ON A SPECIAL PRESENTATION OF MATRIX ALGEBRAS

Authors : Geir AGNARSSON, Samuel S MENDELSON

Pages : 13-42

Doi:10.24330/ieja.662946

View : 12 | Download : 8

Publication Date : 2020-01-07

Article Type : Research Paper

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 $R\cong M_{n}insert ignore into journalissuearticles values(S);$ 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 $insert ignore into journalissuearticles values(m+n);\timesinsert ignore into journalissuearticles values(m+n);$ 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 $R\cong M_2insert ignore into journalissuearticles values(S);$. 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 $\gcdinsert ignore into journalissuearticles values(i,j);=1$. Finally we obtain results that fully determine when there are surjections onto $M_2insert ignore into journalissuearticles values(\field);$ when $\field$ is a base field $\rats$ or $\ints_p$ for a prime number $p$.
Keywords : Matrix ring, matrix algebra, finite presentation