On rings generating supernilpotent and special atoms

  • H France-Jackson Nelson Mandela Metropolitan University, Department of Mathematics and Applied Mathematics, Summerstrand Campus (South), P.O. Box 77000, Port Elizabeth 6031, South Africa
  • N Groenewald Nelson Mandela Metropolitan University, Department of Mathematics and Applied Mathematics, Summerstrand Campus (South), P.O. Box 77000, Port Elizabeth 6031, South Africa
Keywords: &#42-ring, &#42&#42-ring, supernilpotent and special radicals, supernilpotent and special atoms, special and weakly special classes, essential ideal, essential closure

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.
Keywords: *-ring; **-ring; supernilpotent and special radicals; supernilpotent and special atoms; special and weakly special classes; essential ideal; essential closure

Quaestiones Mathematicae 28(2005), 471–478.
Published
2005-11-25
Section
Articles

Journal Identifiers


eISSN: 1727-933X
print ISSN: 1607-3606