The authors study radicals which coincide on artinian rings with Jacobson semisimple rings or equivalently with von Neumann regular rings. We say that two radical classes \(\sigma\) and \(\delta\) coincide on a class \(\mathcal C\) of rings in the weak sense if \(\gamma\cap{\mathcal C}=\delta\cap{\mathcal C}\). The radical classes \(\gamma\) and \(\delta\) coincide on \(\mathcal C\) in the strong sense if \(\gamma(A)=\delta(A)\) for every ring \(A\in{\mathcal C}\). Coincidence in the strong sense implies coincidence in the weak sense but not conversely. The authors give exact lower and upper bounds for strong coincidence. For weak coincidence the exact lower bound is that for strong coincidence. The authors determine the smallest homomorphically closed class which contains all radicals coinciding in the weak sense with the von Neumann regular radical on artinian rings. However, they do not know even the existence of the upper bound for weak coincidence. The authors also show 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 \(A\).