Suppose that A is the algebra C^m([0,1]) of the complex valued m times continously differentiable functions on [0,1] with the norm ||f || = å_{k = 0}^m[1/k!] sup_{x Î [0,1]} |f^(k)(x)|. Let x₀ Î [0,1] and I = {f Î A ; f(x₀) = f¢(x₀) = 0}. Then I is a closed two-sided ideal contained in only one maximal ideal; i.e. {f Î A : f(x₀) = 0 }. Note that the maximal ideals of A are of the form I_x = {f Î A ; f(x) = 0}, x Î [0,1].
A conclusion is that C^m([0,1]) is not spectral synthesis, i.e. it has a closed two-sided ideal which is not the intersection of maximal ideals containing this ideal.

Comment. The disk algebra contains a nonmaximal prime ideal, namely {0}.