milies and of Mathis, stating that a derivation of a ring of upper triangular matrices of a C-algebra (n × n matrices over A) is a sum of an inner derivation and a derivation induced from a C-derivation of A. By an example we show that an extra assumption is needed for proving the above result of automorphisms of upper triangular matrices. Finally we consider automorphisms of subrings of n × n matrices over a commutative ring C, where entries over the diagonal are from C and below the diagonal are taken from a nil ideal. We prove that all such automorphisms are inner