The row rank of a subring of a matrix ring

Abstract:

For $R$ a subring of an $n \times n$ matrix ring ${M_n}(\Delta )$ over a division ring $\Delta$, we examine an invariant called the row rank of $R$. Roughly speaking, the row rank of $R$ is the largest integer $k$ such that $R$ contains all $k$-rowed matrices over a left order in $\Delta$. The row rank of $R$ is then an integer between 0 and $n$; and we will see that row rank $R \geq 1$ means that ${M_n}(\Delta )$ is the maximal left quotient ring of $R$, while row rank $R = n$ signifies that ${M_n}(\Delta )$ is the classical left quotient ring of $R$. Thus row rank provides a link between maximal and classical quotient rings for rings of this type. A description of the subrings $R$ with row rank $R \geq k$ is obtained which subsumes and generalizes earlier theorems of Faith-Utumi and Zelmanowitz, respectively, for the cases row rank $R = n$ and row rank $R \geq 1$.