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.