矩阵代数有限直和的最小生成元个数的几个公式
Some formulas for the smallest number of generators for finite direct sums of matrix algebras
摘要：
得到了元素在有限域中的矩阵代数的有限直和的最小生成元个数的一个渐近上界。这产生了一个上界的类似数量的整数矩阵环。我们还得到了元素在有限域中的2 × 2矩阵代数的有限直和的最小生成元个数的精确公式，并由此得到了2 × 2整数矩阵环的有限直和的类似数量的公式。我们注意到环$\bigoplus_{i=1}^k M_{n_i}(\mathbb{Z})^{n_i}$的生成集可以用作任何矩阵代数$\bigoplus_{i=1}^k M_{n_i}(R)^{n_i}$的生成集，其中$R$是双边为1的结合环。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
