Talk:Ring (mathematics)

Possible topics to add to article

I think that the article has reached a stage at which it is appropriate to add some more advanced topics in ring theory. However, ring theory is such a vast subject that it is difficult to assert what exactly should be included in the article. One possibility would be to choose a well-known general textbook on algebra (such as Lang's famous text) and select the most important topics from there. However, that would exclude several important topics in ring theory that are too advanced to be mentioned in a general algebra textbook. I would be interested to hear whether anyone has views on which topics we could add. I have composed a list below of possible topics which I think could be added (some may be perhaps too specialized), but if anyone has any other suggestions they are very much welcome. I have split the list into "commutative ring theory", "noncommutative ring theory" and "applications of ring theory" and indicated the "importance level" (in my opinion) in terms of stars (* indicates "low importance", ** indicates "mid importance" and *** indicates "high importance"); of course, importance is subjective and measures the importance of including the topic in this article, and I would appreciate hearing the views of others. Note that some topics may already be covered to some extent in the article, and the lists are of course not intended to be comprehensive; they merely serve to give some representative suggestions for possible topics.

Commutative ring theory:

"Basic" topics:

• Module (***)
• Prime ideal (***)
• Primary ideal (**)
• Primary decomposition (*)
• Lasker-Noether theorem (*)
• Noetherian rings (***)
• Artinian rings (**)
• Localization (**)
• completion (*)
• Nakayama's lemma (**)
• Integral extensions (**)
• The "Going up", "Going down", "Lying over", "Lying down" etc. and other fundamental theorems in integrality (*)
• Dedekind domains (**)
• Hilbert's basis theorem (**)
• Hilbert's Nullstellensatz (**)
• Hilbert's syzygy theorem (*)

"Advanced" topics:

• Homological algebra, the most important concepts such as the tensor product, the "Tor functor", the "Ext functor", resolutions etc. (***)
• Krull's principal ideal theorem (*)
• Krull dimension (**)
• Flat modules (*)
• Hensel's lemma (*)
• Grobner Basis (*)
• Cohen-Macaulay rings (*)
• Gorenstein rings (*)

Noncommutative ring theory:

"Basic" topics:

• Jacobson radical (**)
• Jacobson density theorem (*)
• Central simple algebras (*)
• Brauer group (*)
• Dimension theory (*)
• The Wedderburn-Artin structure theory (**)

"Advanced" Topics:

• Polynomial identity rings (*)
• Goldie's theorems (*)
• The Golod-Shafarevich theorem (*)

Applications of ring theory:

"Basic" topics:

• Algebraic number fields (***)
• Class number formula (**)
• Dirichlet's unit theorem (**)
• Local ring of a variety (**)
• Various algebraic constructions in classical algebraic geometry (**)
• Burnside's theorem on groups of order divisible by no more than 2 primes (**)

"Advanced" topics:

• Scheme theory (**)
• Sheaf theory (**)
• Sheaf cohomology (**)

PST 03:23, 27 May 2010 (UTC)

It seems like the Ring theory article might be a better place for these topics, it article is getting a bit on the long side and the other one could do with some expansion. --Salix (talk): 13:57, 27 May 2010 (UTC)

Thanks Salix! I certainly do not intend to write a section on each of these topics (that would, as you say, make the article too long). Rather, we could write a brief paragraph outlining some of the major results in certain areas (for instance, a well-written paragraph outlining commutative algebra without going into too much detail). However, I certainly agree that the article Ring theory should be expanded along these lines and I hope to do so at some point in the future. PST 03:21, 28 May 2010 (UTC)

disagreement on definition

Planetmath defines a ring without the identity element. Some reasons for this are historical and if the identity is a requirement in the definition, morphisms must preserve identity elements. —Preceding unsigned comment added by 72.93.178.132 (talk) 03:21, 30 June 2010 (UTC)

Could the start be made more readable for a non-mathmatician ?

I'm not a mathematician, but I have a BSc in Electrical and Electronic Engineering, an MSc. Microwaves and Optoelectronics and have a Ph.D in Medical Physics. So I'm not a total idiot at maths. But just reading the first paragraph, I was totally lost! If someone with a science Ph.D gets lost in the first paragraph, something is wrong.

