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On coatoms of the lattice of matric-extensible radicals

Published online by Cambridge University Press:  17 April 2009

Halina France-Jackson
Halina France-Jackson
Affiliation:
Department of Mathematics and Applied Mathematics, Summerstrand Campus (South), PO Box 77000, Nelson Mandela Metropolitan University, Port Elizabeth 6031, South Africa e-mail: cbf@easterncape.co.uk
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A radical α in the universal class of all associative rings is called matric-extensible if for all natural numbers n and all rings A, A ∈ α if and only if Mn(A) ∈ α, where Mn(A) denotes the n × n matrix ring with entries from A. We show that there are no coatoms, that is, maximal elements in the lattice of all matric-extensible radicals of associative rings.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 72 , Issue 3 , December 2005 , pp. 403 - 406
Copyright
Copyright © Australian Mathematical Society 2005

References

[1]Andrunakievich, V.A. and Ryabukhin, Yu. M., Radicals of algebras and structure theory, (in Russian) (Nauka, Moscow, 1979).Google Scholar
[2]Booth, G.L. and France-Jackson, H., ‘On the lattice of matric-extensible radicals’, Acta Math. Hungar. 101 (2003), 163–172.CrossRefGoogle Scholar
[3]Gardner, B.J. and Stewart, P.N., ‘Prime essential rings’, Proc. Edinburgh Math. Soc. 34 (1991), 241–250.CrossRefGoogle Scholar
[4]Gardner, B.J. and Wiegandt, R., Radical theory of rings (Marcel Dekker, New York, 2004).Google Scholar
[5]Puczylowski, E.R., ‘Hereditariness of strong and stable radicals’, Glasgow Math. J. 23 (1982), 85–90.CrossRefGoogle Scholar
[6]Puczylowski, E.R. and Roszkowska, E., ‘On atoms and coatoms of lattices of radicals of associative rings,’ Comm. Algebra 20 (1992), 955–977.CrossRefGoogle Scholar
[7]Snider, R.L., ‘Lattices of radicals’, Pacific. J. Math. 40 (1972), 207–220.CrossRefGoogle Scholar