Talk:Ring (mathematics)

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"Constructions" section too big[edit]

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)

Non-empty[edit]

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[edit]

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)

History section[edit]

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)

Ring (mathematics)[edit]

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.