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Let $R$ be a commutative ring with unity which satisfies d.c.c. on radical ideals. then does $R$ satisfy a.c.c. on radical ideals ? If this is not true in general, then what happens if we also assume $R$ is a local domain and/or that $R$ has finite Krull dimension ?

NOTE: If $R$ has finite Krull dimension, then obviously $R$ satisfies a.c.c. on prime ideals.

By the answer below of David Lampert, it follows that only assuming local domain is not enough to guarantee a.c.c. on radical ideals.

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Answer (2) here Valuation ring satisfying either a.c.c. or d.c.c. on prime ideals gives a counterexample to the first 2 questions (a radical ideal in a valuation ring is prime).

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  • $\begingroup$ yes because a radical ideal in a valuation ring is prime $\endgroup$ – David Lampert Nov 20 '18 at 1:07

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