Talk:Krull's Theorem

This is the first proof I've written. I have a couple questions. First, I know that the section where we show that $I$ is a proper ideal does not meet the house style. But using the house style would expand it so much, and I think an expanded version would just interfere with the overall flow of the proof. Is there some good way to compress these kinds of sub-proofs? Maybe indentation, or collapsible boxes? Second, is there a better way to introduce the results of the sub-arguments? If I were writing a straight proof, I would just charge right ahead and leave out comments like "$I$ is a proper ideal" until we actually have established that. But proofs are way more comprehensible if you have some idea of what each subsection is trying to accomplish before you dive into reading it. So, is there a house standard for these kinds of guiding comments? Jlb (talk) 22:32, 9 January 2016 (UTC)


 * I have moved your comment here into the Talk page, where it is better suited.


 * There are several aspects to house style, some of which I have fixed:
 * No underscores in links, e.g.  should be  .  This is rigorously enforced for a number of reasons, the best one being that maintenance becomes so much easier when it is possible to search for all instances of a string and know there are no underscores.


 * When linking a proof, it is appropriate to use just the link, with its capitalisation as given, e.g.  not  . (The same does not apply to links to Definition pages which have a display string given, so, for example, group would be correct.


 * When a page is a subpage of another, e.g.  you will often find there is a redirect containing the "actual" name of the concept which is preferred, that is, in this case:  .  Again, this is done for long-term maintenance reasons -- it is easier to move pages around which are part of a larger definitional structure if they are linked to via a short-form redirect -- then all we need to do is change the destination of the redirect rather than trawl through the pages that use the full form. When pages are moved multiple times this can be significant.


 * At the bottom,  is to be placed. It puts the Halmos symbol in place. We do that rather than use the "blacksquare" $\LaTeX$ code directly because if we decided to change the form of our "end-of-proof" symbol to something else (not immediately likely, but you never know), we do it in one place and job done.


 * There is an extra line inserted between the end of one section and the beginning of the other, to space it out better.


 * Sections of proofs are better set up using section headings rather than just putting them in bold. We have stylistic purposes for bold already, and we try to avoid section headings as one of them. MediaWiki does section headings perfectly well, so we exploit that functionality and thus ensure stylistic consistency across the wiki.


 * Every time a concept is invoked, a link is placed to the page defining it, even if that concept appears multiple times on the same page. (People who have grown up with Wikipedia may have problems with this, for whatever reasons I don't know.)  There are cogent reasons which can be found in various discussions of the rules scattered somewhere around in the talk pages. I have started going through this page and filling them in, but I have not checked it rigorously, hence the template has been left up. Most importantly, note that at least one Category invocation needs to be added to all pages, for apparent reasons. I have added this page to "Ideal Theory"


 * Now for the main point. It all comes down to stylistic preference. You say "I think an expanded version would just interfere with the overall flow of the proof", I say "many bunched up statements all on one line with no gaps between them is unreadable and ugly." You may wish there be room to compromise, but unfortunately those wishes are in vain.


 * house style demands a line space between all sentences, and for all sentences to be simple statements (or at be a simple instance of an implication). It is a thesis of mine that information is most easily assimilated when there is a nice big gap between each statement, and that each statement is simple. This, as I say, is completely non-negotiable.


 * If you are concerned that arguments are too long, and they need to be bunched up so as to see the whole thing on one page (I used to work with software engineers who believed that cramming everything on one page was an acceptable way of presenting their programs stylistically, and to cut a long story short, their point of view did not prevail), then consider breaking up large arguments into smaller sections which can be defined as separate proof pages in their own right. For example, maybe "Every non-empty chain of ideals has upper bound" would be an appropriate page to extract into its own page and establish as a proof, which can then be linked to from this page. Then the proof would be simply: a) $P$ is non-empty as it has $0$ in it, b) link to "every non-empty chain of ideals has upper bound", c) $P$ then satisfies the conditions so as to be able to apply Zorn's Lemma (these conditions can be stated somewhere above), therefore d) the money shot.


 * Finally, it does of course need to be pointed out that once a page has been posted up to it is the property of anyone to edit as they see fit, and however much you like the style to be bunched up, if someone then decides that it looks better spread out will be quite at liberty to do so. On the other hand, it does not go the other way -- if an argument looks too spaced out to you and you think it would be better bunched up, your edits are likely to be reverted as they will then contravene house style. --prime mover (talk) 23:27, 9 January 2016 (UTC)


 * Thanks, this was helpful. I expanded out the argument, adding more organizational statements and putting some of the intermediate statements in headers. I agree that it's much more readable when spread out. (I still do wish these subsections could be collapsible: if you don't understand a statement, expand out the justification. If you see why it's true, just keep reading.) I also added more links to to various math terms. Does the updated version look good? -Jlb (talk) 03:51, 10 January 2016 (UTC)


 * Totally good.
 * I hear what you say about collapsible sections, and we do have an extension that allows for this, but I have never got to grips with how they are implemented (and besides I prefer to have a page which has everything on it without having to expand it out, but that's just me).
 * You might want to put a message on Lord_Farin's talk page as he did the work on this and may be able to help. Mind, he's not as active as he used to be so isn't on line all the time. --prime mover (talk) 10:36, 10 January 2016 (UTC)--prime mover (talk) 10:36, 10 January 2016 (UTC)

Make this considerably shorter by linking to other pages
I strongly suggest to make this shorter; e.g. using Increasing Union of Ideals is Ideal (but beware: that page only does the case of a sequence of ideals). The work done here to prove that the union is an ideal is not in vain; it can safely be copied to a page who proves that separately. --barto (talk) 13:38, 3 February 2017 (EST)


 * Exactly the thing. --prime mover (talk) 15:09, 3 February 2017 (EST)


 * Done. The moderation comments at the top may now apply to Union of Chain of Ideals is Ideal and Union of Chain of Proper Ideals is Proper Ideal. --barto (talk) 06:49, 29 July 2017 (EDT)

Ring with unity
The proof broke at the step where it was shown that the upper bound is proper. Indeed, in a ring without unity, the union of a chain of proper ideals may equal the entire ring. So I added the condition that $R$ must have $1$. --barto (talk) 08:42, 29 July 2017 (EDT)