13
$\begingroup$

A debate in comments was raised by the question

Showing that the only idempotents in $R$ are zero and one

the relevant part of which is

Let $R$ be a ring with $1$ and suppose $R$ has no zero divisors. Show that the only idempotents in $R$ are $0$ and $1$.

One of the answers went straight by assuming the ring $R$ is commutative, which, in my opinion, is too strong an assumption about the question at hand.

The aim of this metaquestion is thus establishing a general policy. It's true that conventions about naming rings are various and contradictory around the world, and I find the answer perfectly acceptable (albeit using a sledgehammer, in the particular case), provided “If the ring is commutative…” is added at the beginning.

To me, for example, rings are not commutative unless stated and I can't get from the question any hint about the OP considering this hypothesis.

| |
$\endgroup$
  • 5
    $\begingroup$ You should probably also add: should rings be considered associative? Should they be rings with $1$ and homs preserving $1$? In any case use determines meaning. And in my experience, in a general level forum like this, ring almost always means commutative ring with 1 and homs preserving 1. Otherwise many answers would be wrong. $\endgroup$ – Bill Dubuque Mar 4 '14 at 15:47
  • 16
    $\begingroup$ @BillDubuque Non associative rings are much more specialized and probably the poster would mention it. I consider your comment not at all constructive. Indeed I avoided mentioning the presence of $1$ (which was stated in the question). $\endgroup$ – egreg Mar 4 '14 at 15:51
  • 4
    $\begingroup$ @Bill: I disagree. Care to give an example of an answer that would be wrong, unless you tacitly assume commutativity? IIRC in the ring related questions I've been a party of (admittedly not too many), this has hardly ever been an issue. Either the ring has been implicitly specified (when all and sundry will immediately know whether it is commutative or not), or the OP stated it in the question. Wisely so IMHO. $\endgroup$ – Jyrki Lahtonen Mar 4 '14 at 16:04
  • 10
    $\begingroup$ The tag-wiki for the tag (rings) says, that the tag (rngs) should be used for questions which do not have unity. It also says that rings are not necessarily commutative. (I do not claim that the tag-wikis should be taken as guidance in general - they can of course be changed - but I thought this is worth mentioning.) $\endgroup$ – Martin Sleziak Mar 4 '14 at 16:04
  • $\begingroup$ @Jyrki Do you also disagree about associativity too? $\endgroup$ – Bill Dubuque Mar 4 '14 at 16:11
  • 3
    $\begingroup$ This probably depends on culture - for example, I would not assume that "ring" means commutative, but I would assume it was associative (I didn't even realize this was up for discussion!). I would probably assume it had $1$ by accident, although I don't think this should be part of the definition. If I was on the ball I would leave a comment to the OP to clarify, but it would be easy to not realize I was adding an assumption (particularly in the case of associativity, where I have never seen a definition omitting it!) $\endgroup$ – mdp Mar 4 '14 at 16:12
  • $\begingroup$ @Bill: No. Amending "my creed" accordingly :-) $\endgroup$ – Jyrki Lahtonen Mar 4 '14 at 16:12
  • $\begingroup$ @JyrkiLahtonen Do you disagree that by far and away the majority of rings discussed in MSE and sci.math are commutative? $\endgroup$ – Bill Dubuque Mar 4 '14 at 16:13
  • 7
    $\begingroup$ @Bill: Yes. A majority, but not a vast majority. Most of the questions about rings are about subrings of complex numbers or of subquotient of polynomial rings over a field or residue class rings of integers. But there are many questions also about group rings and matrix rings. The context makes it clear in a vast majority of cases. The questions, where there is no strong contextual inference are IMHO in a small minority. Therefore there is no default assumption about commutativity, and in those cases it should be specified, when/if needed. $\endgroup$ – Jyrki Lahtonen Mar 4 '14 at 16:18
  • 3
    $\begingroup$ I'm glad to see that we agree about that. In any case, we cannot force conventions on questioners. They will use whatever they are familiar with. The best we can do is to encourage questioners to be more explicit when there are possible ambiguities due too differences in conventions (and lack of adequate context to unquestionably disambiguate) $\endgroup$ – Bill Dubuque Mar 4 '14 at 16:31
  • 9
    $\begingroup$ And to elaborate on Bill's comment, in some cases, questioners sometimes don't even know about the other conventions, so we can't reasonably expect them to post unambiguous questions. I make it a point to mention that there are two usual conventions when prompting for clarification. $\endgroup$ – user14972 Mar 4 '14 at 16:57
  • $\begingroup$ @egreg OK: I think that resolves my comment: thanks $\endgroup$ – rschwieb Mar 4 '14 at 19:04
  • 5
    $\begingroup$ Why not query the OP with a comment if their is a question? Or if you write an answer, clearly indicate what assumptions on the ring one is making. $\endgroup$ – Baby Dragon Mar 5 '14 at 20:15
  • 1
    $\begingroup$ I see somebody voted to close this as primarily opinion-based. LOL. What do you expect with a question tagged with discussion and policy? $\endgroup$ – Jyrki Lahtonen Mar 7 '14 at 11:20
  • 1
    $\begingroup$ @BillDubuque Obviously rings are associative. Any married person can tell you that. ba dum... tsssss $\endgroup$ – corsiKa Mar 7 '14 at 17:19
26
$\begingroup$

