Talk:Ring (mathematics)/Archive 4

Zero a zero-divisor?
I know this is a convention thing. My "personal" preference is to regard zero a zero divisor. One advantage is that the complement of all zero-divisors does not contain zero (and this is important for the construction of total quotient ring). Is this position a minority? (I will also look at other sources.) -- Taku (talk) 17:42, 24 December 2012 (UTC)


 * That is the definition I prefer as well (if only the editors' persuasions decided the opinion, life would be so much easier!). I look forward to seeing what you find out about each version's popularity. I've always been curious about it. Rschwieb (talk) 15:09, 26 December 2012 (UTC)
 * I prefer also the convention that zero is a zero-divisor. With is convention we have the following definitions of properties:
 * A domain is a ring without non-zero zero-divisors
 * The complement of the set of the zero-divisors is multiplicatively closed
 * The total quotient ring is the set of fractions whose denominator is not a zero-divisor
 * Every nilpotent element is a zero-divisor
 * The list is certainly not complete, but only the first assertion would be slightly simpler with the opposite convention. Also the definition of zero-divisors is simpler if zero is a zero-divisor (otherwise twice "non-zero" in the same sentence). D.Lazard (talk) 17:13, 26 December 2012 (UTC)

Ok, I did some research and this is what I found: (It must be noted that Matsumura and AT are both influenced by Bourbaki). So, I suppose the conclusion is that abstract algebra texts prefer a zero-divisor to be zero, but other more upper-division texts disagree. Ahhhh, given the disagreement in literature, I would like to suppose we editors can impose our preference. This is imporant since this will make this article compatible with the rest of articles throughout Wikipedia. (This is for example why this article does not assume a ring to be unital, since, apparently, that's not the case elsewhere in Wikipedia.) -- Taku (talk) 15:15, 4 January 2013 (UTC)
 * A zero-divisor is nonzero: Lang, Cohn, Hungerford, Knapp, Eisenbud, Dummit-Foote, Wikipedia's Zero divisor
 * A zero-divisor may be zero: Isaacs, Bourbaki, Matsumura, Atiyah-Macdonald, some other parts of Wikipedia (e.g., total quotient ring).

Keeping the brakes on too much detail
I notice the "Topics in..." sections have been transforming themselves into Reader's Digest versions of the articles they point to. I really like the function of the sections (to offer up good links to core ideas and theorems) but they are currently cancerously bloated with too much detail. Can we dial it back a bit again? Rschwieb (talk) 14:41, 4 January 2013 (UTC)


 * Yes, growing like a cancer or something :). And yes I agree it's functioning very nicely. The reason for details is the desire to give enough feeling: we shouldn't just link relevant articles. That cannot be called an exposition. -- Taku (talk) 15:20, 4 January 2013 (UTC)


 * Thanks. I think this was a good call. We're also agreed on avoiding naked link lists. Rschwieb (talk) 16:05, 4 January 2013 (UTC)

Yet Another Bad Introductory Paragraph to a Math Entry
This article has one of the worse introductions to a mathematics entry I have read (a common fault of many mathematics articles on wikipedia). The first four paragraphs are verbose, unnecessarily complicated, and have little useful informative content. There are too many links to other mathematical concepts, which if the reader understands, ought already knows what a ring is. They appear to have been written by author/s who are showing off a rudimentary knowledge of mathematics, rather than trying to explain the concepts.

All that needs to be said is something a long the line "A ring as an algebraic structure that generalizes the integers."

The third paragraph ... fuck the third paragraph. Who writes this appalling shit? Who would want to reading this appalling shit?

— Preceding unsigned comment added by 86.27.193.180 (talk • contribs)  14:41, 4 January 2013‎ (UTC)


 * I have rewritten the first paragraph of the lead, hoping that the new version is better. D.Lazard (talk) 14:37, 6 January 2013 (UTC)
 * I have also removed from the third paragraph the things that may be understood only by experts in ring theory.
 * I'll be bold and remove the last paragraph of the lead. D.Lazard (talk) 17:59, 6 January 2013 (UTC)

A few thoughts.
 * The first para looks very good. (this is not my article, so I cannot thank you but I like it.)
 * The second para, very boring but fine otherwise.
 * The third para. I didn't like the old one, which was a hack job like a college essay. The new one is still problematic; it's not accurate! For example, the noncommutative ring theory is much more than operator algebras and ring of differential operators (coincidentally, they are categorically ignored by the current version of this article, which has to be rectified in the future.)
 * The 4th. I'm very ambivalent about that one. The larger question is how much non-associative "ring" should be discussed here. Categorically ignoring them is not "historically" correct. -- Taku (talk) 18:39, 6 January 2013 (UTC)

I'll give my try; please don't hesitate to make any further edits you can think of. -- Taku (talk) 12:11, 7 January 2013 (UTC)

merging ring theory into this article
I actually proposed this years ago. I understand the idea is to use ring theory to cover more advanced materials. But that's not happening right now. Unless someone (absolutely not me) wants to write such an article, I don't see a point of having a separate ring article. Thoughts? -- Taku (talk) 04:55, 6 January 2013 (UTC)


 * As a general principle I would expect an article on Ring (mathematics) to concentrate on telling the reader about rings as mathematical objects: their structure, properties, morphisms, associated objects; whereas Ring Theory should describe the theory, in terms of historical development, applications, relationships to other theories, key results, comparisons with other theories. However this is more a general question.  Perhaps a discussion at Wikipedia talk:WikiProject Mathematics?    Deltahedron (talk) 19:13, 6 January 2013 (UTC)


 * It has nothing to do with advancedness: The difference between theory and object is very clear. The former tells you what sorts of questions the field asks and answers, and the latter should tell you about the object itself and its features and varieties. I think this is the same thing that Deltahedron is getting at. On the positive side, maybe spelling this out will help us cut out unnecessary duplication between the two articles? Rschwieb (talk) 20:18, 6 January 2013 (UTC)

When I said "advanced", I was thinking of the past discussion: the idea was to adopt the structure of group (mathematics) and group theory to this article and ring theory. (presumably in part this was to make the main article elementary enough so it can pass FA.) Anyway, I will start a thread in the project page; makes sense. -- Taku (talk) 20:12, 7 January 2013 (UTC)

Sorting out ring theory and ring (mathematics)
Sorry I have been away for so long since my promise to discuss sorting out the "ring theory" and the "ring mathematics" articles. For obvious reasons, I think the "Topics in commutative/noncommutative ring theory" sections are excellent candidates for sections that should live in "ring theory" rather than this article. It doesn't have to be a complete excision, though, if someone sees some vital content. Rschwieb (talk) 15:46, 7 February 2013 (UTC)


 * (I have been quite busy and couldn't do much about this issue.) Yes, I think most of stuff in the section can safely go to ring theory, perhaps except a section on a central simple algebra. It's an important class of a noncommutative ring (am I biased?). Maybe we can move it to "special kind of rings". -- Taku (talk) 03:55, 8 February 2013 (UTC)

So, I have done some structure-type edits. As is clear, some new sections are utterly incomplete, but, structure-wise, the article seems fairly close to be complete. (I think "morita equivalence" and other module-theoretic invariants belong here. "Applications" section should also be expanded.) -- Taku (talk) 19:15, 16 February 2013 (UTC)

Second example
Instead of having the field of fractions of an arbitrary ring being the second example, perhaps something less abstract and more familiar is better: Like the ring of 2 by 2 matrices over the real numbers? This would also naturally dispel the notion that all rings are commutative. Mark M (talk) 09:49, 21 February 2013 (UTC)


 * That's a good idea: having a noncommutative (instead of a commutative one) example makes sense.
 * On a related note, I have a problem with the first example. To begin with, is it as an abelian group the same as $$\mathbf{Z}/4\mathbf{Z}$$? according to the definition, apparently, it's not. According to the text, it's supposed to be a subset of Z, which gives a wrong intuition. Also, it spent too much space on checking the axioms, which is a trivial matter. The more interesting features of the ring are not discussed like having a nilpotent. -- Taku (talk) 12:58, 21 February 2013 (UTC)
 * I think we can assume some group theory on a reader's part. Then we can just say the cyclic group $$\mathbf{Z}/4\mathbf{Z}$$ also has a ring structure that is compatible with the underlying group structure. This is probably less confusing. -- Taku (talk) 13:19, 21 February 2013 (UTC)


