Talk:Ring (mathematics)/Archive 3

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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)

Planetmath is correct. The definition given in this article under "formal definition" conflicts with normal mathematical usage. It also conflicts with the less formal definition given in the introduction, and with the table of definitions immediately following the introduction. I question whether the "some authors" who define a ring without requiring a multiplicative identity do not in fact constitute a very large majority. Certainly if you were to talk or write about "rings", and what you said applied only to rings with unity, you would have to make this quite clear. If a single word is required to describe a ring with unity, mathematicians such as the author are free to come up with one, provided that it has not been done already. But let this word be at least pronounceable, as opposed to the "rng" that we are apparently supposed to use to describe what is usually referred to as a ring. Perhaps "uring" might pass these days. JHJ47 (talk) 00:47, 26 September 2011 (UTC)

This terminology does not conflict with normal mathematical usage in that there is no normal mathematical usage. Being a number theorist, my rings have unity and are commutative. If I wish to speak of non-commutative rings, I might say that, or use the term "algebra". If I want to speak of a non-commutative ring without unity, I'd probably call it a non-unital algebra. People in different fields mean different things when they say "ring". This article has simply picked one of those meanings. RobHar (talk) 01:58, 26 September 2011 (UTC)

Actually, the article has picked out two of those meanings, with major sources for both, but it is good to have a specialist weigh in. Rick Norwood (talk) 21:02, 26 September 2011 (UTC)

Whatever the rights and wrongs of this question, the inconsistency between the definitions in various places, noted in my first edit, need to be attended to. As regards the question itself, consider the following scenario. In some specialized branch of mathematics, the only groups that feature are Abelian groups. Workers in this branch of mathematics, talking and writing among themselves, could perhaps reasonably assume that when one of them says "group" s/he means "Abelian group", and perhaps even, when writing textbooks on their own special subject, define "group" to have this special sense. If on rare occasions a more general concept of group is required, one could then talk about a "possibly non-commutative group" or even coin a new word for it. But it would be a mistake for such an expert to write a general article on groups, giving a definition that requires a group to be Abelian, with a comment that some authors did not require commutativity. To do so would sow confusion where none had previously existed, and render explanations necessary whenever the properties of groups were under discussion. Whether this analogy is exact enough to be persuasive, I leave to you. A key question seems to me to be, "which references were writing about abstract algebra in general, and which were writing about some specialized field such as number theory?" 222.153.145.205 (talk) 02:56, 27 September 2011 (UTC) Sorry, that 222.153.145.205 was me (JHJ47). I forgot to log in. JHJ47 (talk) 03:28, 27 September 2011 (UTC)

My point, which perhaps I didn't make clearly, was that the notion of a ring does not have a universally accepted definition in mathematics. So that one has to pick a definition (which as you mention hasn't been done here, see however MOS:MATH#Mathematical conventions where it may be saying that rings are not assumed to be commutative on wikipedia). There was a time when a "field" need not have been commutative, but non-commutative fields are now always called something else (skew-fields, or division rings). That has not happened with rings as far as I believe. Maybe I'm biased because I'm a number theorist, but number theorists certainly have plenty of non-commutative rings that they study, they just don't call them that. If I may add, it seems like the term "ring" comes from the number theory side of things and did originally mean something with commutative multiplication. Anyway, given what is said at MOS:MATH#Mathematical conventions, I think you can assume that wikipedia assumes that rings are not-necessarily commutative and you can make the appropriate adjustments.RobHar (talk) 03:28, 27 September 2011 (UTC)

I think the definition should be changed to be consistent with the others articles that refer to a ring. See Characteristic (algebra) and Glossary of ring theory. They all asume a ring has unity, namely that it is a monoid with multiplication. If one wants to talks of "rings without unity" one should call them like that. — Preceding unsigned comment added by 186.22.56.37 (talk) 17:21, 25 May 2012 (UTC)

