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关于生成矩阵环的矩阵对及其表示

On pairs of matrices generating matrix rings and their presentations

B. V. Petrenko, S. N. Sidki

备注：
33 pages. One typo has been corrected: in the first line of the proof of Theorem 2.19 on p. 16, $G_n(\mathbb{F}_q)$ was replaced with $M_n(\mathbb{F}_q)^2-G_n(\mathbb{F}_q)$. No other changes have been made
发表信息：
Journal of Algebra, 310 (2007), no. 1, 15--40
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摘要：

设$M_n(\mathbb{Z})$为整数元$n$乘$n$矩阵环，$n \geq 2$ .本文研究了生成$M_n(\mathbb{Z})$为环的对$(A,B) \in M_n(\mathbb{Z})^2$的集合$G_n(\mathbb{Z})$.我们使用几个演示的$M_{n}(\mathbb{Z})$与发电机$X=\sum_{i=1}^n E_{i+1,i}$和$Y=E_{11}$，以获得以下结果。\begin{enumerate} \item Let $k \geq 1$. Then the rings $M_n(\mathbb{Q})^k$ and $\bigoplus_{j=1}^{k} M_{n_j} (\mathbb{Z})$, where $n_1, ..., n_k \geq 2$ are pairwise relatively prime, have presentations with 2 generators and finitely many relations. \item Let $D$ be a commutative domain of sufficiently large characteristic over which every finitely generated projective module is free. We use 4 relations for $X$ and $Y$ to describe all representations of the ring $M_{n}(D)$ into $M_{m}(D)$ for $m \geq n$. \item We obtain information about the asymptotic density of $G_n(F)$ in $M_n(F)^2$ over different fields, and over the integers. \end{enumerate}

英文摘要：

Let $M_n(\mathbb{Z})$ the ring of $n$-by-$n$ matrices with integral entries, and $n \geq 2$. This paper studies the set $G_n(\mathbb{Z})$ of pairs $(A,B) \in M_n(\mathbb{Z})^2$ generating $M_n(\mathbb{Z})$ as a ring. We use several presentations of $M_{n}(\mathbb{Z})$ with generators $X=\sum_{i=1}^n E_{i+1,i}$ and $Y=E_{11}$ to obtain the following consequences. \begin{enumerate} \item Let $k \geq 1$. Then the rings $M_n(\mathbb{Q})^k$ and $\bigoplus_{j=1}^{k} M_{n_j} (\mathbb{Z})$, where $n_1, ..., n_k \geq 2$ are pairwise relatively prime, have presentations with 2 generators and finitely many relations. \item Let $D$ be a commutative domain of sufficiently large characteristic over which every finitely generated projective module is free. We use 4 relations for $X$ and $Y$ to describe all representations of the ring $M_{n}(D)$ into $M_{m}(D)$ for $m \geq n$. \item We obtain information about the asymptotic density of $G_n(F)$ in $M_n(F)^2$ over different fields, and over the integers. \end{enumerate}

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