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Remark 15.25.5. Let $R$ be a ring. When does $R$ satisfy the condition mentioned in Lemmas 15.25.1, 15.25.2, and 15.25.4? This holds if

  1. $R$ is local,

  2. $R$ is Noetherian,

  3. $R$ is a domain,

  4. $R$ is a reduced ring with finitely many minimal primes, or

  5. $R$ has finitely many weakly associated primes, see Algebra, Lemma 10.66.17.

Thus these lemmas hold in all cases listed above.

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