This article also assumes that a ring has a unity (a footnote mention that some authors do not assume unity). Thus I do not see what has to be changed. D.Lazard (talk) 19:43, 25 May 2012 (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)

Just restating what already said, but this article is completely inaccessible to anybody that doesn't already know the subject. Honestly i wonder if it is useful for anybody, since I guess (hope) that matematicians do not learn this subject on Wikipedia. — Preceding unsigned comment added by 82.150.248.29 (talk) 11:59, 16 May 2012 (UTC)

The first sentence of the article: "In mathematics, a ring (sometimes called associative ring,see below) is an algebraic structure which generalizes the main properties of the addition and the multiplication of integers, real and complex numbers, as well as that of square matrices." What does "that" refer to? Could that be clarified? (Is "as well as that of" -> "and" correct?) Olli Niemitalo (talk) —Preceding undated comment added 09:26, 27 June 2012 (UTC)

Fixed as suggested, as this is correct. — Quondum^☏ 12:57, 27 June 2012 (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)

While I certainly appreciate the point @HansAdler is making, I wanted to provide an alternate viewpoint. I'm a graduate student in physics, I'm not an expert in mathematics (by any means), but lets say I'm versed in the basics. The 'ring' introduction, as it currently reads, is not only opaque, but virtually indecipherable because of its precision, use of jargon, and lack of pedogogical introduction. The introduction, as given by User:Rick Norwood,

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.

is incredibly clear and informative. If such a pedagogical, introductory-introduction isn't included in 'the introduction', I strongly recommend in be include somewhere.... e.g. a 'background section. All Clues Key (talk) 05:10, 3 September 2012 (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)

I think we just have different views on this: I view 4 paragraphs as an upper limit not a recommendation, and this topic as one that's relatively straightforward mathematically so does not need a long introduction. And I think historic information is unnecessary in the introduction, maybe due to my education e.g. history was sometimes mentioned but never thought important and examined. I also tend to think in terms of paragraphs, and the sentences on history don't seem to fit in with the mathematical content. I must say though it does now look much better than a few days ago, with the first two paragraphs in particular a much better introduction.--JohnBlackburne^words_deeds 11:56, 29 August 2010 (UTC)

I think Rick's kind of introduction is a far better idea for the first paragraph of the article than the current one. My suggestion (about three sentences) would be: 1. Start by explaining that this topic is about a generalisation of the concept of numbers in mathematics. (Now the most uneducated lay-reader has a genuine grasp of what this article is about, even if they read no further.) 2. Specifically, that it has addition and multiplication but not division. (This is understandable to almost everybody, no need to get technical about multiplicative inverses yet. The key is trying to capture the distinguishing essence that is specific to the topic, in a way that can be immediately understood without any background study, even at the risk of sacrificing minor precision e.g. element vs. number) 3. That it is a logical intermediate between "groups" and "fields". (This is useful because these two topics are already familiar to every first-year math student, noting that rings are a more esoteric topic generally encountered later, but even any layperson would be able to follow the links and understand either of those two topics at the most introductory level, and thereby by analogy has an idea of what kind of thing this page is dealing with too.) 4. If there is any important everyday application or a technology that used rings for its invention, this would be the place to mention it in passing. (So the reader should encounter the paragraph saying that rings are a hugely popular abstract mathematical subdiscipline and have had relevence to physics, before they hit any technical details, rather than after. Also start an article subsection "applications" to flesh out what these contributions are.) 5. New paragraph. (Now, by all means, proceed with introducing the topic as you would to an audience with the prerequisite background to be able to truly understand the topic. Ideally still keep it brief, and break most of it out into the first subsection, where it can be quickly passed over by experts who are just using the article for reference anyway.) Cesiumfrog (talk) 00:13, 20 December 2010 (UTC)