 * No, I don't think we should assume some group theory on the reader's part. Rings are sometimes introduced before the concept of a group (indeed, that was the case for me). Also, see the "widest possible audience" discussion above. Mark M (talk) 13:52, 21 February 2013 (UTC)
 * I agree with Mark. The whole point of the example is to give an explanation of the definition on a simple concrete case for readers not familiar with any abstract algebra.—Emil J. 14:05, 21 February 2013 (UTC)
 * I do not think that identifying the elements of $$\mathbf{Z}/n\mathbf{Z}$$ with non negative integers lower than n gives a wrong intuition, if one emphasize that the operations are not the same. This may be done by denoting $$\overline x$$ the element of $$\mathbf{Z}/n\mathbf{Z}$$ represented by $$x.$$ On the contrary, many readers could have had courses of computer science or cryptography, and in these matters, one never talk of congruence classes. Moreover, in the computer programs, the modular integers are systematically represented by integers. By the way, modular computation being fundamental in computer science, it could be useful to generalize the first example to an arbitrary modulus. This would even be simpler than the present presentation, if the properties of modulo operation are linked to modulo operation or Euclidean division. D.Lazard (talk) 14:36, 21 February 2013 (UTC)

Oh, sometimes it's much more productive than just edit :) All 3 especially Lazard makes a very good point. (I don't have to repeat what was said.) I still find the presentation subpar. For example, do really people use a multiplication table? (as in the 1st grade.) If it's for a broad audience, maybe it makes sense to discuss a bit more on a modular arithmetic. (once this is clear the ring axioms are basically automatic on the contrary to what might be suggested in the article.) -- Taku (talk) 15:29, 21 February 2013 (UTC)

Proposed merge
I have made a proposal for the merger of noncommutative ring into this one, basically because the former is essay-like and has virtually no content. This does not preclude having such a separate article in the future, but for now I fail to see the point of the article (hence, the proposal.) -- Taku (talk) 14:18, 17 February 2013 (UTC)


 * As much as I'd like to see noncommutative ring as an independent article, there is not a lot there and I don't have enough ideas to fill an article... so I guess I agree with that action. Rschwieb (talk) 16:14, 18 February 2013 (UTC)


 * I also agree with merging in noncommitative ring. While we're at it, I would also support merging in pseudo-ring, which seems unnecessary given the scope of this article. Mark M (talk) 21:43, 19 February 2013 (UTC)


 * I like the idea of Mark a lot. "non-unital" idea is not covered here precisely because it's already in Pseudo-ring (and I think it's a nice article). But, as precisely pointed out, do we really have to do it in a separate article? Since we're keeping ring theory, there seems like enough room for the topic here. Somewhere "definition and illustration" section, say. -- Taku (talk)


 * Something about putting pseudoring in here too seems to spoil the broth to me. I think a case can be made that they are relatively obscure, and are probably best left in their own article. Don't get me wrong: I do like them! But why do we have to cram so many different animals into a single article... Rschwieb (talk) 14:45, 20 February 2013 (UTC)


 * Well, at the moment our definitions of ring and pseudo-ring are exactly the same, so I object to your use of the word "different"! :-) Mark M (talk) 14:52, 20 February 2013 (UTC)


 * Point taken, although that strikes me as an argument to make the pseudo-ring article go away rather than to merge it. The sections in pseudo-ring do not look like very promising additions to ring (mathematics). Rschwieb (talk) 16:56, 20 February 2013 (UTC)
 * I think adding a few examples of rings without identities would be a good idea; which is basically what Pseudo-ring is about. Mark M (talk) 17:13, 20 February 2013 (UTC)

I would like to change my position on the merger of "pseudo-ring". The reason is that I believe we need an article on the topc: there are a lot of technical matters that occur when there is no unity; e.g.,myou be careful about the definition of a Jacbson ring (cf. modular ideal.) As we have learned (or I have), it's better to leave technical matters from main articles such as this one. -- Taku (talk) 01:31, 25 February 2013 (UTC)


 * In that case, perhaps we should insist on there being an identity element as one of the axioms of a ring. I would be fine with that, as long as we make clear that some authors don't include that as one of the axioms. Mark M (talk) 10:01, 25 February 2013 (UTC)


 * The problem is that it doesn't seem that Wikipedia insists a ring has unity. In my view, it's very important to pick the convention that is consistent with other parts of Wikipedia; it would be utterly confusing if the meaning of "ring" varies across articles. Since we cannot adopt Wikipedia to this article, we have to adopt this article to Wikipedia. We also cannot expect the readers to read the definition section first; this isn't a textbook. It's safer to say a "ring with unity." Finally, the focus of this article is already the unital case. -- Taku (talk) 12:23, 25 February 2013 (UTC)


 * Okay.. then maybe you are suggesting the scope of the Pseudo-rings article should "rings without an identity element"? My problem is that at the moment the two articles have identical scopes. Mark M (talk) 13:24, 25 February 2013 (UTC)

Rings without an identity?
Can someone explain why the article no longer includes the existence of a multiplicative identity as an axiom? Was this discussed somewhere? Because according to this 2007 discussion, opinion seemed in favour of keeping the identity. And it appears a few months ago it was still an axiom.. so why the change? Mark M (talk) 13:36, 25 February 2013 (UTC)


 * The linked disucssion was clearly testing to see if people favored ring-with-identity as a convention across WP:MATH articles. It was not a discussion about including or excluding the identity axiom from ring (mathematics) (which would be overstepping the bounds of an encyclopedia's duty). To reflect the reality of ring theory textbooks, we should definitely point out that the axiom is required by some and omitted by some, and that much of ring theory is done assuming the existence of an identity. Rschwieb (talk) 14:48, 25 February 2013 (UTC)


 * But up until a few months ago the scope of this article excluded rings without an identity. Now the scope includes them. Why the change? We now have two articles with identical scope (Am I the only one that believes this?).. this one and pseudo-ring. Mark M (talk) 15:27, 25 February 2013 (UTC)


 * I don't think mention of the use of the identity axiom strictly dictates what's in the article, no. The scopes seem to be clear (and unchanged) as far as I can see: ring (mathematics) should mention both types, but should lean toward rings with identity, while pseudo-ring focuses especially on what happens if identities are absent. Rschwieb (talk) 16:20, 25 February 2013 (UTC)


 * What Rschwieb said :) This is the main article and as such cannot omit the discussion of a ring without identity, no matter how a ring is defined here. -- Taku (talk) 01:20, 26 February 2013 (UTC)


 * Great, that's fine with me; per WP:CONCEPTDAB, it's better to have this article as the more general anyway. The problem of having two articles with identical scopes (just different points of view) could be improved by renaming Pseudo-ring to Non-unital ring; I think this would clarify that the focus of that article is on rings that don't have a unit. Would you be in favour of that? It seems rather silly that currently Pseudo-ring basically begins with "A pseudo-ring is the same as a ring". Mark M (talk) 10:02, 26 February 2013 (UTC)


 * The present formulation in the article is even more silly and confusing: "ring (also called pseudo-ring)". It is also WP:OR to decide that a ring and a pseudo-ring is the same thing. IMO, we have to define rings with unity, and at the description of the axiom of unity, to quote that "for some authors, a ring has not necessarily an unit element", and to explain the various terminologies aimed to disambiguate when it is not clear from the context. Also, a section "Pseudo-ring" could be useful to state the main differences between rings and rngs. If one decides to merge, the merge could be in that section. But, please, not confuse the reader by suggesting, in any way, that the content of this article applies to rngs. D.Lazard (talk) 10:45, 26 February 2013 (UTC)


