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 of an associative ring is a uniform insulator for provided for any nonzero . The ring is called uniformly strongly prime of bound if has uniform insulators and the smallest of those has cardinality . 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 and a positive integer , let be the bound of the matrix ring , and let be , where is a subspace of of maximal dimension with respect to not containing rank one matrices. We show that . This implies, for example, that if and only if there exists a (nonassociative) division algebra over of dimension .