Row rank and the structure of matrix subrings (Q2640679)

The authors continue the investigation begun in their article [Proc. Am. Math. Soc. 99, 627-633 (1987; Zbl 0637.15011)]. Let \(M_ n(\Delta)\) be an \(n\times n\) matrix ring over a division ring \(\Delta\) and let R be a subring of \(M_ n(\Delta)\). The authors introduce a concept of row rank of R in \(M_ n(\Delta)\) as follows: row rank R\(=k\) in \(M_ n(\Delta)\) if k is the maximal integer (between 0 and n) such that there exists a left order D in \(\Delta\) and an invertible matrix \(s\in M_ n(\Delta)\) with \(s^{-1}Rs\supset \{(a_{ij})\in M_ n(D)|\) \(a_{ij}=0\) whenever \(i>k\}\). Using this concept the authors show that if \(Q_{\max}(R)=M_ n(\Delta)\) then R is bounded by a blocked triangular matrix ring and row rank R\(=\dim R-\dim B(R)=n-\dim B(R)\) where dim denotes uniform (Goldie) dimension and B(R) is the prime radical of R. Furthermore, the authors show that rank R\(=column rank R=n\) in \(M_ n(\Delta)\) or else row rank R\(+column rank R\leq n\) in \(M_ n(\Delta)\). In this article the authors pay more attention to subrings R of \(M_ n(\Delta)\) with row rank R\(+column rank R\leq n\) and show that R can be approximated by rings of triangular matrices.