This paper deals with the effect
of radicals (in the Kurosh-Amitsur sense) on supplementary semilattice sums of rings
as defined by J. Weissglass (Proc. Amer. Math. Soc., 39 (1973), 471-473). It is shown
that if ℜ is a strict, hereditary radical class, then ℜ(R) = Σα∈Ωℜ(Rα) for every
supplementary semilattice sum R = Σα∈ΩRα with finite Ω. If ℜ is an A-radical ciass
or the generalized nil radical class, the same conclusion holds with the finiteness
restriction removed. On the other hand, if ℜ(Σα∈ΩRα) = Σα∈Ωℜ(Rα) for all finite Ω,
then ℜ is strict and satisfies (∗)R ∈ℜ⇒ the zeroring on the additive group of R
belongs to ℜ, a condition satisfled by both hereditary strict and A-radical
classes.
