In this article we continue investigations on a Kurosh-Amitsur radical theory for a universal class U of hemirings as introduced by O.M. Olson et al. We give some necessary and sufficient conditions that such a universal class U consists of all hemirings. Further we consider special and weakly special subclasses M of U which yield hereditary radical classes P =um of U. In this context we correct some statements in the papers of Olson et al. Moreover, a problem posed there concerning the equality of two radicals#x3F1;#x221E;(S)and#x3F1;#x3B5;(S)and two similar ideals#x3B2;#x221E;(S)and#x3B2;#x3B5;(S)is widely solved. We prove#x3F1;#x221E;(S)#x2287;#x3F1;#x3B5;(S)=#x3B2;#x221E;(S)=#x3B2;#x3B5;(S)and give necessary and sufficient conditions for equality in the first inclusion. This yields in particular that the weakly special class M#x3B5;(U) is always semisimple, a result which is not true for the special class M#x221E;(U).
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