Spectrum of Noetherian ring is Noetherian: Difference between revisions
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{{further|[[Noetherian space]]}} | {{further|[[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== | ==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. | 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. | ||
Latest revision as of 16:34, 12 May 2008
This article gives a fact about the relation between ring-theoretic assumptions about a commutative unital ring and topological consequences for the spectrum
View other such facts
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.