I have enough knowledge to know this is a non-trivial subject, and do not expect the article to be dummed down so a 5-year old can understand it all. But I can't help feeling it could be better written. There's a section in Wikipedia about how to make technical articles understandable. It even covers the fact that this may be particularly difficult in mathematics, in a section on articles that are unavoidably technical but it does have some suggestions, which could perhaps be implemented. To quote, There should be at least a sentence in the lead of the article to give the lay reader some idea of the place the subject holds in mathematics, what (if anything) it is good for, and what needs to be learned first in order to understand the article. A better place for going into technical details might be in the body of the article, after the Table of Contents.

The three terms algebraic structure, abelian group and monoid are all in the first two sentences of the article. I doubt few people who have not studied for a mathematics degree would know what any of those terms are.

I believe the the ring article on Mathworld is a lot more understandable, though I believe a skilled editor could do an even better job. Drkirkby (talk) 11:03, 25 August 2010 (UTC)

For context the current intro reads

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations (usually called addition and multiplication), where each operation combines two elements to form a third element. To qualify as a ring, the set together with its two operations must satisfy certain conditions—namely, the set must be an abelian group under addition and a monoid under multiplicationa[›] such that multiplication distributes over addition. Certain variations of this definition are sometimes employed, and these are outlined later in the article.

I agree that the current introduction is a quite advanced. algebraic structure is there to put the rings in context, as one of a whole family of other structures, each with different definitions. abelian group means that additions has and identity, every element has an inverse, the operation is both associative and commutative. We could expand the intro to include this, however I would say understanding groups is an essential prerequisite to understanding rings, so there is a good case for linking this term rather than spelling it out and introducing 4 more terms into the intro. monoid is a less frequently important operation so expanding this might be of uses. All these details are spelt out in the Definition and illustration section just under the intro.

You might like to read Advice on using Wikipedia for mathematics self-study which has been recently written.--Salix (talk): 17:40, 25 August 2010 (UTC)

Thank you Salix. Since you say understanding groups is an essential prerequisite to understanding rings, that would be useful to state in the introductory paragraph. Further down I just read that the concept of a ring first arose from attempts to prove Fermat's last theorem. Again, that could be moved further up, to make it more accessible, as most people looking at the page will probably know what that is.

It seems that the first paragraph is written to be mathematically rigorous, but little or no thought given to make it accessible to a larger number of readers. I used to work with a guy who was a mathematician, but he always seemed to be able to explain things to me in a way that did not leave me lost after two sentences. Drkirkby (talk) 19:41, 25 August 2010 (UTC)

I would say that understanding groups is certainly not a prerequisite to understanding rings. I could point for example to numerous introductory algebra books that cover rings before delving into groups (e.g. Hungerford's baby algebra book). I feel like Drkirkby is right in saying that the first paragraph is needlessly technical. The point of a ring is to abstract and generalize the addition and multiplication in the integers, not to have a structure that is an abelian group under one operation and a monoid under another. Perhaps something like the following would be a better first paragraph:

A ring is a mathematical concept abstracting and generalizing the algebraic structure of the integers, specifically the two operations of addition and multiplication. Rings appear as a basic concept in most fields of mathematics, though the study of them for their own sake is part of abstract algebra — specifically ring theory.

One could probably also throw in a sentence summarizing the history. In a subsequent paragraph one could delve into the specifics of the algebraic structure, but I think what I've written does a much better job of placing rings in context than what is currently written. RobHar (talk) 22:30, 25 August 2010 (UTC)

Without intending to claim ownership, I should note that a large part of the introduction was written by me. I did put a great deal of effort into making the introduction as accessible as possible - above this section there are several users who requested that the introduction be made accessible, and most of their criticizms were incorporated each time I revised the introduction.

My advice would be to read through the introduction in such a way that you skip terms with which you are not familiar. Put differently imagine you are reading another language!

Unfortunately, this is the best advice I can offer you. I have been editing this article for approximately 1.5 years and quite frequently have been updating the introduction. Your criticizm is very welcome and I shall try my best to incorporate it should I get the time to revise the introduction the near future. (Or another user may perhaps do so.) One thing to keep in mind is that the section directly below the introduction, i.e., the motivation section, has been written so that it should be accessible to virtually anyone who knows basic arithmetic. Therefore, you might gain a better appreciation of the introductory paragraphs if you read the motivation section first. PST 10:00, 26 August 2010 (UTC)

I'll take your advice and just skips the words I don't understand. That might be some help to me. But it is certainly now how the article should be written. The section in the Wikipedia guidelines on articles that are unavoidably technical specifically says the lead should be understandable to the lay reader, and the best place to go into the technical details is after the table of contents. I appreciate that some of the material before the table of contents is not too technical, but before getting there, I'm totally lost. The fourth paragraph is a lot more understandable than the first two sentences. That seems a bit backwards to me. The truth is, people are not likely to ever read to paragraph 4, if they are totally lost by the first two sentences!

