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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The row rank of a subring of a matrix ring


Authors: M. S. Li and J. M. Zelmanowitz
Journal: Proc. Amer. Math. Soc. 99 (1987), 627-633
MSC: Primary 16A42; Secondary 15A30
MathSciNet review: 877029
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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$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0877029-6
PII: S 0002-9939(1987)0877029-6
Article copyright: © Copyright 1987 American Mathematical Society