Not so long ago I have posted an answer to this question, and the OP deleted it immediately after.

I don't want to comment on this behavior, instead I'd like to repost it for the benefit of other readers of M.SE. (I know that users rated >10k can see such questions, but I think it would be nice to have a rating-free feature to help us in such cases.)

  • 2
    $\begingroup$ Immediately here is about 12 hours. Just to put a timeframe to those that can't see the question. $\endgroup$
    – Asaf Karagila Mod
    Aug 5 '14 at 8:12
  • 1
    $\begingroup$ You wrote that the question was immediately deleted. Here, in your question, immediately means almost 12 hours. $\endgroup$
    – Asaf Karagila Mod
    Aug 5 '14 at 8:30
  • $\begingroup$ It seems that Google still has a cached version - link to search. Will this be sufficient for you or should I (or someone else) copy the text of the question here (or somewhere)? $\endgroup$ Aug 5 '14 at 8:53
  • $\begingroup$ BTW I think this should be tagged (specific-question). $\endgroup$ Aug 5 '14 at 8:54
  • $\begingroup$ Maybe I'm writing this from my phone and can't quite help you with that. $\endgroup$
    – Asaf Karagila Mod
    Aug 5 '14 at 8:54
  • $\begingroup$ @Martin: Maybe it should be just undeleted? $\endgroup$
    – Asaf Karagila Mod
    Aug 5 '14 at 8:55
  • $\begingroup$ @AsafKaragila ok, I have voted to undelete. Two more votes needed. At the same time I have copied all the relevant information to an answer. $\endgroup$ Aug 5 '14 at 9:01
  • $\begingroup$ @Martin: One more. $\endgroup$
    – Asaf Karagila Mod
    Aug 5 '14 at 9:07
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    $\begingroup$ Undeleted. ${}$ $\endgroup$ Aug 5 '14 at 9:29
  • $\begingroup$ Thanks Asaf, Martin, and Gerry! $\endgroup$
    – user26857
    Aug 5 '14 at 9:30
  • $\begingroup$ Some suggestions in the comments here meta.math.stackexchange.com/q/8852/27978 $\endgroup$
    – copper.hat
    Aug 5 '14 at 16:22
  • $\begingroup$ @ZhenLin Are you sure the OP and that BenjaLim (who actually changed his username long time before deletion) are one and the same user? Furthermore, the question here is about a deleted question, not about a deleted account. $\endgroup$
    – user26857
    Aug 6 '14 at 21:16
  • $\begingroup$ It seems like a remarkable coincidence otherwise! (Yes, I only just remembered that he changed his username before that.) $\endgroup$
    – Zhen Lin
    Aug 6 '14 at 21:29

I have copied all the relevant information here. If this was not correct thing to do, I can simply delete this answer.

Question: Why does every maximal ideal closed in $\mathfrak{a}$-topology imply that $\mathfrak{a} \subseteq \text{Jac}(A)$?


I must be missing something very simple, but suppose that every maximal ideal $\mathfrak{m}$ of a Noetherian ring is closed in the $\mathfrak{a}$-topology on $A$. Then why does this imply that $\mathfrak{a} \subseteq \text{Jac}(A)$, the Jacobson radical of $A$? I know if there is an ideal $\mathfrak{a}$ not contained in some maximal ideal $\mathfrak{m}$ then for every $i > 0$ we have $$ \mathfrak{m} + \mathfrak{a}^i = A.$$

How does this contradict $\mathfrak{m}$ being closed? I can't see the existence of $x \notin \mathfrak{m}$ with no open neighborhood about it.

Answer: An ideal $m$ is closed in the $a$-adic topology iff $m=\cap_{i\ge 1} (m+a^i)$. As you observed, this shows that $m+a^i\ne A$ for any $i\ge 1$. If $m$ is maximal this means that $a\subseteq m$.


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