BTW, just to fill you in on my motivation for wanting this. I spend some of my spare time developing the open-source Sage mathematics software. My interest is in using Sage in preference to the expensive closed-source products like Mathematica, which I've used quite extensively. My contributions have been mainly on porting to Solaris and on improving quality control. Sage is particularly strong in the field of number theory (way ahead of Mathematica I'm told). An understanding of rings and fields is pretty essential to using Sage for many problems. So I was rather hoping Wikipedia might at least give me some understanding, but I was totally lost after just two sentences. Fortunately, there do seem to be some web articles that cover this topic with less buzz words, and written so one does not need a maths degree to understand the contents of the first two sentences.

I don't know your background, but I expect you have a maths degree and would not be surprised if you studied for a Ph.D in a similar area. You need to realise only a very very small subset of the population have that background, yet there are many more like myself, who would like to get some understanding of a subject like this, but who will not appreciate it when they can't understand the first two sentences. Drkirkby (talk) 11:18, 26 August 2010 (UTC)

I agree with you and wish it were possible to make this article as accessible as possible. A couple of users, including me, are trying our best to do exactly this. But doing something like this is hard work - it cannot be done overnight. I want to try my best to help you and surely others do to. The best suggestion I can give you if you want to learn about rings is to either take a proper algebra textbook and learn about the subject from there. (If you want recommendations, please feel free to contact me on my talk page and I would be more than happy to oblige.) Or, you could read the article Group (mathematics). (Which is far, far better written than this one - although I am one of the primary contributors to this article, I am not in any way pretending that the article, at this stage, is of the same high quality as Group (mathematics).)

You are obviously highly qualified and and the topics in which you have a background are indeed connected to mathematics in many ways. You should understand that it is in no way your fault that this article is not immediately accessible to you - it is more the fault of the contributors of the article. We will try our best to improve this article, and hopefully, one day, it will become very good. But for now, the suggestions I gave you in the above paragraph seem to be the best possible. (Yes, I am a mathematician, and indeed I do read the literature on ring theory and research the subject - however, it is not my primary research speciality. But I will admit that I know virtually nothing about medical physics! I would not be able to read an article on a topic of analogous difficulty to "ring (mathematics)" in medical physics.) PST 04:34, 27 August 2010 (UTC)

Rewrites to the introduction

Recently, there has been a serious attempt to rewrite the introduction by User:Rick Norwood. I wish to thank Rick Norwood for this. While I do not wish to give the impression that I think Rick Norwood has not put any thought into writing the introduction, it is important to note that a great deal of thought is necessary to write an introduction. I expect this will take at least one month (perhaps much, much more), since there needs to be not only a general consensus that the introduction addresses all of the concerns above, but the introduction also needs to conform to certain Wikipedia standards, e.g., WP:MOS.

That being said, I do wish to note some of my concerns of the revised introduction which I hope Rick Norwood will address. The first paragraph reads as follows:

Mathematicians classify sets of numbers in various ways. One such classification scheme is called abstract algebra, and it differs from ordinary algebra in considering more complecated mathematical constructs than the ordinary numbers of arithmetic. Three important classes of numbers considered in abstract algebra are groups, rings, and fields. A ring can be formed from a group by the addition of a second operation, usually called multiplication. A field is more specialized than a ring in that it has certain additional properties. For example, in a field every number except zero has a reciprocal. This is not necessarily the case in a ring. The classic example of a ring is the ring of integers.

My concerns are the following:

• The first two sentences appear to be setting context for what an algebraic structure is. I believe that this is really important and should be there before "algebraic structure" is mentioned - thank you Rick! However, the point of view taken is that algebraic structures are types of "number systems". This is true in some sense, but if this view is to be emphasized, it needs to be a prominent theme of the entire article. Perhaps more importantly, it needs to be a view shared by several important references. Therefore, I suggest that the first two sentences are changed so as to still give context for the rest of the paragraph, and to still explain intuitively what an algebraic structure is, but to also propogate a view that can be backed up by several reliable sources.