 * Thanks D.Lazard, that was more the response I was expecting. But it is not original research to note that many authors do not assume rings have multiplicative identities. For an extensive list of sources and what conventions they use, see this big table from a 2007 discussion on this talk page. I would be happy with not assuming a unit in this article, but list as a separate extra / optional axiom the multiplicative identity. Mark M (talk) 11:15, 26 February 2013 (UTC)
 * I agree that some, and even many, authors do not assume identity. But to conclude that rings and rngs are the same is original synthesis. D.Lazard (talk) 11:38, 26 February 2013 (UTC)
 * Agreed; what do you suggest we do? Mark M (talk) 11:47, 26 February 2013 (UTC)
 * I do not know if the majority of text books assume identity or not. But in peer reviewed publications, when "ring" is used without precision, it means always "unital ring". Therefore my suggestion is to define rings as rings with unity, and, just after the axiom of identity element, to note that this axiom is not required by some or many authors, to list the various terms that are used to disambiguate, to clarify that, in this article, the rings are unital unless the contrary is explicitely stated, and to link to a section "Pseudo-ring" (or to the eponymous article) for the maindifferences in the properties. D.Lazard (talk) 12:04, 26 February 2013 (UTC)
 * Do you have a source for the statement "in peer reviewed publications, when "ring" is used without precision, it means always 'unital ring'"? Are you sure this is universally so? Are you sure it doesn't vary in different parts of the world? JamesBWatson (talk) 13:27, 26 February 2013 (UTC)

Actually it seems to me that in most articles I have read, it *is* used with precision, and usually the result is "here, all rings have identity." It's just that lots of work gets done where we know maximal ideals exist and we can talk about free modules with obvious bases. This excepts very specific articles I have focusing on rings lacking unity, but again they are precise.

Surely a request for a source on what happens across hundreds of thousands of articles cannot be taken seriously. Anyhow, I have to agree with D.Lazard that by and large people are using rings with identity. Rschwieb (talk) 15:07, 26 February 2013 (UTC)
 * I agree that it is used with precision in articles about rings. But in articles that are not about rings, I am not sure, for example in a sentence like "Let us consider the ring of continuous functions". By the way, I can think only to one use of non unital rings outside pure ring theory: the "ring" of continuous function with compact support, used in theory of distributions. It could be interesting to look on the common usage in this context. An example (which may not been considered as a source): The word ring is used several times for unital rings in Distribution (mathematics), without precision, but the smooth functions with compact support are not qualified of ring nor of pseudo-ring, only of module over the full ring of the smooth functions. D.Lazard (talk) 16:15, 26 February 2013 (UTC)
 * I cannot deny that most of the articles I read are on rings and modules :) I agree with everything above! Rschwieb (talk) 16:27, 26 February 2013 (UTC)


 * My real world experience is with commutative rings and I have almost never seen anyone or any article discussing a commutative ring without unity. I'm not so sure about the noncommutative case. My impression is that people sometimes use "algebra" to discuss possibly nonunital situation while requiring a ring to be unital. I prefer not to speculate more. In any way, in light of the fact there is no universally accepted definition, it is up to Wikipedia editors to pick a definition (of course their reasoning should still be based on sources.) The standard way to establish the consensus is to use polling. I can set up one if there is no objection. -- Taku (talk) 19:50, 26 February 2013 (UTC)

Polynomial ring
I strongly suggest to remove the last paragraph before section Matrix ring, which begin by "Finally, there is a coordinate-free way to define ...". Firstly, it is too technical. Secondly, this construction is not that of the ring of polynomials, but that of the ring of polynomial functions, which is not the same. That is, the polynomial ring may be identified with S(V) and not with S(V*). D.Lazard (talk) 22:57, 22 February 2013 (UTC)


 * I have to agree it's too technical. I moved the materials to ring of polynomial functions and left a link. The construction should still be mentioned in the section, I think. -- Taku (talk) 23:26, 22 February 2013 (UTC)


 * I have removed the following:
 * If S=R[t], then f(x) is the composition f∘x of f and x.
 * If S=R, then the map $$x \mapsto f(x)$$ is the polynomial function defined by f.
 * basically because I couldn't understand them. For the 1st one, x is not a function; so it doesn't make sense. For the second, I think it's out of context. We agreed that it's simpler to just discuss polynomials (not polynomial functions). -- Taku (talk) 03:47, 24 February 2013 (UTC)

Requiring a ring to have unity by definition
It has been suggested above that we define a ring to be unital in this article. Since authors of algebra texts differ on this requirement, it is necessary for Wikipedia editors to pick a definition of a ring for this article. (This is under the assumption that this article defines a ring at all; see options below.) This polls is to gauge editors' positions on this matter. Note Have fun with voting! -- Taku (talk) 18:58, 27 February 2013 (UTC)
 * 1) Many articles in Wikipedia link to this one. It is reasonable to assume that a reader reading another article expects to find a definition of a ring in this article. In other words, the definition given here will not just be used here but may be used in other parts of Wikipedia.
 * 2) Manual of Style/Mathematics says that a ring is unital by convention; cf. Talk:Ring_(mathematics)/Archive_1
 * 3) The definitions of algebra over a field and algebra over a ring are different matters.
 * 4) This is about the definition of a ring. The fate of pseudo-ring (merger, deletion, renaming) is a different matter.
 * 5) No matter how a ring is defined, this article will still mention the presence of different definitions; this poll has nothing to do with this. The non-unital situation would be covered to a certain extent if not the main focus.
 * In favor of requiring a ring to have unity by definition
 * My reason is simple. I almost blindly follow Bourbaki and this is his (their) definition of a ring. -- Taku (talk) 18:58, 27 February 2013 (UTC) Clearly I should stay neutral on this issue. -- Taku (talk) 17:53, 2 March 2013 (UTC)
 * We do not write articles only for algebraist, but also for all mathematicians and physicists. While in papers about rings both conventions are used, in texts on other subjects, "ring" always mean unital ring. For an example, see distribution (mathematics), where "ring" is used several times for unital rings, but the non unital ring that appears naturally in the theory is never qualified as "ring". (See may posts above). D.Lazard (talk) 20:17, 27 February 2013 (UTC)
 * In favor of permitting a ring not to have unity
 * Since this article is essentially a WP:CONCEPTDAB, we should be including the most general definition in this article. Mark M (talk) 19:25, 27 February 2013 (UTC)
 * I agree with Mark M that we need to give the most general definition. Also there are non-unital rings that come up in practice:
 * $$ k\mathbb{Z}$$ for k > 1
 * Ideals of rings are rings, but don't necessarily have a unity
 * $$C^*$$-algebras don't necessarily have a unity
 * Some authors call rings without an identity a rng (ring without the i), but to me rng means random number generator. --Mark viking (talk) 20:39, 27 February 2013 (UTC)


 * We should not give a definition of a ring at all since there is no universally accepted definition; e.g., discuss rings without actually defining them
 * Others

Discussion

 * Comment - This poll does not have the authority to decide something as sweeping as this across so many articles. Consider the policy WP:NPOV, and look at this huge list of sources.. it would obviously violate NPOV to arbitrarily choose one source (such as Bourbaki) and go with that. It would be better to clearly explain the conventions within the article; then readers might have a better idea of what "ring" means in their context. Mark M (talk) 19:25, 27 February 2013 (UTC)
 * I disagree. We should choose "one definition" and use it throughout Wikipedia: it would be very confusing if different articles use a different definition of a ring. "I" based my vote on Bourbaki; that's just my opinion. Each editor is entitled to his/her opinion. -- Taku (talk) 19:37, 27 February 2013 (UTC)
 * One ring to rule them all!Naraht (talk) 20:23, 27 February 2013 (UTC)


