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    • CommentAuthortheojf
    • CommentTimeDec 21st 2011 edited

    My immediate reason for starting this thread is that the question currently has two votes to close as "too localized". I think it is a very reasonable question, especially for an early graduate student. But I don't particular care about this question (which certainly could be improved). Rather, I wanted to voice my opinion that MO should continue to support questions from young graduate students.

    • CommentAuthorKevin Lin
    • CommentTimeDec 21st 2011 edited

    The "Why not real names?" thread is a bit tl;dr for me, but one thing that young graduate students could do to identify themselves as such would be to use their real names and/or to identify themselves as such in their user profiles. (Which I see the author of the question that Theo links to has done.)


    Too, uhm, localized? I guess this is just somebody trying to make a word pun.

    • CommentAuthorYemon Choi
    • CommentTimeDec 21st 2011

    As one of the people who voted to close: firstly, the reason I thought it should be closed is that it's frankly a bit below the level I think MO should take as its average; and the "too localized" option seemed the closest fit.

    I am inclined to think that if one knows what rings with and without identity are, one should have been shown the construction of "adjoining a 1" - perhaps I am misjudging where such things occur in one's 1st degree. If one has seen the construction, then it shouldn't be such a big jump when looking for the left adjoint to try this particular construction (rather than say SAFT or GAFT).

    This is not meant to dissuade the OP from posting other questions, of course. I just felt that this one was a case of "could have thought a bit harder" - though maybe I'm biased, since the unitization is a common device for functional analysts and I guess it might not get mentioned explicitly in algebra courses.


    Theo, I think it would be a bad policy to allow any question from young grad students. It's the content of the question that should matter, not the question-asker's background. This question in particular would probably be a better fit at


    one should have been shown the construction of "adjoining a 1"

    I don't know of a single undergraduate or graduate subject to (normally) contain this construction. Of course, there are many places in an algebra course where it could reasonably fit in as an exercise, but it is not too often used as such.

    It's indeed a question more suitable for M.SE, but whether a question fits into MO or not should be obvious to the author before asking, not in hindsight, and this one isn't really distinguishable from a legit MO question before one knows the answer.

    • CommentAuthorYemon Choi
    • CommentTimeDec 21st 2011 edited

    Darij: the unitization of a Banach algebra almost always crops up in any course on such objects (so that one can use basic ring-with-identity theory to get e.g. Gelfand-Mazur, Gelfand representation etc). Granted, that is the direct sum of the algebra with the ground field, but the jump from there to rings is a small one. I thought the unitization of a ring-without-identity might have been in Beachy's notes on Abstract Algebra? perhaps I have misremembered.

    As for hindsight: I take your point, but still feel that the original question doesn't require one to know something from a book, or a particularly obscure trick. You want to embed a ring without identity, R, into some ring S that does have an identity, and you want the universal such one. So given any map from R to some other T with identity, what would the extension to S look like? well, it would have to send the 1 of S to the 1 of T, and at this point you look at the unital subring of S containing R and go "oh look, that would have worked".

    I'm not saying that the OP should have immediately known the answer, but I do think it is a bit basic for MO. My understanding is that we still close questions where the questioner doesn't know that the question is not-at-MO-level - admittedly they are often worded more poorly and less thoughtfully than this one, but where does one draw the line?

    • CommentAuthorquid
    • CommentTimeDec 22nd 2011

    @Kevin Lin: your suggestion assumes that many/most people refer to the user profile. In my observation this is not the case. IMO the only way around this is to include the information directly in the body of the question; the status of early graduate student, not the real name, as personal motivation.

    The suggestion to use real names if anything works against what Theo wants. Because, either one is in the situation of 'violation' of it or one runs the risk of public humiliation.

    @Theo: +1.


    I agree with Theo, and I think Darij puts his finger on it when he says:

    whether a question fits into MO or not should be obvious to the author before asking, not in hindsight, and this one isn't really distinguishable from a legit MO question before one knows the answer.

    Also, remember that we don't all follow a linear progression through a single mathematical speciality. Fields medallists might want to ask "early graduate level" questions about subjects other than their own. And anyway, the concept of "graduate level" is pretty flaky: see my comment here.