Posting this as an answer to test how widespread my beliefs/definitions are :-)

My creed (largely those of a loyal disciple of Jacobson's BA I-II):

  1. A ring is not necessarily commutative.
  2. The multiplication of a ring is associative.
  3. A ring has a multiplicative neutral element (often denoted by $1$). The lesser structures are rngs.
  4. A homomorphism of rings maps the neutral element to the neutral element.
  5. But OTOH an integral domain (and a field) is commutative by default.
| |
$\endgroup$
  • 8
    $\begingroup$ I'm currently TA for an algebra course in which 3 and 4 are dropped, and being convinced that this is a good definition (in particular, it means that all ideals are themselves rings). But I'm not fanatical about that. I stand firmly behind 1 and 2. $\endgroup$ – mdp Mar 4 '14 at 16:17
  • 2
    $\begingroup$ What conventions prove most convenient for an abstract algebra textbook need not necessarily bear any relation to the conventions that prove most convenient for a general-level math forum. $\endgroup$ – Bill Dubuque Mar 4 '14 at 16:20
  • 1
    $\begingroup$ @MattPressland I find that removing $1$ just shows an artificial similarity between rings and groups, where subring corresponds to subgroup and ideal to normal subgroup. When you start requiring that ring morphisms take $1$ into $1$, this similarity is lost. $\endgroup$ – egreg Mar 4 '14 at 16:28
  • $\begingroup$ @egreg But we don't require this! (For one thing, neither the domain or the codomain need have a $1$!). $\endgroup$ – mdp Mar 4 '14 at 16:29
  • 3
    $\begingroup$ @MattPressland Yes, I understand it. Try doing sensible module theory on rings without $1$. ;-) $\endgroup$ – egreg Mar 4 '14 at 16:30
  • $\begingroup$ @egreg No thanks! (It doesn't come up in the course). I understand there are pros and cons. ;) $\endgroup$ – mdp Mar 4 '14 at 16:30
  • 10
    $\begingroup$ @BillDubuque While I agree with your sentiment that definition should be determined by usage, I think it would be harmful for the site to diverge from the language used by the wider mathematical community (for example, it would be confusing to new users). While this doesn't help for whether rings have $1$ or not, where there is no general consensus, I'm not aware of anybody who thinks all rings are commutative all of the time, so I think it is best to continue to use "commutative ring" as a special case of "ring". $\endgroup$ – mdp Mar 4 '14 at 16:33
  • 1
    $\begingroup$ @Matt I don't propose to attempt to enforce any convention. Rather, I merely point out my observations about such usage in general level math forums such as sci.math and MSE, from a few decades of experience. Personally, in such forums, I adopt conventions that conform to usage, since that proves most convenient. When there is possible ambiguity I usually state my reading, as I did in the question at hand, by saying $R$ is a domain in my answer (domains are commutative by most definitions). $\endgroup$ – Bill Dubuque Mar 4 '14 at 16:46
  • $\begingroup$ @BillDubuque I am similarly not proposing such a thing. Stating your reading seems like a perfectly good approach (I think I have also done this) although my usual approach is to ask a clarifying question to the OP before answering. Happily, these approaches are not mutually exclusive! My only suggestion that involves actual community action is that, if somebody writes "ring", and it emerges later that they wanted to assume it was commutative, their post should be edited to say "commutative ring" to increase clarity when other people read the post later. $\endgroup$ – mdp Mar 4 '14 at 16:53
  • 6
    $\begingroup$ @BillDubuque You have surely noted that the question doesn't say domain, but no zero divisors, which, IMO, is a clear indicator that commutativity is not assumed. $\endgroup$ – egreg Mar 4 '14 at 16:58
  • 1
    $\begingroup$ @Matt That seems like a fine approach to me. Usually the context is enough to disambiguate. In this case it was not (I don't think that the mention of idempotents need imply that the ring is noncommutative since idempotents also play a key role in commutative rings e.g. they are intimately connected to the Chinese Remainder Theorem, factorizations into coprimes, etc). In any case, I think we should allows answerers to resolve ambiguity however they find useful to do so (even if they did not correctly infer the author's intent, it often leads to enlightening answers). $\endgroup$ – Bill Dubuque Mar 4 '14 at 17:00
  • 4
    $\begingroup$ @egreg The only thing that is "clear" to me is that you and I happen disagree on some subjective matters about conventions - which has little relevance to the general question posed in this meta thread. In any case, I think that arguing about conventions is one of the most fruitless arguments that one can engage in. $\endgroup$ – Bill Dubuque Mar 4 '14 at 17:06
14
$\begingroup$

