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Noetherian topological space

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Suppose $R$ is a commutative unital ring. Are the following two concepts equivalent?

* Spec R is a noetherian topological space

* R is noetherian

[Sent this via PM mail as well; sorry about the duplication.]
Apparently not. According to Problem 2.13 (p. 80) from Hartshorne, the latter condition implies the former, but the implication is one-way: Spec(R) can be noetherian even when R is not. I can't supply proofs of these results offhand.

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