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RADICAL RELATED TO SPECIAL ATOMS REVISITED
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 91 / Issue 2 / April 2015
- Published online by Cambridge University Press:
- 14 October 2014, pp. 202-210
- Print publication:
- April 2015
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- Article
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- You have access
- Export citation
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A semiprime ring
$R$ is called a 
$\ast$-ring if the factor ring 
$R/I$ is in the prime radical for every nonzero ideal 
$I$ of 
$R$. A long-standing open question posed by Gardner asks whether the prime radical coincides with the upper radical 
$U(\ast _{k})$ generated by the essential cover of the class of all 
$\ast$-rings. This question is related to many other open questions in radical theory which makes studying properties of 
$U(\ast _{k})$ worthwhile. We show that 
$U(\ast _{k})$ is an N-radical and that it coincides with the prime radical if and only if it is complemented in the lattice 
$\mathbb{L}_{N}$ of all N-radicals. Along the way, we show how to establish left hereditariness and left strongness of important upper radicals and give a complete description of all the complemented elements in 
$\mathbb{L}_{N}$.
