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

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 $\bigoplus_{i=1}^k M_{n_i}(\mathbb{Z})^{n_i}$ may be used as a generating set of any matrix algebra $\bigoplus_{i=1}^k M_{n_i}(R)^{n_i}$ where $R$ is an associative ring with a two-sided 1.