There is a basic theorem in the geometry of schemes saying that the Spec of a Noetherian ring is a Noetherian topological space. It can be formulated as the ACC condition implies the ACCR condition (the ascending chain condition for radical ideals). I'm wondering if the converse is true.


In case it is tangentially relevant, there is also the ACCP condition (the ascending chain condition for prime ideals). Clearly ACCR implies ACCP, but ACCP does not imply ACCR, e.g., an infinite product of $\mathbb F\_2$.


If it isn't clear, we assume AC all the time.



Ascending chain condition on radical ideals
The converse is not true, i.e., there exists a non-noetherian affine scheme whose underlying topological space is noetherian. (Examples: Valuation rings of rank 2.)


More results and examples about such questions can be found for example in the following articles:



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> J. Ohm, R. L. Pendleton, [*Rings with noetherian spectrum,*](http://projecteuclid.org/euclid.dmj/1077377788) Duke Math. J. 35 (1968), 631-639.
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> W. Heinzer, J. Ohm, [*Locally noetherian commutative rings,*](http://www.ams.org/journals/tran/1971-158-02/S0002-9947-1971-0280472-2/) Trans. Amer. Math. Soc. 158 (1971), 273-284.
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