On pairs of matrices generating matrix rings and their presentations
Abstract
Description
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}
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
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