My immediate reason for starting this thread is that the question http://mathoverflow.net/questions/84058/are-all-commutative-rngs-ideals-of-commutative-rings 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.

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

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 math.stackexchange.com.

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?

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.

@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).

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.

Math.stackexchange.com would have been a great fit for this question.

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?

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.

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. http://mathoverflow.net/questions/78596/

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. http://mathoverflow.net/questions/83714/

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. http://mathoverflow.net/questions/82910

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

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.)

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

@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'.

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.

+10^{10}