• The rest of the paragraph is very good but perhaps it can be written slightly more formally, while still retaining an "informal style".

• The last sentence of the paragraph is slightly debatable. "Ring of integers" has other meanings in algebra, especially algebraic number theory, and since ring of integers primarily concerns the more general version of "ring of integers", the last sentence could potentially be more confusing than helpful to a layman. The best suggestion I can offer is to modify this sentence slightly while still retaining the essential idea. For instance, "The classic example of a ring is the set of all integers together with the two operations of addition and multiplication." could work but more thought might be necessary.

Of course, as is evident from my above criticizms, there are not too many problems in the introductory paragraph. In particular, Rick Norwood has done an excellent job and this should too be emphasized. PST 22:02, 26 August 2010 (UTC)

However, I notice that Rick Norwood has made identical edits to Group (mathematics) and Field (mathematics). One of the primary contributors to Group (mathematics) reverted the edit - [1]. While I think that Rick's edits were good, this revert by an experienced editor made me wonder whether there needs to be a general consensus that these edits are good. (And e.g., address the above concerns.) This needs to occur before the edits are reinstalled since this article is quite frequently viewed and we probably do not want to make big changes to it before they are fully discussed on the talk page, agreed upon, and referenced. When that is done and the added material is written in an encyclopaedic tone, the edits can immediately be reinstalled. Hopefully this can be done quickly since as I mentioned above, the edits were very reasonable. PST 04:50, 27 August 2010 (UTC)

My long experience editing Wikipedia tells me that if we wait a month to make a change, no change will ever be made.

Should we call the elements in a ring "numbers"? I didn't use the word lightly, and of course I know that I'm using "numbers" in a very broad sense. But I could not think of another word that would convey anything to the lay reader. The correct word, "elements", probably suggests carbon and oxygen, or maybe earth, air, fire, and water.

I'm happy with your change from "ring of integers" to "integers together with the two operations..."

I strongly believe that there should be some uniformity among the articles "group", "ring", and "field". It won't be easy. In the past, I've treid to get some uniformity among the articles "parabola", "hyperbola", and "elipse" without success.

Rick Norwood (talk) 12:07, 27 August 2010 (UTC)

Rick, apologies for all the confusion with the reverts. I think that, at least for the time being, your revision of the article should be kept. I have reverted my revert of your revisions. I think that we should maintain your revisions. Perhaps if we discuss some of the minor points of your revision that needs fixing we can do so here. I have to be honest - my general view is that there are some (perhaps minor) stylistic problems with your introduction. However, since I greatly appreciate your work and since these can probably be easily fixed, I feel that your revision should be kept, and we will tweak it as necessary to fix up the introduction on a whole. How does that sound? PST 13:29, 27 August 2010 (UTC)

Thanks. I've made some changes as discussed above. I am not sure about including this in the introduction: "A ring can be formed from a group by the addition of a second operation, usually called multiplication. A field is more specialized than a ring in that it has certain additional properties. For example, in a field every number except zero has a reciprocal. This is not necessarily the case in a ring." Should we take it out? Rick Norwood (talk) 16:00, 27 August 2010 (UTC)

Unfortunately the introduction as it stands has two serious problems. The article's title needs to be the subject of the first sentence: the reader should not have to skip a paragraph and sentence to find it. See e.g. WP:BOLDTITLE. And the introduction is too long: it was on the long side at four paragraphs and is too long now. This is especially important for a technical subject - a reader should be able to quickly read the introduction to get an overview of the topic. Too much technical detail is very discouraging and makes the article far less accessible. --JohnBlackburne^words_deeds 16:18, 27 August 2010 (UTC)

I agree that the current introduction has serious problems. Wikipedia is not a children's encyclopedia. Article leads should be as accessible as possible, but this includes accessibility by an article's most likely audience. It is not OK to dumb things down beyond reason, to the point that those who actually have the necessary prerequisites for understanding the subject of an article have to wade through walls of text, searching for tiny bits of relevant information that may be hiding somewhere in the haystack. Why is the following in the lead of the article on rings, rather than the article group (mathematics), field (mathematics), algebra over a field, division ring, abstract algebra, ...?

