In Atiyah-Macdonald, the Going-Down Theorem (listed as Theorem 5.16) assumes the integral extension to be a domain, which is due to the usage of a lemma (listed as Proposition 5.15). However, it seems to me nowhere in the proofs of both theorems used this assumption, so I wonder if it can be removed.
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to quote the proof in A & M, "$K$, the field of fractions of $A$". – Angina Seng Sep 22 '19 at 08:12
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ya, but the extension B doesn't have to be a domain, does it – Zhenyi Chen Sep 23 '19 at 19:49