 * Comment - the place to define conventions for many articles is the Manual of Style for mathematics. And waddaya know - it already has been done for rings! Under Manual_of_Style/Mathematics, we have the sentence: "A ring is assumed to be associative and unital. There is an exception for rings of operators, such as * algebras, B* algebras, C* algebras, which we do not assume to be unital." RockMagnetist (talk) 01:50, 28 February 2013 (UTC)
 * Sound very reasonable except that no one is, apparently, following the manual of style, including the main article on a ring. -- Taku (talk) 07:01, 28 February 2013 (UTC)
 * I think it's fine to assume by default that rings are unital and associative, which is what the Manual of Style says (and this was reinforced in this discussion from five years ago). This poll, on the other hand, is suggesting that rings must be associative and unital.. this is not something I will agree with (per WP:NPOV). Moreover, (as discussed above) in the article Ring (mathematics), we have to be especially sensitive to the fact that different definitions are used by different sources. But this doesn't mean we have to change the default convention across Wikipedia articles (and contrary to what Taku says, as far as I can tell, the vast majority of articles are following this convention). Mark M (talk) 09:24, 28 February 2013 (UTC)
 * No, you misunderstood the poll completely. This is about whether a ring is unital or not by default. I will clarify this above. (I didn't know there was the same poll before.) -- Taku (talk) 12:23, 28 February 2013 (UTC)
 * Perhaps I misunderstood what you intended to ask; but I simply responded to the question as it was written. Am I in favour of requiring a ring to have a unity? No. Am I in favour of having articles by default using the convention that rings have a unity? Yes. Fortunately, that's already the case anyway. Mark M (talk) 13:13, 28 February 2013 (UTC)
 * I think you're being dense. "Clearly" the 2nd one is intended. (I feel like a politician.) It's about the definition of a ring; that's all. For instance, that's what the section title says. -- Taku (talk) 14:25, 28 February 2013 (UTC)
 * And this is why Polling is not a substitute for discussion. Mark M (talk) 21:24, 28 February 2013 (UTC)
 * You want to discuss what? -- Taku (talk) 21:33, 28 February 2013 (UTC)
 * Comment Authors of algebra texts differ on this requirement, and thus it is up to Wikipedia editors to pick a definition for this article. This statement is patently inconsistent with the most basic ideas of editing wikipedia. Editors do not weed through the established definitions of everything and then "pick one." If there are more than one prominent definitions out there, then you represent them all appropriately. I'm on the same page as Mark in saying that I think under the current MOS policy, we represent rings with and without identity just fine. It is difficult to see what is being suggested here that is not already being done. Without qualification rings can be assumed to have identity, and with precision you can make exact statements. I do not see how the main article failed to follow it, as described above. Certainly the main article must talk about this point with precision, since conventions vary throughout the literature. Frankly, I am surprised anyone who has edited wikipedia as long as you have would suggest such a thing. Rschwieb (talk) 19:58, 1 March 2013 (UTC)
 * Do people actually read the notes attached to the poll? Of course, all definitions will be presented without prejudice. (See my talk page as well.) This poll is about picking a definition of a ring that is used in this article. Not about eliminating any discussion of the non-unital case. The current definition of a ring that apppears in this article does not require a ring to have the multiplicative identity. Some editors including myself believe this is not a good definition. Thus, we're having the poll to see if the change of the definition has enough support. That's all. I really want to know what can be done to make this clearer. Anything else has nothing to do with the poll. -- Taku (talk) 23:31, 1 March 2013 (UTC)
 * Ok, maybe this is what was meant. Do you suggest that we should not give a definition of a ring at all? since whatever definition we choose that would be a non-NPOV? I think the demerits (utter confusion due to th lack of the clear definition) outweighs any advantage obtained by being neutral. But that's one position (I suppose); thus, I added this option above. -- Taku (talk) 23:50, 1 March 2013 (UTC)


 * Comment - I too am bemused by this poll. Since when do Wikipedians vote on the definition of a technical term? Follow the sources. If there are sources stating that rings are commonly assumed to be unital, then say that it is the default and cite them. RockMagnetist (talk) 00:22, 2 March 2013 (UTC)
 * Because there is no "default definition"; authors of algebra textbooks do not agree on whether a ring is unital or not (as noted above as well as in the article). The discussion would not get us anywhere since there is no "right" definition; the quickest and fairest way (or so I thought) to pick one among the definitions is through the polling. -- Taku (talk) 00:46, 2 March 2013 (UTC)

In the good old, bad old days, there was often a bull goose mathematician who could "fix" mathematical notation and terminology by ukase. Those days are long gone. Much as we might wish Wikipedia was prepared to step into the power vacuum and decide what the "correct" definition of a ring is, mathematicians who wouldn't even listen to Bourbaki certainly aren't going to listen to Wikipedia. Rick Norwood (talk) 14:45, 2 March 2013 (UTC)
 * (How many times do I need to assure this?). This poll is "not" about fixing the definition of a ring in the world. But about fixing one that is given in this article. (After all, the talk page is not a discussion forum.) Any definition that appears in literature or in textbooks will be presented in the article. No definition will be exterminated as clearly stated above. But it is also reasonable that this artices give a primary definition of a ring as it does now. Many articles in Wikipedia define things even if there is some disagreement in sources; for example, some authors allow varieties to be non-irreducible. But that's not the definition Wikipedia uses. It is resonable to assume that rings are not exceptions. We give all standard definitions but pick one as primary one. -- Taku (talk) 17:51, 2 March 2013 (UTC)


 * Both rng/pseudo-ring and unital/unitary ring are excellent definitions of "ring". Readers looking for either will come to this article, and some readers will not even know what they are looking for.
 * Just like every other publication, we have to make up our collective mind on how to use the term "ring" in Wikipedia. Ideally, we should be consistent not just on this article but throughout the encyclopedia.
 * WikiProject Mathematics actually has a normative document for this kind of situation. WP:WikiProject Mathematics/Conventions says that rings are unital except that this article does not currently conform to this convention. I don't know if this (still) describes the status quo correctly for the other pages.
 * As I see it, we have three major options, not two:
 * ring vs. unital ring
 * pseudo-ring vs. ring
 * pseudo-ring vs. unital ring
 * The last option is the least elegant but also the least confusing for experts. They will always know what we mean. We could even split the present article in this way and make ring a redirect to pseudo-ring with a headnote mentioning unital ring. Every reference to "ring" would immediately be recognisable as a reference needing disambiguation one way or the other.
 * I am personally happy with each of the three options. The only thing I consider unacceptable is inconsistency between, let alone within, articles. We should settle on one convention, to be fixed at WP:WikiProject Mathematics/Conventions and implemented consistently, and settle on it after a community discussion that should be wider than WT:WikiProject Mathematics/Conventions. The natural location for this is WT:WPM, as it has over 700 watchers (a lot more than this article). Hans Adler 18:47, 2 March 2013 (UTC)
 * (See below too). Just to repeat myself once again, this poll is not about figuring out the relative supremacy of ring definitions. It's OR, POV. But the point is that this poll has nothing to do with that matter. For the last point, I dont't think the place of the discussion is particularly relevant: the discussion could take place in relevant places and we can just disseminate the results at the end. I already advertised the poll in the project talk page and that should be enough. -- Taku (talk) 14:29, 7 March 2013 (UTC)

Technical difficult with editing ring (mathematics)
Hi all,

I have been experiencing very weird editing difficult with ring (mathematics). Basically, I tried to submit an edit but then either it times out or I get an error page. I'm not sure if it is my Internet connection or something else. But the problem is limited to this particular article; not even its talkpage exhibits a problem. Is there anyone having the same problem with this particular page? (There was a similar problem back in 2005 or 6 when there was shortage of server capacity. But that was a Wikipedia-wide problem.)

