In a recent paper van Leeuwen and Heyman constructed a supernilpotent radical class using the class of almost nilpotent rings. Using a similar construction, for any class C satisfying the following four properties we obtain a superrnlpotent radical class containing C.
(N1) C contains the class Z of all zero rings.
(N2) C is hereditary.
(N3) C is homomorphically closed.
(N4) If A and A/I are elements of C for some ideal I of a ring A, then A ∈ C.
Every supernilpotent radical class P clearly satisfies these conditions. For any such radical class we define the class of almost radical rings and use these to construct a new radical class P2 which contains the given one. Also, we give a characterization for dual supernilpotent radicals.
