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Yesterday the moderator Alex Gruber decided to merge this question Example of integral domain with infinitely ascending chain of ideals. which explicitly asks for an example of a non-noetherian ring other than $K[X_1,\dots,X_n,\dots]$ to this one Example of a ring with an infinite inclusion chain of ideals which asks for an example of non-noetherian ring.

As a matter of fact, the question Example of a ring with an infinite inclusion chain of ideals was closed as off-topic for obvious reasons and then deleted. (The example $K[X_1,\dots,X_n,\dots]$ can be found in at least two other older threads on M.SE and at some moment I've provided links to these threads. A new one is Neither Artinian nor Noetherian rings. Also Ideals of a polynomial ring in infinitely many variables which are not finitely generated.) I don't why the moderator has reopened it, but I'd be glad to know his reasons.

While I can agree that this answer could count as a different example from the usual one, the answerer has not done a great job: I can't see any "bi-infinite" (whatever it means) chain of ideals in that ring, and I also can't see a proof showing that the chain (what chain?) is strictly ascending.

On the other side, in the other thread I've posted this answer which satisfied all the requirements: the ring is not noetherian (and this is easily seen) and it's not even isomorphic to $K[X_1,\dots,X_n,\dots]$ (with proof).
Now my answer looks strange: why am I proving that the given example is not isomorphic to $K[X_1,\dots,X_n,\dots]$? The present question asks only for an ascending chain of ideals! (This is why I've to delete it.)

I've posted this in order to find out the moderator arguments for merging the two questions and because my flags telling the above story had no concrete echo so far. (I'm happy that at least some of them were not declined.)

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    $\begingroup$ I have added (specific-question) tag, since I assume that you are asking mainly about this particular case. (I.e., the linked question does not serve merely as an illustration of a more general issue which you want to discuss.) $\endgroup$
    – Martin Sleziak
    Feb 29, 2016 at 9:41
  • $\begingroup$ The third paragraph seems unnecessary and quite unrelated to the rest of this question. Maybe you could post a gentler version of this as a comment to the answer. $\endgroup$
    – user160629
    Feb 29, 2016 at 11:40
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    $\begingroup$ Whatever the argument for merging, I voted to close the question on main. It is exemplary bad. $\endgroup$
    – quid Mod
    Feb 29, 2016 at 12:31
  • $\begingroup$ I thought you'd know it, the revision history shows who closed/reopened/deleted/undeleted a question. $\endgroup$
    – Daniel Fischer
    Feb 29, 2016 at 15:04
  • $\begingroup$ @AlexanderGruber Thanks for not posting an answer to this question, and for provoking some mess with your uninspired merging. $\endgroup$
    – user26857
    Mar 5, 2016 at 15:23

1 Answer 1

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I flagged this post for merging, so I will try to provide my original rationale here. However, I now realize I was mistaken. I shall be more careful in the future when suggesting a merge, since undoing a merge is nearly impossible. (It has to be done by an SE employee, it can not be done by a mod)

In my further explanation, I will refer to Example of a ring with an infinite inclusion chain of ideals. (This one does include $K[X_1, X_2, \cdots, X_n]$) as question 1, and to Example of integral domain with infinitely ascending chain of ideals. (This one excludes $K[X_1, X_2, \cdots, X_n]$, and this post was merged into question 1.) as question 2.

First, my rationale for the undeletion of question 1: Actually it is quite simple. I though the answers given were worth retaining, and therefore I voted to undelete. I would never delete a question with good answers, and if I come across a deleted question with good answers, I vote to undelete it.

Now, my rationale for the merging flag:

  • I did not take into account that an example other than $K[X_1, X_2, \cdots, X_n]$ meant an example not isomorphic to $K[X_1, X_2, \cdots, X_n]$.
  • Therefore I thought that it was just one example that doesn't belong to question 2, but does belong to question 1.
  • Also, merging the two questions didn't create any duplicate answers.
  • Therefore I'd seem better to have one large collection of the examples instead of two questions with examples. Further, this would also reduce the risk for the answers from being deleted again.

However, point 1 in this list is an important one. My apologies for missing it.

I'd like to say a few more words in defense of Alexander Gruber. The OP wrote:

I don't why the moderator has reopened it, but I'd be glad to know his reasons.

For the simple reason that it was impossible to merge the question without reopening it.

  • Question 1 was closed as a duplicate of Question 2. Therefore, the post had to be reopened.
  • Then Question 2 was closed as a duplicate of Question 1.
  • Then Question 2 was merged into Question 1.
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    $\begingroup$ The question is just bad, everybody that reopens it should at the very least improve the questions. By now it very much seems it was not even an honest question, which in part explains why it is as bad as it is. Moreover the idea to create a list of non-noetherian rings with ideal witnessing it is not a good one (way too broad). $\endgroup$
    – quid Mod
    Feb 29, 2016 at 19:11
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    $\begingroup$ @quid Why do you think it was a dishonest question? Because the question author was deleted? Besides, the other question did not show much effort either (only that it could not be a principal ideal domain). $\endgroup$
    – wythagoras
    Feb 29, 2016 at 19:13
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    $\begingroup$ For this there could be many reasons, but note the account having given the accepted answer, and look what else they did that day (pay special attention to the timespans that elapsed between Q&A). $\endgroup$
    – quid Mod
    Feb 29, 2016 at 19:24
  • $\begingroup$ On the other question, it is not great either, but relatively better. It asks for integral domains. (The other example does not even make precise if it is about commutative rings or not.) It excludes the most common example, even though in a clumsy way. It shows some thought. But, if its up to me, now in this convoluted situation I'd just delete both, problem solved. $\endgroup$
    – quid Mod
    Feb 29, 2016 at 19:31
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    $\begingroup$ @wythagoras Thanks for your detailed answer. But the one who merged the questions was the moderator Alex Gruber not you. $\endgroup$
    – user26857
    Feb 29, 2016 at 19:50
  • $\begingroup$ @user26857 You are welcome, but I feel that it is my fault just as much as his since I flagged it. Maybe the moderator should have declined my flag. Regarding the second comment, I meant that I already understood that there was a sockpuppet. $\endgroup$
    – wythagoras
    Feb 29, 2016 at 19:53
  • $\begingroup$ @wythagoras +1 (I'm still waiting for an answer posted by the moderator who merged the two questions, but he is usually absent from M.SE, so this could take a long while.) $\endgroup$
    – user26857
    Mar 1, 2016 at 15:30

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