    Some of the rest of the thread just linked to is relevant too, and I'll quote Kevin Walker again:

    In the early days of MO I remember people often saying that the ideal MO question is one which you (an expert in field X but a newcomer to field Y) would have difficulty with but could be answered quickly and easily by an expert (in field Y).

    That might not be particularly relevant to this question, but it is relevant to the general issue of what "level" we insist on.

    Finally, let's not overlook the fact that the OP wasn't merely asking for a left adjoint to the inclusion Ring --> Rng. He was also asking whether/why the unit is injective. This is more substantial, and there are probably ways of seeing it that don't involve an explicit construction of the adjoint. By way of comparison, it's a fact that for all but two exceptional finitary algebraic theories, the unit of the free-forgetful adjunction between the category of algebras and the category of sets is monic. This can be proved with no explicit construction.

    Yemon Choi: "As one of the people who voted to close: firstly, the reason I thought it should be closed is that it's frankly a bit below the level I think MO should take as its average; and the "too localized" option seemed the closest fit."

    If we wouldn't allow for below average questions, MO would collaps to a single open question. ;-)
    • CommentAuthorquid
    • CommentTimeDec 22nd 2011

    @Michael: while what you say is somehow amusing, it is not really linked to what Yemon said as he said that it is below what (my emphasize) 'MO should take as its average' as opposed to below what the average is (which in principle could be below/above/equal to well perhaps even incomparable to what, in Yemon's opinion, should be the average).

    @General: Also while I expressed my general agreement with Theo, and also have nothing against this question, I would also like to say that this does not mean in any way that I think somebody voting to close this was in any sense 'wrong'. I think one general issue is often that, along with what Tom Leinster said including the quote by Kevin Walker, is that in some sense a reasonable MO is simple for somebody here (to me that's sort of the entire point of the website). So, that a reasoning 'this is trivial for me it should be closed' is not always reasonable; while I think it happens sometimes also in case where it is perhaps not reasonable (actually I rather know than think that it happens, because I did it; as sometimes the dynamic is like this).

    Perhaps a test for the apropriateness of a question [purely meant for mathematical questions] could be how ones reaction towards a collegue (in a different research area) would be when she or he would knock at ones office door and say:

    I think your interests and areas of knowledge are closer to this problem I have. Do you happen to know an answer to this question?

    If even the idea of this happening for a particular question is rather inconceiveable, then the question is likely really off-topic. However, if this is not so, then the question is why should it be off-topic here. Even if one in theory could answer (in the sense that it would be true): there is no need to ask me, just ask one of the better undergraduates taking a course on subject S.

    Because, regarding mathematical question [soft is soemthing else entirely], my impression of the idea behind MO is that it is so to say 'one giant math department' and it thus seems apt to not impose stricter standards than one would have in personal mathematical conversations with collegues (if they happen with the specific intent of asking a serious question).


    @Yemon: okay, I can't argue against that having never successfully mastered a functional analysis course. But ironically, I rarely see this thing where I'd expect it to see, that is, in algebra courses. The answer to the question is indeed extremely simple and elementary, but requires a little bit of thinking-outside-the-box and is unexpected if you are used to thinking of rings as homogeneous rigid blocks (and you do get this kind of thinking if you read a bad or superficial introduction to algebra, or try to apply your "fields" intuition to rings).

    I find quid's knock-at-the-door test a really good one, although a bit subjective, because there is much variance in what we would find appropriate (depending on time, mood and personality).


    I agree with Theo. While the question might be approaching the too-low-level threshold, I think it's safely on the high side of that threshold.

    Tom Leinster saved me the trouble of making a point about mathematicians well beyond graduate school asking questions about fields where they have little background. So instead let me make a related point: Experts in field X are in a poor position to judge whether questions in field X are too elementary. Too much knowledge clouds our ability to distinguish things which are truly obvious from things which are merely obvious in hindsight.

    Occasionally I come across a question on MO which seems way too elementary to me, and yet for some strange reason has garnered no votes to close. This only happens for questions in my particular areas of expertise. In these cases I think the many non-voters-to-close are correct and my judgement has been clouded by too much familiarity with the subject.

    • CommentAuthorbsteinberg
    • CommentTimeDec 22nd 2011 edited
    I was one who voted to close. I must confess I was unaware this was a grad student. My reasoning was that the level of this question seemed comparable to many others that were closed in MO as being more appropriate to mathstackexchange. In fact the construction of the unitalization of a ring can be found easily via a websearch since older algebra books (before Bourbaki influenced everybody to assume rings have a 1 and before category theorists decided to remove the "i" from rings without identity) will have the construction.

