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Let R = Σα∈Ω Rα be a semilattice Ω-graded ring and P a radical. Conditionsare found under which P(R) = Σα∈Ω P(Rα) if and only if P(Rα)Rβ+RβP(Rα) (∈) P(Rαβ)for all α,β∈Ω. Examples are given to show that some conditions cannot be removed.