Mathematicians classify sets of numbers in various ways (a mathematician would use the word "element" here instead of the word "number"). One such classification scheme is called abstract algebra, and it differs from ordinary algebra in considering more complicated mathematical constructs than the ordinary numbers of arithmetic. Three important classes considered in abstract algebra are groups, rings, and fields.

And does anybody really think that the following example of hand-holding is appropriate for an encyclopedia?

The definition of a ring is necessarily technical.

Actually, it's not just inappropriate, I guess it's also blatantly false from the POV of most readers who have the appropriate level of mathematical maturity for reading this article.

As an example from an unrelated field, let's look at the lead of small interfering RNA. It assumes that the reader knows what RNA is, what a molecule is, and what a nucleotide is, and that's just as it should be. A reader unfamiliar with any of these concepts has no chance to understand the article in any meaningful way and should follow the links to learn about the prerequisites first. I am not personally familiar with RNA interference, but it's clear to me from the lead of small interfering RNA that it is another crucial prerequisite and that I would have to read about it first before continuing with the article I am (in this hypothetical case) interested in. I have no doubt that it would be possible to rewrite the lead of small interfering RNA so that it starts with bees and flowers, progresses through Gregor Mendel's experiments, gives a rough description of the basics of chemistry, etc., but that would be totally inappropriate. Hans Adler 18:19, 27 August 2010 (UTC)

I made similar changes to the other articles but they were reverted by people who agree with Hans Adler, whose point seems to be that anyone who comes to this article will already know what a ring is and want further information on the subject. I'm not sure that's true. I made the edit in response to someone who compalined about the article's inaccessability, and I think the attitude that anyone who doesn't know math (and chemistry) should go back to grade school is harsh. "Wikipedia is not a children's encyclopedia." Someone who looks this article up on Wikipedia instead of, say, Mathworld may very well have heard a mathematician use the word and wonder what it means. (I remember John Campbell asking if, when mathematiciand talk about open and closed setd, is that like a lawyer talking about an open and shut case.) But your points about getting the word in the first sentence and keeping the introduction brief are well taken, and I'll edit accordingly. Rick Norwood (talk) 20:15, 27 August 2010 (UTC)

I've shortened the introduction. I think it could be shorter still. Also, I think a better picture could be found. Rick Norwood (talk) 20:27, 27 August 2010 (UTC)

This seems to be just a variation of the old Royal Road problem. Basically, our well written technical articles (obviously not all our technical articles are well written, and this problem does need addressing) have four kinds of readers:

1. Those who basically know most of the contents and just come here for reminders.
2. Those who don't know the contents but have the necessary background for understanding them.
3. Those who don't have the necessary background for understanding an article, but enough to realise that this is the case.
4. Those who are so ignorant that they think when they don't understand anything without bothering to follow the links and learn about the basics first, then it's because the article is badly written.

If we try to make category 4 happy, we are going to fail and alienate categories 1–3. Hans Adler 20:30, 27 August 2010 (UTC)

It's still too long and still doesn't say what a ring is in the first sentence: having it bolded further down where the definition actually appears is confusing and against WP:BOLDFACE, and it should also not be italicised. Everything up to the second "A ring ..." could be replaced with "In mathematics, " to deal with this, though it would still be a bit long. No ideas on a better image – it doesn't add much but it's an abstract topic that's not usually presented visually. Maybe the image could be moved down and associated with a related example. --JohnBlackburne^words_deeds 20:41, 27 August 2010 (UTC)

Some of it was fixed while I was typing, but it still is too long: it should be no more than four paragraphs, especially as the concept of a ring is not a complex one and should be more easily introduced. And the definition is still split over two paragraphs. For me the definition given in the second paragraph would make a pretty good introductory one, as it's precise but uses mostly elementary language. The example in the first paragraph is too early as without formal definitions it's difficult to know what it means. And anyone who wants to know what abstract algebra is can follow the link. So I would replace the first two paragraphs with:

In abstract algebra, a ring is a set with two binary operations (usually called addition and multiplication), where each operation combines two elements of the set to form a third element in the set. To qualify as a ring, the set must satisfy the ring axioms: it must be an abelian group under addition and a monoid under multiplication, and multiplication must distribute over addition. An example is the set of integers with the usual addition and multiplication.

