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ON SUPERNILPOTENT NONSPECIAL RADICALS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 78 / Issue 1 / August 2008
- Published online by Cambridge University Press:
- 01 August 2008, pp. 107-110
- Print publication:
- August 2008
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- Article
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Let ρ be a supernilpotent radical. Let ρ* be the class of all rings A such that either A is a simple ring in ρ or the factor ring A/I is in ρ for every nonzero ideal I of A and every minimal ideal M of A is in ρ. Let
be the lower radical determined by ρ* and let ρφ denote the upper radical determined by the class of all subdirectly irreducible rings with ρ-semisimple hearts. Le Roux and Heyman proved that 
is a supernilpotent radical with 
and they asked whether 
if ρ is replaced by β, ℒ , 𝒩 or 𝒥 , where β, ℒ , 𝒩 and 𝒥 denote the Baer, the Levitzki, the Koethe and the Jacobson radical, respectively. In the present paper we will give a negative answer to this question by showing that if ρ is a supernilpotent radical whose semisimple class contains a nonzero nonsimple * -ring without minimal ideals, then is a nonspecial radical and consequently 
. We recall that a prime ring A is a * -ring if A/I is in β for every 
.
be the lower radical determined by 
is a supernilpotent radical with 
and they asked whether 
if 
. We recall that a prime ring 
.