Some formulas for the smallest number of generators for finite direct sums of matrix algebras

R. V. Kravchenko; B. V. Petrenko

paper

Some formulas for the smallest number of generators for finite direct sums of matrix algebras

arXiv:math/0611674

Abstract

We obtain an asymptotic upper bound for the smallest number of generators for a finite direct sum of matrix algebras with entries in a finite field. This produces an upper bound for a similar quantity for integer matrix rings. We also obtain an exact formula for the smallest number of generators for a finite direct sum of 2-by-2 matrix algebras with entries in a finite field and as a consequence obtain a formula for a similar quantity for a finite direct sum of 2-by-2 integer matrix rings. We remark that a generating set the ring $⨁_{i = 1}^{k} M_{n_{i}} (Z)^{n_{i}}$ may be used as a generating set of any matrix algebra $⨁_{i = 1}^{k} M_{n_{i}} (R)^{n_{i}}$ where $R$ is an associative ring with a two-sided 1.

29 pages. We found the generating function for gen_{m,2}(q), and we added an appendix containing a 2-generator presentation for a finite direct sum of matrix algebras over an infinite field