--JohnBlackburne^words_deeds 20:52, 27 August 2010 (UTC)

On first sight this looks like an excellent proposal. Hans Adler 21:13, 27 August 2010 (UTC)

I've done it - replaced what's there with the above. It still looks a bit long as an introduction, I think as it goes into too much detail in the last two paragraphs, though it's not obvious to me what should be cut. Perhaps the historic information could be moved down to its section as it's not necessary for the definition and is still easily found by anyone interested in it ?--JohnBlackburne^words_deeds 21:33, 27 August 2010 (UTC)

Yes the introduction is somewhat long but this is normal for articles in the transition to becoming "good articles". The current article is far from it, of course. I agree that we should discuss these things - however, in my mind, the most important thing is to look at the quality of the article and its coverage. (See the first discussion in this talk page at the very top.) The introduction is about the same size as that of Group (mathematics), so I feel, and this is only my opinion, that there is no need to discuss cutting a couple of lines out in the introduction when the rest of the article is seriously lacking for the more advanced mathematical readers. We can discuss it, of course, and I am more than happy to do so, but I think we should take advantage of the recent surge of activity in the talk page to discuss some more significant problems with the article. PST 23:26, 27 August 2010 (UTC)

The first thing I would like to say is that the current revision has gone "back to square one". Before Rick Norwood made his edits, the article looked like this. JohnBlackburne's edits were very good, although it appears that they just returned the article to an old state.

The sad part is that this article has not progressed very much for roughly 20 months. (I have been watching this article for about that long.) I really do want people to come and contribute this article - but it is becoming such that whoever comes to try, fails, and goes away. And we are talking about experienced contributors here. (At least 5 have failed in the last 20 months after making major attempts.) Now it has mainly been me reverting their edits and telling them that their edits have certain serious problems that cannot be fixed. I really dislike doing this - I know they put much effort into trying - but I simply cannot hide the truth of what their edits really are.

This is exactly the case with Rick Norwood here. First, I reverted his edits. (As well as his similar edits to field (mathematics); another user reverted his duplicate edits to group (mathematics).) But then I began wondering - OK, his edits have serious problems (as described above by Hans Adler and JohnBlackburne). These were glaring at me. But if I continue to revert edits, no matter what they are, of people who are making serious attempts to improve the article, I began wondering what would be the future of this article. Although most of the current introduction, as well as the rest of the article, seems to have been written by me (I hope this does not sound haughty - many other people gave suggestions and helped too, and without their suggestions this article would not be possible), I cannot continue to "protect" this article so to speak. I might retire from Wikipedia tomorrow! I really do not know. Therefore, I decided, that in the best interests of this article and Wikipedia, Rick's revisions should be kept. My hope was that this would encourage future editors to contribute.

But what was going to happen was inevitable. Rick's revisions had serious problems and that cannot be denied. Consequently, his edits were reverted. I do not have any say in whose edits should be kept - the community does. I gave my best attempt to encourage others to contribute to this article. In the end, I am inclined to agree with Hans Adler and JohnBlackburne. However, I hope that this does not deter other editors to contribute to this article in the future. In my mind, this is the most significant problem the article has at the moment. The sad part is that we cannot simply "accept" every edit that is made to this article, and most edits do have serious problems. (My edits have problems as well, of course!)

At the very beginning of this talk page, I have suggested some possible topics that could be added to this article. Perhaps those should be our primary inspiration now. There is absolutely no mention of topics like "scheme" or "sheaf" and similarly nothing (or very little) about the algebraic number theory side of ring theory. Currently, real mathematics readers are being deterred from reading this article. PST 23:26, 27 August 2010 (UTC)

I've had another go at it myself, to get the length down to what I think's reasonable without losing any of the mathematical content. The main changes were

• moving the historical information out - I looked at merging it with the History section but the only thing that was in the lede and not the section and so was moved was the date and reference, which I tidied up slightly.
• merging the following two statements, which in practice meant picking one (the second) and removing the unencyclopaedic bit in the middle, but don't have any particular preference – they both seem a bit imprecise.
  • Ring theory studies those properties which must be true for any set that obeys the ring axioms.
  • Modern ring theory—a very active mathematical discipline—studies rings in their own right.
• reorganising what's left into three paragraphs.

