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

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

2006-11-22

arxiv.org/abs/math/0611674v7

Mathematics

Rings and Algebras

29 pages. We found the generating function for gen_{m,2}(q), and we added an appendix containing a 2-generator presentation fo

Scientific paper

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.

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