# Deleted question. How to recover some information?

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.)

• Immediately here is about 12 hours. Just to put a timeframe to those that can't see the question. – Asaf Karagila Mod Aug 5 '14 at 8:12
• You wrote that the question was immediately deleted. Here, in your question, immediately means almost 12 hours. – Asaf Karagila Mod Aug 5 '14 at 8:30
• 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)? – Martin Sleziak Aug 5 '14 at 8:53
• BTW I think this should be tagged (specific-question). – Martin Sleziak Aug 5 '14 at 8:54
• Maybe I'm writing this from my phone and can't quite help you with that. – Asaf Karagila Mod Aug 5 '14 at 8:54
• @Martin: Maybe it should be just undeleted? – Asaf Karagila Mod Aug 5 '14 at 8:55
• @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. – Martin Sleziak Aug 5 '14 at 9:01
• @Martin: One more. – Asaf Karagila Mod Aug 5 '14 at 9:07
• Undeleted. ${}$ – Gerry Myerson Aug 5 '14 at 9:29
• Thanks Asaf, Martin, and Gerry! – user26857 Aug 5 '14 at 9:30
• Some suggestions in the comments here meta.math.stackexchange.com/q/8852/27978 – copper.hat Aug 5 '14 at 16:22
• @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. – user26857 Aug 6 '14 at 21:16
• It seems like a remarkable coincidence otherwise! (Yes, I only just remembered that he changed his username before that.) – 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 $\mathfrak{a} \subseteq \text{Jac}(A)$?

tagged:

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 $a\subseteq m$.