# On a special presentation of matrix algebras

## TL;DR

This paper characterizes when certain rings are matrix rings, focusing on the case of 2x2 matrices over a ring S, and describes the structure of related universal algebras with specific relations.

## Contribution

It provides a complete structural description of universal algebras over commutative rings with specific relations, especially when gcd(i,j)=1, and determines conditions for surjections onto 2x2 matrix rings over fields.

## Key findings

- Complete structure of A-algebras with relations x^iy+yx^j=1 and y^2=0 when gcd(i,j)=1.
- Conditions for surjections onto M_2(F) over Q and Z_p.
- Characterization of rings as 2x2 matrix rings via specific elements and relations.

## 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 $af^m+f^nb=1$ 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 $x^iy+yx^j=1$ 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 $p$.