    In fact, things like constructing left and right adjoint for similar situations to this (and maybe even this example) were exercise in Bergman's category theory class at Berkeley. The OP gives in the question the "abstract nonsense" approach to constructing it and the meat of his question is why is the unit of the adjunction monic. I guess had he prefaced the question by saying "I am a grad student working in logic/category theory and I need to understand for my thesis whether the unit of the adjunction from non-unital rings to unital rings is monic, I have found a presentation for the left adjoint but I do not see why it is monic can you help me?" Even then I would prefer the statement that I looked at Jacobson, Kaplansky, Herstein,Cohn and Lam and could not find a construction (which is unlikely to be the case since I am sure at least one of those). In other words, it should be clear that the left adjoint of the forgetful functor from non-unital rings to rings should be well understood and that a little googling will find it.

    But I do feel bad since this is a grad student and I would have give more leniency had I noticed it. If it is closed, I would be willing to reopen.

    ASIDE: sorry as a semigroup theorist, I can't bring myself to use the terms "rng" or "rig" or one day I will find my self studying "grp"s (no offense to Gerhard and let me be thankful this holiday season "i" doesn't in the word group anywhere :) :). Nb. I am a fan of category theory and Bill Lawvere so do not take this the wrong way!
    • CommentAuthorMariano
    • CommentTimeDec 22nd 2011
  would have been a great fit for this question.


    Mariano, you probably won't find many people disagreeing with your statement, but I think it's missing the point:

    1. Theo said in his opening statement that he didn't particularly care about this question.

    2. The fact that a question is a good fit for math.stackexchange doesn't mean it's a bad fit for us. There's bound to be some overlap. There should be. The alternative is that there's a gap between the two sites, into which some questions will fall.

    3. As Darij pointed out, the judgements we can make after a question is answered shouldn't be important. It's how it looks at the time of asking that matters.

    • CommentAuthorMariano
    • CommentTimeDec 22nd 2011

    Dear Tom,

    I would also have made the observation that M.SE would have been a good/better fit for the question before the question was answered. I do not mind at all the presence of the question, and closing it now that it has been answered seems to me slightly pointless, really :)

    I also think there is bound to be some overlap, and that in fact it is probably good that there be one. But this particular question does not strike me as a "graduate student" question really. Ins't this a standard thing to see in an undergraduate course in which one first meets rings?

    • CommentAuthorquid
    • CommentTimeDec 22nd 2011 edited

    Mariano says

    I do not mind at all the presence of the question

    To me this is an important point. For some questions I can see why some (sometimes including me) do mind to have them on MO, or how having them could have negative consequences. For this question I cannot see this at all. So, if there is no reason against it, why not have it.

    And there is also another aspect that can be a bit complicated but in this case is clear (positively), and is related to Yemon's question where one draws the line. To me it also makes a considerable difference wether some user might occassionally ask also something (in a well phrased way) that is perhaps just a bit too simple, or whether one rather gets the impression the user's general math background is somehow insufficient for MO. In the latter case I think it makes sense to direct (in a diplomatic way, preferably) the user in its entirety so to say to math.SE instead of MO. Here, this seems clearly not case.

    To insist that individual users split their activity over the two sites just to obtain slightly better 'fits' seems to impose an unecessary burden, on purely practical grounds. (One could, but does not have too IMO, reconsider this after changing to SE2.0 when this is more conveniently possible.)


    I agree with Theo, Tom, and Kevin Walker. On the whole I find MO less interesting than it was it's first year, and I'd like there to be more questions of this level.

    • CommentAuthorbsteinberg
    • CommentTimeDec 22nd 2011
    I agree. I will stick to closing only spammy/HW questions or things that are really undergrad level.
    • CommentAuthorgrp
    • CommentTimeDec 22nd 2011 edited
    I would like to point out three examples involving question closure; there is a lot of grey area in terms of question content for deciding whether to close a question or deem it appopriate for MathOverflow, and sometimes mistakes are made.