-- Taku (talk) 18:14, 7 March 2013 (UTC)


 * I tried a dummy edit (add a few spaces) to the article and it returned a wikimedia error page. The edit did show up on the page, however, and I was able to revert, too. --Mark viking (talk) 19:21, 7 March 2013 (UTC)


 * Yeah, I had exactly the same sort of difficulties several days ago when I left that clipped off edit summary. It was pretty strange... at the same time I was not having any trouble at all posting at the WP:Math page. Rschwieb (talk) 21:17, 7 March 2013 (UTC)


 * Oh, so it's not just me. Still weird. Anyway, I think I will post a note at Village pump (technical). Thanks for the responses. -- Taku (talk) 01:22, 8 March 2013 (UTC)
 * I have had similar problems with, at least, two other pages, but never with talk pages. In some case I get an error message, but when I came back some time later, it appeared that my edit was finally saved. D.Lazard (talk) 12:04, 8 March 2013 (UTC)
 * Try the trick I described at WP:VPT. -- Red rose64 (talk) 13:42, 8 March 2013 (UTC)
 * I see has damaged the page to the point it is impossible to actually edit the formulas and have them work.  (Bear with me, this is not irrelevant.)  The reason he did so was to improve rendering, but it makes actually editing imppossible.  — Arthur Rubin  (talk) 17:33, 10 March 2013 (UTC)

Requiring a ring to have unity by definition: follow-up
So, it went well :) I think the complication the poll expericed is the fact that, when I started it, I was not aware of the convention noted in the manual of style and the past poll. I agree that, as some pointed out, the poll was somehow redundant given the clear statement given in the mos. On the other hand, the mos also says that "However, each article may establish its own conventions". Thus, the issue still remains: we need to pick a definition of a ring for ths article. And, as I repeat, it is important to be aware that the readers who are following links to find a definition of a ring would find one here not in the mos. Hence, it is imporant to pick one not only for this article but for other articles linking this one.

I'm not too sure about the course of action: the simplest approach is to adopt the convention in the mos here. This would resolve inconsistency for example. Objections? -- Taku (talk) 14:18, 7 March 2013 (UTC)


 * Sorry for being so slow to respond, but I'm going to pick up down here so that we don't have to jump around. There's clearly some misunderstanding among us all, so I'm trying to take a step back.
 * It looks like we all agree that "all definitions should be included" and that "unity is required more often than not" is going to appear in the article, and we all agree that the MATH:MOS stance is something we all like.
 * Here is the issue. Every time I look at the article, I see that it presents all definitions and is consistent with the MOS. But you insist that a definition needs to be picked and you want to be consistent with the MOS. I cannot figure out what you are getting at. As far as I can tell, the article already does all of those things. There are four lights, and you insist "we need to have four lights". We say "but there are four lights" and you say "I just think the time has come to put in a fourth light".
 * When you say "pick a definition to use," I do not understand you. Since all definitions are represented in the article, and you insist that "we need to pick a definition", there is no way in English to interpret that in any other way except "some definitions are going away". You have had to repeat many times "I'm not trying to get rid of definitions", because we are at a loss of understand what you have said in any other way.
 * I think the best way to break the impasse is if you provide the changes you want to make right here. We should have tried that a long time ago :) Rschwieb (talk) 22:00, 7 March 2013 (UTC)
 * Good idea, Rschwieb! Let's see what edit you want to make, Taku. RockMagnetist (talk) 22:08, 7 March 2013 (UTC)


 * ok. "picking" may be a wrong word choice. But I think I'm correct to understand that we're effectively "picking a definition to use in Wikipedia." The rational is given above: the MOS is not a place where a reader would be looking for (a) definition(s) of a ring; but this article is. Of course, as a reference, we mention all the conventions that are used in textbooks and literature. But, in the end, we also have to present "our choice" for the simple reason that any mathematical statement is meaningless if it contains an undefined term. In articles other than this, the problem is solved by linking this article. In this article, we "must" define a ring (based on texts or literature, of course).
 * In any way, the change I propose (as suggested before) is the insertion of something along the line
 * In this article, by convention, a ring is required to have the multiplicative identity. The Manual of Style of Wikipedia, which mathematics articles in Wikipedia mostly follow, also has the same requirement.
 * Nothing murderous :) -- Taku (talk) 12:19, 8 March 2013 (UTC)


 * OK, well that certainly clears up a lot. I had some minor concerns about how that might affect discussion about rings without identity, but after reviewing the article, I feel like those could easily be addressed with pointers in the right places. After thinking about it, it seems like it would be a good way to broadcast the convention in the MOS, since non-editors are for the most part unaware or indifferent about the MOS.
 * Here is my variant of what Taku proposed: "In this article and other algebra articles, unless otherwise stated, a ring is assumed to be an associative ring with multiplicative identity. This is a convention adopted in the Wikipedia Mathematics Manual of Style to reflect the most frequent meaning of "ring". Articles and sections which deviate from this convention are free to specify clearly what is meant in that context."
 * This is a little less blunt about what is going on, and a little less dictatorial in its appeal to the MOS. Let me know what you guys think. Rschwieb (talk) 13:45, 18 March 2013 (UTC)


 * I just want to say I'm fine with the wording, with the remark that we need a source to back up the claim that the unital assumption is common; editors' feelings are not enough. -- Taku (talk) 23:51, 20 March 2013 (UTC)
 * While that last comment is literally true, I think the task is not really source-finding but more of book-tallying and data mining. Any volunteers? Rschwieb (talk) 13:39, 21 March 2013 (UTC)


 * I've already pointed to the discussion from 5 years ago that tallied about 60 books regarding whether or not they assumed rings have a unit (here). It appears to be split 50-50. So I think the current wording of the definition is fine.. "Some authors require these 7 axioms, others also require this 8th one". It's really the only way to be neutral. I don't know the plan regarding conventions in the rest of the article; but it seems that we should wording things in a way that is sympathetic to readers for whom rings don't have a unit (again, WP:NPOV). Mark M (talk) 14:38, 21 March 2013 (UTC)
 * Wow! Perfect! I'm glad to see someone already tackled that problem :) I'm a little surprised the without unity was this high. I intend to comb through the books there later on to see if that is or isn't because of the particular books. Thanks for posting that! Rschwieb (talk) 19:23, 21 March 2013 (UTC)

Non-empty
If a subring must be a ring, which must have a "0" (additive identity), I would think it must be non-empty. — Arthur Rubin (talk) 19:05, 14 July 2013 (UTC)
 * Certainly. JamesBWatson (talk) 09:23, 15 July 2013 (UTC)
 * By the way, my thanks to JamesBWatson. When I made my edit, I didn't understand what a fuss was about. -- Taku (talk) 23:38, 15 July 2013 (UTC)
 * Glad you got it straightened out. Please assume some good faith about my abilities in the future. I'm not in the habit of making pointless edits (although I do manage to make mistakes :) ). Rschwieb (talk) 13:59, 16 July 2013 (UTC)

Notation for matrix ring
The article currently writes Rn to denote the n by n matrix ring over R. But it is much more standard to write Mn(R). Ebony Jackson (talk) 23:21, 16 November 2013 (UTC)


 * Agreed. In fact, I've just made the notion change. -- Taku (talk) 15:25, 17 November 2013 (UTC)

"Constructions" section too big
I see that the Constructions section remains bloated with reexplanations of things already detailed in other articles. Ideally most of these could be reduced to a sentence (or three) giving a heuristic idea of what they do or where they are used. Rschwieb (talk) 13:12, 27 April 2013 (UTC)


 * I disagree. This is a general survey article and as such most of materials would be duplicates of other articles in a digested form. Every article in Wikipedia is ideally self-contained. In other words, the readers shouldn't have to follow the links to understand concepts. Thus, telling a certain construction appears in such and such context would not simply do since the readers cannot understand the concept then. In terms of analogy, it's like the history section of Japan would be a complete duplicate of materials in other places in Wikipedia. It's merely an abbreviated version but still adequate for the purpose. Here, 3 sentences are just not enough to give an adequate treatment. As for the overall length of the article, it doesn't seem to be a problem. For example, Starbucks is much longer than the article. (I wonder if anyone really reads the whole article :). Since the article has to be both comprehensive and accessible, the current length is justifiable. (Or at least this is my view.) -- Taku (talk)


