Suppose that A is the algebra C^m([0,1]) of all m times continously differentiable complex-valued functions on [0,1] with the norm ||f || = å_{k = 0}^m[1/k!] sup_{x Î [0,1]} |f^(k)(x)|. Let I = {f Î A ; f(0) = f¢(0) = 0}. Then [A/I] is not semisimple, since assuming f_° to be f_°(x) = x, then f_°² Î I and so (f_° + I)² = f_°² + I = 0, hence r(x) = lim_n ||(f_° + I)ⁿ||^[1/n] = 0. Therefore f_° + I Î Rad([A/I]). But f_° + I ¹ 0. So that [A/I] is not semisimple.