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ON α-LIKE RADICALS

Part of: Radicals and radical properties of rings

Published online by Cambridge University Press:  16 June 2011

H. FRANCE-JACKSON
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)
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Abstract

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A radical ρ is called prime-like if for every prime ring A, the polynomial ring A[x] is ρ-semisimple. Let α be a radical satisfying the polynomial equation α(A[x])=(α(A))[x] for every ring A. A radical γ is called α-like if for every α-semisimple ring A, the polynomial ring A[x] is γ-semisimple. In this paper, we study properties of α-like radicals. We show that α-likeness is a generalization of prime-likeness and extend some results concerning prime-like radicals. This allows us easily to find distinct special radicals which coincide on simple rings and on polynomial rings, which answers a question put by Ferrero.

Keywords

prime-like radicalα-like radicalspecial radicalAmitsur property of radicalspolynomially extensible radicals

MSC classification

Secondary: 16N80: General radicals and rings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 1 , August 2011 , pp. 111 - 115
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

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