A note on uniform bounds of primeness in matrix rings

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 X of R we have xXy ^ 0 whenever 0 ^ x, y € 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 x n matrix ring over F.1991 Mathematics subject classification (Amer. Math. Soc): primary 16N60; secondary 17C55, 17C6O.

Strong primeness in matrix rings

“…Define a division pseudoalgebra over a division ring D to be a D-bimodule D M D with a multiplication : M 2 → M which is left linear in its first argument, right linear in its second and such that for y ∈ M \{0}, x ∈ M , each of the equations y w = x, w y = x has exactly one solution w ∈ M . For D = F a field, this idea coincides with the usual concept of a (not necessarily associative) division algebra over F [7]. For D = F a field, every n-dimensional vector space over F is isomorphic to F n , so every n-dimensional division algebra over F is isomorphic to F n with an appropriate vector multiplication; this proves [6, Theorem 1.2(ii)].…”

“…is thus equivalent to the question "What is the smallest number of bilinear equations y T A (p) x = 0 that one needs to force y = 0 or x = 0?". In the following result, parts (i) and (ii) without the conditions on the bounds come from [4, Proposition II.1] and [9,Lemma 9] respectively; the conditions on the bounds are obtained by adapting proofs from and using ideas in those articles and [7]. Theorem 2.…”

“…The ideas of Proposition 1 are used in the proof of [7,Theorem 3]. The result and argument are adapted here, using ideas from [8].…”

“…, B (2n−2) = E (n−1,n) − E (n,n−1) . (These are close variants of matrices used in the proof of [7,Theorem 4]…”

“…Van den Berg [5] sharpened this result, showing that the bound of uniform strong primeness of M n (D)always lies from n to 2n − 1inclusive. Curiously, its exact value is not determined solely by n, but also depends on subtle algebraic features of the ground division ring D. Indeed, it has been shown that the bound of uniform strong primeness of M n (F)is 2n − 1if F is an algebraically closed field [5, Proposition 8], and n if and only if there exists a (possibly nonassociative) division algebra over F of dimension n ([6, Theorem 1.2], [7,Theorem 11]); this means, for example, that the bound of M n (Q), with Q the field of rationals, is always n, for there exists, for every n, an irreducible polynomial of degree n over Q and thus an n-dimensional field extension of Q. The bound of uniform strong primeness of M n (F)can, however, lie strictly between n and 2n − 1as examples in this paper and earlier papers show; the ring of 3by 3 matrices over the reals, for example, is uniformly strongly prime of bound 4. This paper continues the work of Beidar, Wisbauer and the second author [5][6][7] on bounds of uniform strong primeness in matrix rings.…”