I also moved the image and template into the large gap next to the TOC as that seemed sensible, swapping their order as they look better with the larger template on top, but I'm still unsure where the image should go: as Brews notes it's connected to more advanced examples of ring theory, and the connection might be clearer with a simpler image.--JohnBlackburne^words_deeds 19:00, 28 August 2010 (UTC)

A quick search on commons turned up another image which I've put in its place, with a link to Polynomial ring as that establishes the connection, but even without that it's clearer I think as adding and multiplying polynomials is high school math.--JohnBlackburne^words_deeds 19:20, 28 August 2010 (UTC)

Thanks JohnBlackburne! Your changes were very good. A few things come to mind:

• The image is great! One concern is that in ring theory, one does not add and multiply two polynomials pointwise - one adds and multiplies polynomials as formal objects. The reason for doing this is that, over a finite field for example, many "formally different polynomials" could be equal as functions defined on the field. I know this is a small and bothersome technicality that few people would consider, but it is certainly a concern, in my opinion, if mathematically incorrect statements are propogated in the introduction itself. On the other hand, I like the image very much. It might be non-trivial to find a similar image that is mathematically correct in all contexts. The earlier image depicting addition in projective space is correct but is simply not intelligible to many people. That might include, for instance, undergraduate students learning ring theory for their first time. (And who might not know projective geometry.) I think we need to think about this more - another possibility could be addition on an elliptic curve (which is an abelian variety) but that does not make much sense since this article is about rings, and not groups! (Note that while the image does not explicitly describe the addition and multiplication of two curves (which might not be obvious, especially to a lay reader), the only possible interpretation seems to be addition and multiplication pointwise, especially since the graphs are depicted geometrically. And as I mentioned above, this is not the way polynomials are added and multiplied, in general.)

• Where did the history section in the introduction go? Could you please add it back to the history section? I think this is important. My other concern is that, while Alain Connes is no doubt a very professional mathematician, to have his name in the introduction but not David Hilbert's or Emmy Noether's name, does little justice to the subject. I feel that the history section should be added back - the introduction is for that sort of thing - describing the history of the concept is one important part of the introduction.

• I agree the introduction should be short, but now, at least in my opinion, it is too short! I am not necessarily suggesting that the introduction should be identical to that of group (mathematics), but the introduction in the latter article is much bigger than this one, and since the latter article is a featured article, it seems reasonable to believe that the introduction should be slightly longer. And as I mentioned in my second point above, there are several key points missing in the introduction. We should not sacrifice quality of writing and coverage for brevity, in my opinion. Also, per Wikipedia:LEAD#Length, if the article size is more than 30,000 characters (which it is - I think the size is double this), the lead should be about 4 paragraphs. There are certain things that probably should be deleted in the article, but it is certainly not the case that half of the article should be deleted. Since the article contains 60,000 characters, the introduction needs to be bigger. (The article size is definitely correct - in fact, it needs to be longer. For example, the article group (mathematics) is more than 3/2 this article size and it is a featured article. Please do not think that we need to trim the article, at least at this stage!)

I think the above three points need to be addressed. However, if I get a chance, I will try to address some of them myself. PST 00:23, 29 August 2010 (UTC)

I have revised the introduction. While at first glance, it might look long, this is deceiving since per WP:LEAD, it can be 4 paragraphs if necessary. (And "ring" is a concept where a slightly longer introduction may be necessary.) Compare also to the introduction of group (mathematics). I think that the current introduction also prepares the article for future expansions. Ultimately, the article will grow bigger, and there is no point in trimming the introduction now and then expanding it as the article grows. We should monitor the growth of the article and decide on this basis how to improve the lead. However, this is just my opinion. Other comments are welcome. PST 00:59, 29 August 2010 (UTC)

Figure in introduction

The lead figure is striking as an artistic creation, but it is not accessible to the lay reader. It introduces in its caption a bunch of technical terms (projective space, geometric addition, algebraic geometry) that are not relevant as explanation of the figure. Instead, the caption is hijacked to become a marginal note to connect to other topics.

How the figure relates to geometric addition is not explained, either in the caption or the text. The connection to the topic Ring is not spelled out, although one might surmise that "geometric addition" is some form of "addition" that is one of the two binary operations of a ring. A lay reader will know possibly what "addition" is in high-school algebra, maybe in vector algebra, but I'd say "geometric addition" is a concept they never will have encountered, and they will be unable to see how the figure relates in any way to addition of any kind.

So, bottom line, if the figure is retained, a digression to explain its content is necessary. Brews ohare (talk) 17:18, 28 August 2010 (UTC)