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A PRIME ESSENTIAL RING THAT GENERATES A SPECIAL ATOM

Part of: Radicals and radical properties of rings

Published online by Cambridge University Press:  02 November 2016

SRI WAHYUNI ,
INDAH EMILIA WIJAYANTI  and
HALINA FRANCE-JACKSON
SRI WAHYUNI
Affiliation:
Jurusan Matematika, FMIPA UGM, Sekip Utara, Yogyakarta, Indonesia email [email protected]
INDAH EMILIA WIJAYANTI
Affiliation:
Jurusan Matematika, FMIPA UGM, Sekip Utara, Yogyakarta, Indonesia email [email protected]
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 email [email protected]
*
✉ [email protected]
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Abstract

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A special atom (respectively, supernilpotent atom) is a minimal element of the lattice $\mathbb{S}$ of all special radicals (respectively, a minimal element of the lattice $\mathbb{K}$ of all supernilpotent radicals). A semiprime ring $R$ is called prime essential if every nonzero prime ideal of $R$ has a nonzero intersection with each nonzero two-sided ideal of $R$ . We construct a prime essential ring $R$ such that the smallest supernilpotent radical containing $R$ is not a supernilpotent atom but where the smallest special radical containing $R$ is a special atom. This answers a question put by Puczylowski and Roszkowska.

Keywords

prime essential ringprime radicalspecial and supernilpotent radicalsspecial and supernilpotent atomsleft hereditary radical

MSC classification

Primary: 16N80: General radicals and rings
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 95 , Issue 2 , April 2017 , pp. 214 - 218
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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