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Starting with a class J! of £2-groups, necessary and sufficient conditions on J( are given to ensure that the corresponding Hoehnke radical p (determined by the subdirect closure of Jl as semisimple class) is a radical in the sense of Kurosh and Amitsur; has a hereditary semisimple class; satisfies the ADS-property; has a hereditary radical class or satisfies pN n / C pi and lastly, have both a hereditary radical and semisimple class or satisfies pN n / = pi.

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A * -primitive involution ring R is either a left and right primitive ring or a certain subdirect sum of a left primitive and a right primitive ring with involution exchanging the components. An example is given of a left and right primitive ring which admits no row and column finite matrix representation. We characterize * -primitive involution rings in terms of maximal * -biideals. A * -prime involution ring has a minimal left ideal if and only if it has a minimal * -biideal, and these involution rings are always * -primitive.