Radicals coinciding with the von Neumann regular radical on artinian rings

We study radicals which coincide on artinian rings with Jacobson semisimple rings or equivalently with von Neumann regular rings. Exact lower and upper bounds for strong coincidence are given. For weak coincidence the exact lower bound is that for strong coincidence. We determine the smallest homomorphically closed class which contains all radicals coinciding in the weak sense with the von Neumann regular radical on artinian rings, but we do not know even the existence of the upper bound for weak coincidence. If a radical gamma coincides with the von Neumann regular radical on artinian rings in the strong sense, then gamma(A) is a direct summand in A for every artinian ring A.