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- W2957827910 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 $ntimes n$ matrix ring, so $Rcong M_{n}(S)$ for some ring $S$, if and only if it contains a set of $ntimes n$ 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 $Rcong M_2(S)$. 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 $p$." @default.
- W2957827910 created "2019-07-23" @default.
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- W2957827910 date "2019-07-11" @default.
- W2957827910 modified "2023-09-27" @default.
- W2957827910 title "On a special presentation of matrix algebras" @default.
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