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ON PRIME-LIKE RADICALS

Part of: Radicals and radical properties of rings

Published online by Cambridge University Press:  02 March 2010

S. TUMURBAT  and
H. FRANCE-JACKSON
S. TUMURBAT
Affiliation:
Department of Algebra, University of Mongolia, PO Box 75, Ulaan Baatar 20, Mongolia (email: stumurbat@hotmail.com)
H. FRANCE-JACKSON*
Affiliation:
Department of Mathematics and Applied Mathematics, Nelson Mandela Metropolitan University, Summerstrand Campus (South), PO Box 77000, Port Elizabeth 6031, South Africa (email: cbf@easterncape.co.uk)
*
✉For correspondence; e-mail: cbf@easterncape.co.uk
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Abstract

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A radical γ is prime-like if, for every prime ring A, the polynomial ring A[x] is γ-semisimple. In this paper, we study properties of prime-like radicals. In particular, we give necessary and sufficient conditions for a radical γ containing the prime radical β to be prime-like. This allows us to easily find distinct special radicals that coincide on simple rings and on polynomial rings, which answers a question put by Ferrero. It also allows us to reformulate a long-standing open problem of Gardner in terms of prime-like radicals.

Keywords

prime-like radicalspecial radicalAmitsur property of radicalspolynomially extensible radicalssubidempotent radicals

MSC classification

Secondary: 16N80: General radicals and rings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 1 , August 2010 , pp. 113 - 119
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

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