I'm interested in intuition about the affine schemes of rings with a non-vanishing Jacobson radical.
In the ring of real-valued continuous functions on a topological space the Jacobson radical is reduced to zero. Thinking of an arbitrary ring as the global sections on an affine scheme the situation is quite different, but there is also the addition of generic points (more stuff) and a very weak topology (fewer continuous functions).
In short: How does a non-vanishing Jacobson radical manifest itself geometrically? What happens, and how should I think about this? If $A$ is a ring with a non-vanishing Jacobson radical $J$, how does for instance the space $Spec(A/J)$ look compared to $Spec(A)$?