 * I think what we can do is to move some constructions to ring theory. For instance, I think "polynomial ring" is a must for this article, but "group ring" may be omitted since the latter is basically a generalization of the former. In terms of balance, we need to add more concrete examples; e.g., universal enveloping algebra. -- Taku (talk) 14:46, 27 April 2013 (UTC)
 * Isn't a polynomial ring more like a semigroup-ring (oops, as rings are assumpted to be with identity, it's a monoid ring)? It's not Z, it's N.  Yes, group ring should should be ommitted.  — Arthur Rubin  (talk) 20:06, 27 April 2013 (UTC)
 * Correct. A semigroup generalizes a group ring and a polynomial ring is a special case of a semigroup ring. -- Taku (talk) 20:54, 27 April 2013 (UTC)


 * @Taku You're right: ideally one would not have to leave the page to understand concepts on it. Of course, that is not realistically possible, and we can and do leave the page all the time in almost every article! Reduplicated material is often badly written (as I think part of the group-ring stuff is here.)
 * Your overall argument is not even consistent. You said at the beginning "this is a general survey article" and at the end you say it "has to be comprehensive." The two of us definitely agree here that it is a survey article, but survey articles are by definition not comprehensive. This is because good writers recognize that comprehensiveness is impractical in a small space and an impediment for the reader.
 * I think we also agree on this: there should be a brief descriptions, which will necessarily say things in other articles. I didn't say there would be no duplication. I said that the summaries should be reduced. They should alert the reader to the existence and the relevance of the topic at hand. On the other hand, a brief description of a group ring should not run screaming naked afield with this:

"'In representation theory, when G is abelian group, $\delta_s$ is often denoted by $e^s$. Writing the group operation on G additively and omitting *, one then has $e^s e^t = e^{s+t}$; the analogy with the exponential is obvious.'"


 * Arthur even unknowingly reflected one of my main motivations for reducing this stuff: the group ring is explained in a hideous way that obscures its relationship to the polynomial ring. I know both descriptions are valid, but it is not the best here.


 * Anyhow, let me end with a reiteration. The goal here should be to alert the reader to the existence and relevance of these construstions. It should not contain five bullet points detailing its homological properties.
 * I'll try to bring some suggested reductions here, one at a time. Rschwieb (talk) 13:58, 30 April 2013 (UTC)


 * Yes, I need to see more concrete suggestions instead of abstract ideas. (and I do enjoy knowing your thoughts on th article.) But to respond to some of points that are already made, yes, I agree that we should "alert the reader to the existence and the relevance of the topic" but at the same time we should not omit some certain technical details. For example, I was just reading a paragraph on a subring. It looks complicated compared to, say, the counterpart on a subgroup. But I think it is a good one since it tries to explain technical complications. Any article in Wikipedia in fact, it is important that the article is technically correct and to a certain extent self-contained. (maybe except some stuby articles, but for flagship-type articles such as this one.) -- Taku (talk) 13:15, 3 May 2013 (UTC)

I agree that the section goes into too much detail on some topics, which would be better given a brief introduction, accompanied by a link for anyone who wishes to go pursue further information. JamesBWatson (talk) 12:04, 7 June 2013 (UTC)


 * Do other people agree that the inclusion of too much peripheral material is preventing ring from being a good article? A lot of interesting mathematics has been added to the article, but much of it is not directly related to the topic of rings.  Things like "subring" and "ideal" deserve to be mentioned here, but certainly not facts about the Brauer group of a nonarchimedean field!  I think the best surveys are those that say just enough to give readers a feel for the subject and that provide references for those who want to read more.  Another problem with including too much material that is duplicated on more specific pages is that it makes it much harder to keep Wikipedia self-consistent and correct.  It would be great if some of the knowledgeable editors who have added interesting material to this page could move it to more specific pages! Ebony Jackson (talk) 20:07, 24 November 2013 (UTC)
 * I agree with this. From a layman's perspective, this looks like an article on ring theory, and not merely rings. As an example, an article on fields should not try to list and explain all possible subclasses of fields. Wikipedia is a richly linked medium, so we should use the links to avoid duplication. Several sections seem to be overweight in this regard. —Quondum 20:40, 24 November 2013 (UTC)

I think the consensus is clear by now that some degree of downsizing is needed :( But which materials exactly? I for one think, for example, polynomial rings and matrix rings are important enough. I think the problem is ring theory has been under-untilized so far. For whatever reason, the theory article has failed to develop (right now, no one is editing it.) The easiest (and emotionally painless) option is just move some stuff from here to there. -- Taku (talk) 20:50, 24 November 2013 (UTC)


 * Everything in the "structures and invariants" section is an obvious candidate to put at ring theory. All of those topics are about studying entire classes of rings rather than the properties or internal workings that all rings share, or basic ways to build new rings.
 * I still think pruning the Construction section's description way back would also help. There is really no sense in explaining polynomial rings in that much detail. I agree that readers should have access to a short description of the topic. The thing is that the main article is supposed to be able to do that in the lead. We shouldn't have to do it again here. Rschwieb (talk) 18:11, 25 November 2013 (UTC)


 * For the "Polynomial rings" subsection here, I might simply write something like
 * "Polynomials with real coefficients, such as $$3.2 x^2 - \pi x + 5$$, can be added and multiplied. The set of all such polynomials forms a ring denoted R[x].  More generally, if A is any commutative ring, then the polynomial ring A[x] is the set of formal expressions
 * $$a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$
 * where n ranges over nonnegative integers, and a0, ..., an range over elements of A. One can consider also polynomial rings in more than one variable."
 * prefaced with a link to the main article on polynomial rings. Before making any such change, however, some of the material presently in this section should be preserved by moving it to polynomial ring.  In particular, the universal property is important, and not currently mentioned at the polynomial ring page; perhaps it could be the first bullet item in the "Summary of the results" subsection there. Ebony Jackson (talk) 06:49, 30 November 2013 (UTC)


 * I agree that the universal property of polynomial rings is important. It appears (restricted to injective ring homomorphisms and without being named) in the section "Polynomial evaluation". Maybe this article deserve to be reorganized to more emphasizing on this property.
 * On the other hand, I remark that all ring constructions (except matrix and homomorphism ring) are universal properties. I suggest to define (in Ring (mathematics) these constructions by their universal properties, to show that the result is isomorphic to the classical construction given in the main articles, and to refer to the main articles for the description of their properties. It would have the advantage to avoid duplication and to show that ring theory allows to unify various constructions that, otherwise could appear as ad hoc constructions.
 * D.Lazard (talk) 11:30, 30 November 2013 (UTC)


 * That is a nice suggestion. It might occasionally lead to some descriptions that are more difficult for a beginner than the concrete description, so maybe the concrete descriptions should be given briefly as well. Ebony Jackson (talk) 05:57, 3 December 2013 (UTC)


 * I like the proposal a lot myself too. It's important not to put random facts but try to give some coherent picture. The "universal property" should clearly be part of this attempt. As for more specifics, I don't know if I do this myself, but for example, the "localization" section needs to mention of the "universal property" of localization; even from the categorical point of view (cf. localization of a category.) The "Group ring" section, perhaps, should be merged into rings with generators and relations to emphasize the universal aspects. -- Taku (talk) 13:57, 6 December 2013 (UTC)

History section
Where should the history of rings section be? Right now, ring (mathematics) and ring theory both have the history sections and they look similar. Perhaps, in some future, there will be history of rings (it's important), but for now I think the history section at ring theory should be merged into one here, the direction of merger is because this article cannot be without the history section while ring theory can. -- Taku (talk) 14:28, 17 December 2013 (UTC)
 * Perhaps the history of the notion of ring could be at ring (mathematics), while the history of the further development of ring-theoretic ideas could be at ring theory? Ebony Jackson (talk) 16:51, 17 December 2013 (UTC)
 * That seems like the logical division: historical origin of the concept here, and then historical development of the theory there. Rschwieb (talk) 13:53, 18 December 2013 (UTC)
 * Ok, you two are right :) But this should probably be part of what we're going to do about ring theory. In my opinion, the article is not functioning as of this moment, though I can see the argument the future might be brighter? for it. -- Taku (talk) 23:03, 18 December 2013 (UTC)

