Let D be an infinite division ring with centre F and $\mathbb{T}(\mathrm{\infty },D)$ the group of infinite upper triangular invertible matrices indexed by $\mathbb{N}\times \mathbb{N}$ over D. In this paper, we first show that if D is finite dimensional over F, then every element in $\mathbb{T}(\mathrm{\infty },D)$ whose diagonal entries are commutators of $D\setminus \left\{0\right\}$ is a commutator of an infinite diagonal matrix and another infinite upper triangular matrix. Some applications are also shown. For example, if ${\dim }_{F}D\le 4$, then every element in the commutator subgroup $\left[{\mathrm{G}\mathrm{L}}_{VK}\right(\mathrm{\infty },D),{\mathrm{G}\mathrm{L}}_{VK}(\mathrm{\infty },D\left)\right]$ of the Vershik-Kerov group ${\mathrm{G}\mathrm{L}}_{VK}(\mathrm{\infty },D)$ is a commutator.

中文翻译：

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除法代数上无限可逆矩阵的某种分解

设D为中心为F 的无限分割环，$\mathbb{时间}（\mathrm{无穷大},D）$索引为的无限上三角可逆矩阵组$\mathbb{氮}\times \mathbb{氮}$超过D。在本文中，我们首先证明，如果D是F上的有限维，则中的每个元素$\mathbb{时间}（\mathrm{无穷大},D）$其对角线条目是换向器$D\setminus \left\{0\right\}$是一个无限对角矩阵和另一个无限上三角矩阵的交换子。还显示了一些应用程序。例如，如果${\mathrm{暗淡}}_{F}D\le 4$，那么换向子组中的每个元素$[{\mathrm{G}\mathrm{L}}_{VK}（\mathrm{无穷大},D）,{\mathrm{G}\mathrm{L}}_{VK}（\mathrm{无穷大},D）]$Vershik-Kerov 小组的${\mathrm{G}\mathrm{L}}_{VK}（\mathrm{无穷大},D）$是换向器。