If the question is about algebraic geometry or commutative algebra (duh!), you may assume that rings are commutative.
Else you may not.

| |
$\endgroup$
  • 1
    $\begingroup$ And in algebraic geometry, you can even assume they are unital (unless specified otherwise at least). $\endgroup$ – Tobias Kildetoft Mar 6 '14 at 8:07
  • $\begingroup$ And also homs preserve $1$. $\endgroup$ – Levent Jul 5 '16 at 21:01
11
$\begingroup$

I didn't know that a ring was ever assumed to be commutative. I know that there isn't a clear convention about a ring having an identity. Anyway, I think when we come across questions where the definition is unclear, then either the OP should clarify or an answerer should make clear what definition s(he) is using (and maybe even provide an answer/comment to each case). I guess it all comes down to what you are doing.

As Bill Dubuque says in his comment to the other answer: What conventions prove most convenient for an abstract algebra textbook need not necessarily bear any relation to the conventions that prove most convenient for a general-level math forum. In some cases all rings are assumed to be commutative because that is all you want to consider (See for example: these notes). But clearly in other cases they are not (who would not call the set of $n\times n$ matrices a ring?)

That said I looked up the basic definition of ring various places as to whether or not a ring is in general assumed to be commutative

  • Wikipedia says no.
  • Hungerford's Algebra says no.
  • Rotman's An Introduction to Homological Algebra says no.
  • Rotman's Advanced Modern Algebra says no.
  • Lang's Algebra says no.
  • Dummit and Foote's Abstract Algebra says no.
  • Artin's Algebra states that all ring are assumed to be commutative (but states that in general they aren't assumed to be).
  • Jacobson's Basic Algebra I says no.
  • James Milnes' notes Fields and Galois Theory says no.
  • James Milnes' notes Algebraic Number Theory states that all rings are assumed to be commutative.
  • Joseph Gallian's Contemporary Abstract Algebra says no.
  • I.N. Herstein's Abstract algebra says no.
  • J.B. Fraleigh's A first course in abstract algebra says no.

