J. Aust. Math. Soc.  76 (2004), 167-174
On uniform bounds of primeness in matrix rings

Konstantin I. Beidar
  Department of Mathematics
  National Cheng Kung University
  Tainan
  Taiwan
  beidar@mail.ncku.edu.tw
and
Robert Wisbauer
  Mathematical Institute
  University of Düsseldorf
  Germany
  wisbauer@math.uni-duesseldorf.de


Abstract
A subset $\mathcal{S}$ of an associative ring $R$ is a uniform insulator for $R$ provided $a\mathcal{S} b\neq 0$ for any nonzero $a,b\in R$. The ring $R$ is called uniformly strongly prime of bound $m$ if $R$ has uniform insulators and the smallest of those has cardinality $m$. Here we compute these bounds for matrix rings over fields and obtain refinements of some results of van den Berg in this context. More precisely, for a field $F$ and a positive integer $k$, let $m$ be the bound of the matrix ring $M_k(F)$, and let $n$ be $\dim_ F(\mathcal{V})$, where $\mathcal{V}$ is a subspace of $M_k( F)$ of maximal dimension with respect to not containing rank one matrices. We show that $m+n=k^2$. This implies, for example, that $n=k^2-k$ if and only if there exists a (nonassociative) division algebra over $F$ of dimension $k$.
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