    The first one is where I expressed doubt that the question is appopriate for MathOverflow; regardless of my opinion, Noam Elkies came up with a nice solution for it, and the poster did nothing wrong except (in my opinion) provide sufficient motivation.

    The second one is one which I thought should be closed, then I tried to answer it and found it more challenging. There was an initial comment which was later removed, saying why the post was closed.

    The third one is one that probably should be closed, but I include it because of Gerry Myerson's point. Many people who are not high reputation users are unaware of the votes to close/open a question; I would like more transparency in the closing/opening process, and it as simple as leaving a brief remark. Even if I disagree with it, I appreciate Mark Sapir's "Not research. Voting to close" remarks: at least I can see what possible rationale there is.

    Finally, I would like to make MO available to young graduate students. Also old graduate students. Also students whose profession is not necessarily mathematics. Also people outside academia who have an interest in mathematics. Also a wider class of people who can contribute mathematically interesting content. I worry that this thread suggests that MathOverflow be reserved for the research mathematician, or cater primarily to their interests; if one goes too far in that direction, it may go back to a cadre of algebraic geometers helping each other and helping younger people become algebraic geometers; there's nothing wrong in that, but it makes MathOverflow much less useful in my opinion.
    As a helpful suggestion to those who want to see "raising the average level" on MathOverflow, I recommend posting questions and answers of higher level as well.

    Gerhard "Also Available To Research Mathematicians" Paseman, 2011.12.22
    • CommentAuthorMariano
    • CommentTimeDec 22nd 2011 edited

    grp quoth

    I worry that this thread suggests that MathOverflow be reserved for the research mathematician, or cater primarily to their interests

    Hmm. Isn't that the whole point of MO?! (For sufficiently non-idiotic meanings of the word reserved)

    • CommentAuthorquid
    • CommentTimeDec 22nd 2011 edited

    Mariano: did you perhaps mean to say "grp grumped" instead of "quid qouth"? In any case I did not say what you quote, Gerhard did.

    Or, in brief: Mariano mumbled "quid quoth," yet grp grumped.

    (It goes without saying 'grumped' and 'mumbled' is chosen for the alliteration only.)

    • CommentAuthorMariano
    • CommentTimeDec 22nd 2011

    (Heh. Sorry! Fixed...)

    • CommentAuthorgrp
    • CommentTimeDec 22nd 2011
    If alliteration is the main goal, I suggest "griped" as an alternative to "grumped". I appreciate, however, that that was not Mariano's main goal.

    To respond to Mariano: I think there are enough community members who view MathOverflow as a domain primarily if not exclusively for research mathematicians, and that such a viewpoint tends to shift behaviour in the direction from primary to exclusive. I respect such a viewpoint, but hasten to add that MathOverflow is more interesting for such mathematicians and useful for a larger segment of society if MathOverflow is extended to include those with mathematically interesting content to contribute; the previous viewpoint (in my opinion) does not seem flexible enough to admit such an extension.

    Of course, I am also not wanting MathOverflow to turn into (something like the USENET newsgroup) sci.math, or even sci.math.research in its current form (at this writing). The tension that currently exists among different subsets of the community helps to keep the MathOverflow quality level of content from sinking too low to be useful to many and raising too high (meaning here requiring much specialized knowledge) to be of wide interest. I think it is ok for the pendulum of opinion about level to swing if it does so slowly, and not too far.

    Gerhard "Ever Witness A Lassie's Vacillations?" Paseman, 2011.12.22
    • CommentAuthortheojf
    • CommentTimeDec 22nd 2011

    I should mention that I completely agree that this question in particular would be very appropriate for math.stackexchange. Indeed, most of the questions that I hoped to protect when starting this thread would also be appropriate for math.stackexchange. I'm not a user over there (I probably will become one after we migrate and I can link my accounts), so I'm not an expert about the level there. But from what I've read on meta here, I have concluded that it would be very good if the sites do continue to encourage some overlap. Namely, I would like to encourage overlap in "graduate level" (even "early graduate level") questions.

    • CommentAuthorquid
    • CommentTimeDec 23rd 2011

    @Gerhard: okay, good suggestion, final try: Mariano mumbled // ''quid quoth" // Qui? Quoi? // grp griped!