Against usual
It is not necessary to assume that all ring has a multiplicative identity element.--Peiffers (talk) 05:24, 26 July 2014 (UTC)
 * Mathematically not, perhaps, but an editorial choice has to be made about how to organise the material into Rings and Rngs. This was the firstissue raised on this talk page, now in Talk:Ring (mathematics)/Archive 1, back in 2003.  Deltahedron (talk) 07:09, 26 July 2014 (UTC)

Does an ideal have to be defined as an additive subgroup?
The definition currently in the text is "A subset I of R is then said to be a left ideal in R if $$R I \subseteq I$$." I suggested the definition should begin "An additive subgroup I ..." I appreciate that if R is a ring with 1, as is assumed in the article, then the definition as stated implies that I will be an additive subgroup in particular. However, I note that of the texts immediately available to me, Herstein, Jacobson, Lang, Lewis and van der Waerden all include the requirement of being an additive subgroup in their definition of ideal. So perhaps my suggestion is consistent with the reliable sources. Deltahedron (talk) 11:38, 21 September 2014 (UTC)
 * As the text contains the definition of RI, I agree that "additive subgroup" is not formally needed. However, it may be useful, as, without it, the definition is confusing for those who omit to read the definition of RI. On the other hand the present definition is unnecessarily too technical. I suggest "A subset I of R is then said to be a left ideal in R if the sum of two elements of I and the product of an element of R and an element of I always belong to I. Equivalently, I is a left ideal if $$RI\subseteq I,$$ where$$R I$$ denotes the span of I over R, which is the set of finite sums$$r_1 x_1 + \cdots + r_n x_n, \quad r_i \in R, \quad x_i \in I.$$ This isequivalent to say that I is a left submodule of R (viewed as a module over itself)." The definition as submodule is lacking and is fundamental in many applications. By the way, a section on modules is lacking in this article (The section "The action of a ring on an abelian group" may hardly be considered as a section on modules). D.Lazard (talk) 12:33, 21 September 2014 (UTC)


 * I now see that maybe the "ideal" (couldn't resist) definition of an ideal might be something like that of subring: a subset I is an ideal if x + y and rx are in I for all x, y in I and r in R. The parallelism is helpful for the readers to compare the two definitions. The definition "RI\subseteq I" is because it really explains the only requirement is that R acts on I as a ring; i.e., I is a submodule. I'm not too sure about the "module section"; obviously, the readers should go to module (mathematics) to read about modules. There is some ring-theoretic aspects of the modules theory (hence, the emphasis on ring action), but the article shouldn't get too deep into modules, in my opinion. -- Taku (talk) 14:33, 21 September 2014 (UTC)


 * I went ahead and put it back to including the "additive subgroup" because that is the most common wording. We gain transparency, and any parallelism with other definitions is a bonus. Rschwieb (talk) 13:01, 22 September 2014 (UTC)

Redundancy of ring axioms
Interestingly, it is not circular to conclude that commutativity is a consequence of the other axioms. One could probably drop some other axiom (say, right distributivity) instead. That this redundancy is not obvious (as seen by the to-and-fro) suggests that it makes sense to note that the ring axioms are redundant, but singling out commutativity alone for such treatment may not be sensible. —Quondum 05:15, 23 April 2015 (UTC)
 * In my textbooks, commutativity of addition is defined before the axioms of multiplication are defined and before cancellation laws are deduced. It did not make sense to me to say that commutativity of addition follows from the other axioms. If it can be shown that an axiom can be deduced from axioms which follow it, I think that should be stated in a separate section. I think it should also be proven rigorously, since sometimes an axiom only appears to follow from others. For example, the reflexive property of an equivalence relation. — Anita5192 (talk) 05:35, 23 April 2015 (UTC)
 * I have reinserted it under the guise of general redundancy of axioms, with a clearer, simpler proof. I think that it would be inappropriate to single out commutativity as the redundant axiom, but any of several could presumably be omitted. It may be seen that both distributivity laws, multiplicative identity and cancellation all are used. In this guise, whether the omitted axiom is somewhere in the middle hardly matters. But the sheer surprise we all seem to experience when seeing this seems to argue that a note in the article is warranted. Do you think I've met the criteria you've mentioned (separate section, rigorous proof, also made much clearer)? —Quondum 05:53, 23 April 2015 (UTC)
 * PS: My favourite example of redundancy of axioms is for a group. We can define a group as a nonempty associative quasigroup. Note that there is no need to axiomatize an identity or inverses; two-sided division suffices. —Quondum 06:06, 23 April 2015 (UTC)
 * I don’t think Slawekb’s last two edits were helpful, as the fact that commutativity of addition follows from the other axioms is interesting, unlike the trivial fact that one half of the additive inverse axiom is redundant in the presence of commutativity (not to mention that the choice of axioms is now weirdly inconsistent, as the additive identity axiom is still stated as two-sided). It’s not OR. Some references can be found in http://math.stackexchange.com/a/346682, for instance.—Emil J. 11:30, 23 April 2015 (UTC)
 * By the way, the same argument also shows that commutativity is redundant in the axioms of modules (and vector spaces).—Emil J. 11:43, 23 April 2015 (UTC)
 * The axioms we give at vector spaces only assume that the additive inverse and identity are one-sided. So, no, it does not follow that commutativity is redundant with the other axioms.   Sławomir Biały  (talk) 11:51, 23 April 2015 (UTC)
 * You know perfectly well what I meant, and anyway, this is of no concern to this article.—Emil J. 11:54, 23 April 2015 (UTC)
 * I don't think the adversarial tone is constructive. I don't "know perfectly well what [you] meant", and if it "is of no concern to this article", why bring it up in the first place?  Standard sources define rings by three axioms, not eight.  So it makes sense to write the article from that perspective.  None of these sources point out any redundancy in the axioms and, indeed, these three axioms are manifestly independent of each other.  (Although this does not preclude the possibility of axiomatizing a ring in some other way; naturally that would require a good source to include.)  The source mentioned at stackexchange does not directly support the claim made in the article.  Betsch is discussing fields rather than rings, and the statement lacks a proper citation anyway.   Sławomir Biały  (talk) 12:16, 23 April 2015 (UTC)
 * Well, well, well. Much ado about nothing.  I notice that no-one has addressed my opening comments in this thread.  I really don't see why a separate note about axioms being redundant should trigger a flurry of edits on the axioms themselves. There is nothing wrong with axioms being redundant; this is quite normal, as the note mentioned.  And if I was to go through this and other maths articles removing everything that is unsourced, I'm afraid the content would be sorely diminished; I would be interested in comment on the consistency of the application of criteria for inclusion.  —Quondum 14:22, 23 April 2015 (UTC)
 * It's not for want of looking that I removed the content in question. Indeed, all sources that described rings by means of axioms included only three axioms.  No comment was made in the standard places that the axioms are not independent, because this depends on the way the three axioms are unwrapped into properties.  Naturally, the "properties" that a rimg satisies are not generally expecyed to be independent.  No, we shouldn't remove all unrefined content, but when content fails the basic test of WP:V, it should be removed, in my opinion. In this case, it is truly much ado about nothing.  There is not even the barest indication that these issues have been considered anywhere in the literature.  (The nest peopke have come up with is the ضaforementioned stackexchange thread refers to an unnamed paper by Henkel, decades before the concept of a ring was introduces by David Hilbert.  Needless to say, this would make an extremely poor source.)  Sławomir Biały  (talk) 15:33, 23 April 2015 (UTC)
 * Not “nest”, just first. I. M. Isaacs, Algebra: A Graduate Course, AMS, 1994, p. 160. K. D. Joshi, Foundations of Discrete Mathematics, New Age International, 1989, Exercise VI.1.1, p. 405. For modules: S. Mac Lane, G. Birkhoff, Algebra, AMS, 1999, Exercise V.1.6, p. 162. I’m sure you could find more with a bit of effort, this is simply a well known observation.—Emil J. 16:24, 23 April 2015 (UTC)

