Let R be a commutative ring, m>1 an integer and q>1 an odd number. The following questions are addressed: (1) when is every element in ${\mathbb{M}}_n\left(R\right)$ and, respectively, in ${\mathbb{T}}_n\left(R\right)$ a product of q-potents? (2) when is every element in ${\mathbb{M}}_n\left(R\right)$ and, respectively, in ${\mathbb{T}}_n\left(R\right)$ a product of m q-potents? and (3) when is every element in ${\mathbb{M}}_n\left(R\right)$ and, respectively, in ${\mathbb{T}}_n\left(R\right)$ a product of an idempotent and a q-potent?

中文翻译：

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环上的矩阵是 q 势的乘积

令R为交换环，m >1 为整数，q >1 为奇数。解决了以下问题：(1) 中的每个元素何时出现${\mathbb{米}}_n\left(R\right)$并且分别在${\mathbb{吨}}_n\left(R\right)$q势的产物？(2) 中的每个元素什么时候出现${\mathbb{米}}_n\left(R\right)$并且分别在${\mathbb{吨}}_n\left(R\right)$m q -势的乘积？(3) 中的每个元素何时出现${\mathbb{米}}_n\left(R\right)$并且分别在${\mathbb{吨}}_n\left(R\right)$幂等和q -potent的乘积？