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A note on uniform bounds of primeness in matrix rings

  • John E. Van den Berg (a1)

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, yR. 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 × n matrix ring over F.

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References

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[1]Handelman, D. E. and Lawrence, J., ‘Strongly prime rings’, Trans. Amer. Math. Soc. 211 (1975), 209233.
[2]Lambek, J., Lectures on rings and modules (Blaisdell, Toronto, 1966).
[3]Milnor, J., ‘Some consequences of a theorem of Bott’, Ann. of Math. 68 (1958), 444449.
[4]Olson, D. M., ‘A uniformly strongly prime radical’, J. Austral. Math. Soc. (Series A) 43 (1987), 95102.
[5]van den Berg, J. E., ‘On uniformly strongly prime rings’, Math. Japon. 38 (1993), 11571166.
[6]van den Berg, J. E., On chain domains, prime rings and torsion preradicals, PhD Thesis, (University of Natal Pietermaritzburg, 1995).
[7]van der Waerden, B. L., Modern algebra, volume I (Frederick Ungar, New York, 1953).
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A note on uniform bounds of primeness in matrix rings

  • John E. Van den Berg (a1)

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