Ring (mathematics)
A more basic way to learn rings, beyond comparison to modulus, is to investigate Vector space "Definition" (Linear algebra topic) which is ring for simpler objects (linear) and which shows readily hand done ways to determine or create. — Preceding unsigned comment added by 72.219.207.160 (talk) 16:33, 22 December 2013 (UTC)

"Abelian groups" and "monoids" are too technical
This article, and in particular the definition of a ring in this article, should be accessible to a person who does not know what an abelian group is, or what a monoid is. An relevant Wikipedia guideline is Make technical articles understandable. Regarding where we should be on the scale of understandable vs. technical, I think it's a bad idea to follow Bourbaki. By worrying too much about redundancy I think we are at risk of making this article less understandable. I believe introductory textbooks more commonly list in the neighbourhood of 8 axioms, rather than just 3; probably because the 8 easy things are easier to understand than the 3 more difficult things. So I think we should be numbering the axioms as they were before. Mark MacD (talk) 13:27, 23 April 2015 (UTC)


 * We are following not just Bourbaki, but also MacLane and Birkhoff, and Serge Lang. I note that in the previous version, the words monoid and abelian group also appeared. All I have done is to delineate more clearly that there are only three axioms, not eight.   This is well sourced to standard references.  I did not see any introductory treatment that called the eight previously numbered items "ring axioms".  And indeed it seems like our own sloppiness here is the entire source of the confusion in the previous thread.
 * Sławomir Biały (talk) 15:20, 23 April 2015 (UTC)


 * Is it correct to call the three listed criteria "axioms"? Axioms that are defined in terms of more axioms seem to be a misnomer. —Quondum 16:07, 23 April 2015 (UTC)
 * Yes, they are axioms. See the cited sources. (And anyway, this is a complaint that could just as well apply to the earlier revision, which relies on the notion of "set", which satisfies some axioms of its own that we have not included.)  Sławomir Biały  (talk) 16:27, 23 April 2015 (UTC)
 * It was not a "complaint". It was simply a question about terminology. —Quondum 17:04, 23 April 2015 (UTC)


 * When I searched Google books for "Ring axioms", the first 10 hits all listed between 7 and 9 axioms for a ring (which I could see from the snippet view), numbered in a wide range of different ways. My point is that sources differ regarding the how to define a ring, and even when they agree they differ on how to present the definition; recall this epic table in this talk page's archive. So we should be using our editorial judgement for what is best for the reader, rather than simply following a few favourite sources of mathematicians. Mark MacD (talk) 22:39, 23 April 2015 (UTC)

While they are really definitions, many standard sources say axioms.Rick Norwood (talk) 19:57, 23 April 2015 (UTC)

Would it be possible to make the introduction more readable, like the Mathworld one?
I've made the point here before, back in 2010

https://en.wikipedia.org/wiki/Talk:Ring_%28mathematics%29/Archive_3#Could_the_start_be_made_more_readable_for_a_non-mathmatician_.3F

that despite having a BSc, MSc and PhD in science subjects, I found the introduction incomprehensible. In particular, at that time, there were numerous things in the first two sentences which were indecipherable. Now if I look on Mathworld article on rings

http://mathworld.wolfram.com/Ring.html

the introduction is written in a way that is 100 dB easier to understand than what is currently on Wikipedia. To copy just some of it.

''A ring in the mathematical sense is a set S together with two binary operators + and * (commonly interpreted as addition and multiplication, respectively) satisfying the following conditions:

1. Additive associativity: For all a,b,c in S, (a+b)+c=a+(b+c),

2. Additive commutativity: For all a,b in S, a+b=b+a,

3. Additive identity: There exists an element 0 in S such that for all a in S, 0+a=a+0=a,

4. Additive inverse: For every a in S there exists -a in S such that a+(-a)=(-a)+a=0,

etc etc''

The Wikipedia article is a bit more penetrable than it was 7 years ago, but still seems to suffer this problem. Yet another author, can provide an introduction that far easier to read.

Would it not be sensible to make the introduction more like the Mathworld one? Drkirkby (talk) —Preceding undated comment added 01:32, 27 March 2017 (UTC)


 * You can make things easier by removing difficult stuff. The question seems if that's desirable here. The mathworld article is definitely an easier read for those without prior backgrounds in abstract algebra. The Wikipedia article, at the current form, seems to be less interested in actual teaching rings to the uninitiated but is more interested in being informative and comprehensive. It is a matter of trade-off: more information-density decreases the readability (just because the readers need to read more). Wikipedia being an encyclopedia, the priority is the comprehensiveness over an easier comprehension. So, no, I don't believe the mathworld is a right style for Wikipedia. -- Taku (talk) 05:21, 10 April 2018 (UTC)
 * Moreover, what you suggest is essentially to move the section "Definition" to the beginning of the lead. This would contradict MOS:LEAD which says that the lead should be devoted to summarize the content and the context of the article, without technical details. Here, the context is fundamental, because the fact that the concept is fundamental for all mathematics is more important, in an encyclopedic context, than the technical details that the reader can find in any textbook. D.Lazard (talk) 09:17, 10 April 2018 (UTC)

This page needs specific examples of the relevance of rings in physics, for example.
I have a very reasonable background in physics, mathematics and philosophy, so should be able to engage with rings quite well. As with the previous commentator, I find rings nebulous. It is not the mathematics, but that they connect abstractions to abstractions and my brain keeps going, 'Who cares!' An example of how this might play out in cosmology or particle physics would be useful. Rings connect to things that connect to Noether's theorem, so it is important for me to understand this. However, as Wigner pointed out, mathematics has an unreasonable effectiveness in the natural sciences, because maths is epistemologically disconnected from science. Leaping the gap between the two is sometimes a jump too far for many people. — Preceding unsigned comment added by Centroyd (talk • contribs) 16:06, 24 May 2019 (UTC)
 * Many classical groups are found as the units of matrix rings. These groups are used to display symmetry in various systems. Two and three dimensional examples are enough to open the door. — Rgdboer (talk) 23:52, 25 May 2019 (UTC)

Main image
Among other edits, I replaced the main image (the cover page of a paper by Hilbert) with an image of a number line. The previous image was really not doing the page any favours. A picture of the first page of an article is hardly interesting, nor illustrative of what a ring actually is, not to mention that it was illegible at that scale anyway. The number line was the first thing that came to mind, as a way to illustrate the archetypal ring of integers. However, I'm open to replacing this with some other illustration of another example of a ring - maybe an image illustrating matrix multiplication, or a graph of a polynomial. --Jordan Mitchell Barrett (talk) 03:21, 20 December 2019 (UTC)

In §Definition, shouldn't the axiom of multiplicative identity exclude the 0 element?
The axiom of multiplicative identity on the page right now reads: However, a · 0 ≠ a and 0 · a ≠ a for all a in R That is, axiom of multiplicative identity needs to exclude the additive identity element (0).
 * 1) * There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R  (that is, 1 is the multiplicative identity).

In fact, a · 0 = 0 and 0 · a = 0 for all a in R   For proof, see page 5 of — Preceding unsigned comment added by Cpt Wise (talk • contribs) 02:03, 17 January 2020 (UTC)


 * 1 is still a multiplicative identity with respect to 0, so what's there is fine. Since 0 times anything is still 0, you could eliminate 0 from the multiplicative identity requirement for 1 (as Mathworld seems to do), but the two versions of the axioms are equivalent. –Deacon Vorbis (carbon &bull; videos) 02:35, 17 January 2020 (UTC)

There is precisely one ring where the multiplicative identity is the additive identity, i.e. 1 = 0, namely the single element zero ring. It too satisfies both forms of the multiplicative identity axiom, and there is no single element rng without a multiplicative identity, so in ring theory including or excluding the additive identity from the multiplicative identity axiom makes no difference. 2A00:23C6:1482:A100:C9BC:989B:9556:A886 (talk) 09:32, 23 July 2020 (UTC)