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On uniform bounds of primeness in matrix rings

Part of: Rings and algebras arising under various constructions Radicals and radical properties of rings

Published online by Cambridge University Press:  09 April 2009

Konstantin I. Beidar  and
Robert Wisbauer
Konstantin I. Beidar
Affiliation:
Department of Mathematics, National Cheng Kung UniversityTainan, Taiwan, e-mail: beidar@mail.ncku.edu.tw
Robert Wisbauer
Affiliation:
Mathematical Institute University of DüsseldorfGermany e-mail: wisbauer@math.uni-duesseldorf.de
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Abstract

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A subset S of an associative ring R is a uniform insulator for R provided a S b ≠ 0 for any nonzero a, b ∈ R. The ring R is called uniformly strongly prime of bound m if R has uniform insulators and the smallest of those has cardinality m. Here we compute these bounds for matrix rings over fields and obtain refinements of some results of van den Berg in this context.

More precisely, for a field F and a positive integer k, let m be the bound of the matrix ring Mk(F), and let n be dimF(V), where V is a subspace of Mk(F) of maximal dimension with respect to not containing rank one matrices. We show that m + n = k2. This implies, for example, that n = k2 − k if and only if there exists a (nonassociative) division algebra over F of dimension k.

MSC classification

Secondary: 16S50: Endomorphism rings; matrix rings 16N60: Prime and semiprime rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 2 , April 2004 , pp. 167 - 174
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Handelman, D. and Lawrence, L., ‘Strongly prime rings’, Trans. Amer. Math. Soc. 211 (1975), 209–223.CrossRefGoogle Scholar
[2]van den Berg, J. E., ‘On uniformly strongly prime rings’, Math. Japon. 38 (1993), 1157–1166.Google Scholar
[3]van den Berg, J. E., ‘A note on uniform bounds of primeness in matrix rings’, J. Austral. Math. Soc. Ser. A 65 (1998), 212–223.CrossRefGoogle Scholar