I invite other people to insert references into this list.

| |
$\endgroup$
  • 4
    $\begingroup$ Right: since this list contains many undergraduate mainstays which do not assume commutativity, it is evidence that the "most beginners are only thinking about commutative rings" theory is pretty dubious. Naturally classes focusing on commutative rings alone may do so, but that group does not seem to be representative of the whole population. So for best preparation and least confusion, it is good for answerers to be mindful of this. $\endgroup$ – rschwieb Mar 5 '14 at 13:53
  • 1
    $\begingroup$ Nobody claimed that "most beginners are only thinking about commutative rings". $\endgroup$ – Bill Dubuque Mar 5 '14 at 16:11
  • 3
    $\begingroup$ Dear @BillDubuque : Yes, technically nobody has claimed that here, but somebody has mentioned that they feel now and have felt "for decades" that they are justified in assuming beginners' questions on "forums like this" should be answered as if they had meant "commutative" by default. Presumably that person wants to help the larger half of beginners, so they wouldn't take the inconsistent stance of disagreeing with my first comment. $\endgroup$ – rschwieb Mar 7 '14 at 17:04
  • 1
    $\begingroup$ @rschwieb My remarks mean this: when there is inadequate context to infer if the ring is commutative, I often choose the commutative case, since commutative rings are far more common than noncommutative rings in general level forums like sci.math and MSE. Furthermore, I think it is misguided to attempt to enforce any kind of policy to resolve such ambiguities because not too infrequently beautiful answers result from that extra freedom given to the answerer - allowing them to expound on matters that the OP had no idea to ask about, but are closely related, possibly the essence of the matter. $\endgroup$ – Bill Dubuque Mar 7 '14 at 17:16
  • $\begingroup$ I like how Milnes has contradictory conventions. Of course, those assumptions are scoped only to those texts, but still... $\endgroup$ – Emily Mar 12 '14 at 4:39
  • 2
    $\begingroup$ I think commutative rings are more common when asking about a particular ring. When asking about rings in the abstract without stating commutativity, commutativity is never stated (perhaps there is feedback of OPs saying they meant commutative ring in comments?) $\endgroup$ – Jacob Wakem Mar 14 '14 at 2:55
9
$\begingroup$

I agree with all of Jyrki's conventions in the sense that they are the same as mine.

But, look, they are just conventions. They depend on context and thus should be spelled out. In any course I teach, I surely make sure to explain that my rings contain a multiplicative identity (and if I don't, it soon comes up and someone asks). I don't go into a disquisition about why my convention is the right one and the standard one [although I think it is...]; I just explain.

Whether my rings are commutative or not is not a global convention of mine: in what I am doing, more often than not they are, but plenty often they are not. So I remind people more often which way I mean it to be: e.g. by writing "ring" on the board and saying "commutative ring".

So my answer to the question asked is: it is always better if we don't have to assume. If you as an asker mean your ring to be commutative, I think you should either say so or include the commutative-algebra (or whatever it is) tag. On the other hand, if you are asking about rings which are not obviously noncommutative (e.g. so not if you are asking about ideals in matrix rings), then you should specify whether you would like answers to assume that the ring is commutative, not to assume that, or whether you are open to answers to do both.

In my opinion just writing "Let $R$ be a ring...." and not indicating in any way whether you want it to be commutative or not is not a best practice, because it will confuse some people and may lead to undesired answers. Do we need to press this issue any further than that?

| |
$\endgroup$
8
$\begingroup$

I don't think it makes sense to enforce any convention (not least because there is no enforcement mechanism; also, as the discussion here shows, there is no general agreement on conventions).

Answerers should try to answer questions as best they can; part of this involves interpreting the question as best they can. I think people do this, and one should assume good faith on the part of answerers.

If the OP of the question finds an answer limited in scope (e.g. because it presumes that the rings in question are commutative) they are welcome to clarify their question, ask a new more precise question, leave a comment on the answer, etc.

If another user finds an answer limited in scope and would like to leave a more general answer, they are also free to do so.

More experience/mathematically mature answerers are normally aware of the possibility of different conventions/interpretations, and often mention this. Less experienced users may not be so aware, and so it is not reasonable to expect them to be as attentive to such issues. One way they will gain experience, and awareness, though, is by seeing some of the discussions of such issues in the comments/answers that appear here.

In summary: I don't see that there is any real problem, or anything that needs to be done.

| |
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .