On a special presentation of matrix algebras

Abstract

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $n\times n$ matrix ring, so $R\cong M_n(S)$ for some ring $S$, if and only if it contains a set of $n\times n$ matrix units $\{e_{ij}{\}}_{i,j=1}^n$. A more recent and less known result states that a ring $R$ is a complete $(m+n)\times (m+n)$ matrix ring if and only if, $R$ contains three elements, $a$, $b$, and $f$, satisfying the two relations $a{f}^m+f^{n}b=1$ and $f^{m+n}=0$. In many instances the two elements $a$ and $b$ can be replaced by appropriate powers $a^i$ and $a^j$ of a single element $a$ respectively. In general very little is known about the structure of the ring $S$. In this article we study in depth the case $m=n=1$ when $R\cong M_2(S)$. More specifically we study the universal algebra over a commutative ring $A$ with elements $x$ and $y$ that satisfy the relations $x^{i}y+y{x}^j=1$ and $y^2=0$. We describe completely the structure of these $A$-algebras and their underlying rings when $\gcd (i,j)=1$. Finally we obtain results that fully determine when there are surjections onto $M_2(\mathbb{F})$ when $\mathbb{F}$ is a base field $\mathbb{Q}$ or ${\mathbb{Z}}_p$ for a prime number $p$.

32 pages