Spectrum of Noetherian ring is Noetherian

This article gives a fact about the relation between ring-theoretic assumptions about a commutative unital ring and topological consequences for the spectrum
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Statement

The spectrum of a Noetherian ring is a Noetherian space.

Definitions used

Spectrum

Further information: Spectrum

Noetherian ring

Further information: Noetherian ring

Noetherian space

Further information: 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.