    To not just (ab)use this board for showing my dilletantic attemps at writing poetically, something on-topic too:

    I feel I am in some sense rather with Mariano, yet only would prefer to stress mathematics over the mathematicians. To me saying 'research mathematician' has some feel of focusing on what type of job people have. However, that this is not the main point, in particular in view of the add-on qualification, is not some point of disagreement it seems.

    In my opinion MO should be:

    For everybody interested in asking, answering or just reading focused and precise questions and answers on mathematics of a sufficiently high level presented in the form as (or sufficiently close to it) commonly used by (a meaningful subset of) the mathematical community.

    In other words I strongly prefer and in general (exceptions can always be made) would even somewhat insist that an MO-participant is somewhat 'fluent in the language(s) of research mathematicians'.

    Everything that goes under 'soft question' (including funny lists, carrer advice, community standards, tools for this or that, and so on) is IMO in principle off-topic, yet can and perhaps should be addmitted on occassion, as an exception.
    In other words, for a precise mathematical question it should be only closed if there is a reason to do so, in the sense that its existence on MO seems likely to negatively affect it and/or the question is unclear/incomprehensible/unmotivated (in some sense this is implicty in the 'precise').
    By contrast, for evrything else there needs to be a convincing argument why it should be on MO, so a reason to stay open. (And it helps in building the community, can be a convincing argument, so my position does not by itself exclude evry big-list, say. Likewise, it seems of interest to many or of great interest to some and is unlikely to cause problems, can also be a convincing argument. In any case I would prefer not to start a discussion on soft questions in addition, I only added this to make the 'reason to close' -- 'reason to stay open' dichotomy transparent.)

    So what is the sufficiently 'high level'. Following the position I sketched above it does not matter that much. It certainly includes research level mathematics. Whether or not it should include graduate level mathematics is somehow the question. But, at the very least I cannot see a reason against it (I can infact see some reasons for it, but this is not key). So, whether MO is abstractly for it or not, in any case there seems no reasons to excluded it actively (viz. close it).

    Two more points related to this: a well-written graduate level question is IMO for not highly informed observers indistinguishable from a research level question (as said by many in the above discussion). Therefore, all arguments that might be made to the extent that since they are closer on the level scale they are more likely to attract calculus homework problems and related unwanted content, seem insignificant to me. By contrast, poorly phrased graduate level questions (but also poorly phrased research level questions!) can for not highly informed observers, and sometimes even for those, be indistinguishable from basic homework questions.

    Thus, it seems to me that the overall quality of the site, and the volume of unwanted content, will depend much more on the standard of presentation (motivation/background) of individual questions (and answers) than on enforcing some notion of 'level'.

    MO has moved from "of interest to research mathematicians" to "research mathematics". This lead to a strong increase in standards even for clearly specified hard questions. Many research mathematicians are curious about fields distinct from their research. So naturally, questions about this other field are not at the level of research mathematics. This effect is especially strong in fields not universally part of a mathematicians education. For example, many mathematicians are relatively ignorant about the foundation of mathematics, and there were lots of popular questions on logic that were at most undergraduate level in the old days of MO.

    The upside of this change is that MO is now a more effective tool for research. The downside is that MO has become less interesting. In the long run, this may reduce the number of competent mathematicians participating and even make MO less useful as a tool for research.

    Another effect of the change in MO is that even hard answers to hard research questions have sometimes become less interesting. There seems to be an emerging consensus that answers should be geared solely to helping the questioner understanding the answer (see the the discussion on anonymity here and "the background of the questioner"). IMO, one should try to formulate answers in a way that makes them interesting to the wider MO community, even when the brilliant questioner might be helped with a few cryptic remarks uncomprehensible to outside observers.

    The upside of this change is that MO is now a more effective tool for research. The downside is that MO has become less interesting.

    If it really is an either-or (which I doubt) then my vote is on the former. I have lots of places where I can find interesting stuff, even within mathematics, but not so many that are truly useful tools in my day-to-day research.


    @Andrew: +10.



    • CommentAuthorquid
    • CommentTimeDec 24th 2011

    Great we have so much agreement here!

    Just that it does not get lost, or casual observers get a wrong impression, let me quote Andrew, with whom I agree as well, just with adding some emphasize.

    If it really is an either-or (which I doubt) then my vote is on the former. I have lots of places where I can find interesting stuff, even within mathematics, but not so many that are truly useful tools in my day-to-day research.