RADICAL THEORY FOR SEMIRINGS

Abstract

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 ϱ∞(S) and ϱε(S) and two similar ideals β∞ (S) and βε(S) is widely solved. We prove ϱ∞(S) ⊇ ϱε(S) = β∞(S) = βε(S) and give necessary and sufficient conditions for equality in the first inclusion. This yields in particular that the weakly special class Mε(U) is always semisimple, a result which is not true for the special class M∞(U).