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 X of R we have aXb ≠ 0 whenever 0 ≠ a,b ∈ 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 talk is devoted to an investigation of uniform bounds of primeness in matrix rings over fields. The existence of certain n-dimensional nonassociative algebras over a field F decides the uniform bound of the nxn matrix ring over F.