On rings generating supernilpotent and special atoms

Abstract

We study prime rings which generate supernilpotent (respectively special) atoms, that is, atoms of the lattice of all supernilpotent (respectively special) radicals. A prime ring A is called a **-ring if the smallest special class containing A is closed under semiprime homomorphic images of A. A semiprime ring A whose every proper homomorphic image is prime radical is called a *-ring. We show that the class of all **-rings strictly contains the class of all *-rings and that the smallest supernilpotent (respectively the smallest special) radical containing a nonzero **-ring is a supernilpotent (respectively special) atom. This generalizes H. France-Jackson's results concerning supernilpotent and special atoms. A big open question in radical theory asks whether the prime radical coincides with the upper radical generated by the essential closure of the class of all *-rings. We ask the similar question concerning **-rings and note that a negative answer to our question implies a negative answer to the big question. We also give some necessary and sufficient conditions for a prime ring to generate a supernilpotent atom which might help to answer the big open question.