Hostname: page-component-54dcc4c588-smtgx Total loading time: 0 Render date: 2025-09-14T01:35:44.526Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on uniform bounds of primeness in matrix rings

Part of: Radicals and radical properties of rings Jordan algebras

Published online by Cambridge University Press:  09 April 2009

John E. Van den Berg
John E. Van den Berg
Affiliation:
Department of Mathematics and Applied Mathematics, University of Natal Pietermaritzburg, Private Bag X01, Scottsville 3209, South Africa e-mail: vandenberg@math.unp.ac.za
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 × n matrix ring over F.

MSC classification

Secondary: 16N60: Prime and semiprime rings 17C55: Finite-dimensional structures 17C60: Division algebras

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 65 , Issue 2 , October 1998 , pp. 212 - 223
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Handelman, D. E. and Lawrence, J., ‘Strongly prime rings’, Trans. Amer. Math. Soc. 211 (1975), 209–233.CrossRefGoogle Scholar
[2]Lambek, J., Lectures on rings and modules (Blaisdell, Toronto, 1966).Google Scholar
[3]Milnor, J., ‘Some consequences of a theorem of Bott’, Ann. of Math. 68 (1958), 444–449.CrossRefGoogle Scholar
[4]Olson, D. M., ‘A uniformly strongly prime radical’, J. Austral. Math. Soc. (Series A) 43 (1987), 95–102.CrossRefGoogle Scholar
[5]van den Berg, J. E., ‘On uniformly strongly prime rings’, Math. Japon. 38 (1993), 1157–1166.Google Scholar
[6]van den Berg, J. E., On chain domains, prime rings and torsion preradicals, PhD Thesis, (University of Natal Pietermaritzburg, 1995).Google Scholar
[7]van der Waerden, B. L., Modern algebra, volume I (Frederick Ungar, New York, 1953).Google Scholar