Spectrum of Noetherian ring is Noetherian

Statement
The spectrum of a Noetherian ring is a Noetherian space.

Converse
The converse is not in general true. In other words, there can be non-Noetherian rings whose spectrum is Noetherian. For instance, if the quotient of a ring by its nilradical is Noetherian, then the spectrum is Noetherian, even if the ring itself is not Noetherian.

Proof
The key idea is this: a strictly descending chain of closed subsets in the spectrum, gives rise to a strictly ascending chain of radical ideals in the ring. Thus, if the spectrum had an infinite strictly descending chain of closed subsets, then the Noetherian ring would have an infinite strictly ascending chain of radical ideals: a contradiction.