A note on uniform bounds of primeness
in matrix rings

John E. van den Berg
Department of Mathematics and Applied Mathematics
University of Natal Pietermaritzburg
Private Bag X01
Scottsville 3209
South Africa

Abstract:

A nonzero ring R is said to be uniformly strongly prime (of bound n) if n is the smallest positive integer such that for some n-element subset Xof R we have $xXy\neq 0$ whenever $0\neq x,y\in R$. The study of uniformly strongly prime rings reduces to that of orders in matrix rings over division rings, except in the case n=1. This paper is devoted primarily to an investigation of uniform bounds of primeness in matrix rings over fields. It is shown that the existence of certain n-dimensional nonassociative algebras over a field F decides the uniform bound of the $n\times n$ matrix ring over F.