On pairs of matrices generating matrix rings and their presentations

Let $M_n(\mathbb{Z})$ the ring of $n$-by-$n$ matrices with integral entries, and $n\ge 2$. 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 $k\ge 1$. 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 $m\ge n$. \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}