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.


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).

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.