Over a year later, I would like to second what Cesiumfrog said. This article is still very confusing, and unclear. The broad background, efficiently delivered by Rick, is completely missing - and without that context, the article is out-of-reach. Referring to the Royal Road reference by Hans Adler, a type '3' reader (like myself) is stupefied by the use of jargon, and generally doesn't have the requisite months to familiarize themselves with related subjects to trudge through the similarly problematic articles references for background (read: this is a systematic problem in mathematics articles). What is 'lost' by adding an extra couple of sentences on background, is surely gained by the order-of-magnitude larger audience that would be addressed. All Clues Key (talk) 05:15, 3 September 2012 (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)

New picture, new text

I think the new picture is much better than the old. Many readers will be familiar with the idea of the graph of a polynomial.

The length of the lede should depend on how much must be said to give the general reader an idea of what rings are all about. I've taken out some things that seem to me to be too technical for the lede and which are already in later sections of the article. I think the history paragraph in the lede is also too long, but I did not change it.

There is a short article Ring theory that does not seem to have anything not in this article. Should it be merged here?

Rick Norwood (talk) 12:55, 29 August 2010 (UTC)

I agree that most readers will know what a polynomial is. I don't think this new lead figure illuminates in any way what a ring is, however. The caption says polynomials form a ring, but the figure is just a graph of a few polynomials, and is worth a good deal less than a thousand words, in fact worth less than the caption itself (a dozen words). Brews ohare (talk) 18:09, 29 August 2010 (UTC)

An image illustrating geometric addition on a cubic curve in projective space. Ring theory has many important applications in algebraic geometry.

The problem is there is no image of a general ring as it's too abstract. One example is a polynomial ring and that's what the image is suggesting, with a link, as well as a suggestion how it works: adding and multiplying polynomials is pretty elementary, covered here first at at GCSE (age 16), so it's an accessible illustration of how rings are useful. It is a replacement for the one at right, which I think is showing something similar but is far less obvious, at least to me - even with the overlong caption it does not relate the diagram to ring or ring theory.--JohnBlackburne^words_deeds 19:20, 29 August 2010 (UTC)

Could you please see group (mathematics)? While the content of this lede should not necessarily be the same as group (mathematics), the format should, at least to a certain extent, since group (mathematics) is a mathematics featured article. I like the first two paragraphs of the introduction to ring (mathematics) now - there is a balance between encyclopedic brevity and coverage - and I think Rick's changes in this direction were very good.

However, this is not just an article for laymen. This article is for mathematicians as well. There needs to be something about how ring theory evolved and what its implications are in mathematics in the introduction. The 4th paragraph seems best for this, since if the laymen can get through the first three paragraphs comfortably, they should be motivated to read through the rest of the article and skip the 4th paragraph. Hence my inclusion of the 4th paragraph. Again, I emphasize the connection between group (mathematics), a featured article, where the 4th paragraph is similar in structure to this one. Once the history section of this article is expanded to a reasonable size, please feel free to delete the history part of the 4th paragraph. However, deleting it now is not fruitful, in my opinion, since there is no mention of many of the concepts in the 4th paragraph later in the article. (E.g., noncommutative geometry is not mentioned.)

The third paragraph has now been trimmed to one line. In my view, that does not merit a new paragraph. Therefore, I would think that either the third paragraph is merged with one of the first two paragraphs, or the old material in the third paragraph reinstalled. I know there is a reason to be concerned regarding laymen reading this article, and I agree with Rick Norwood that the earlier details in the third paragraph were somewhat technical. However, mathematicians are reading this article as well! It seems fair to me, that since especially nearly everything after the lede is explained to the point of triviality, having about two paragraphs on the context of ring theory in mathematics seems reasonable. Bear in mind, as well, that the history of ring theory is something that should be part of any encyclopedia and that this can be understood to anyone who does not have any mathematical background necessarily. Again, please see group (mathematics). Cheers, PST 00:06, 30 August 2010 (UTC)

I never proposed taking history out of the lede, just (maybe) shortening it a bit.

Can we find a picture that illustrates ring theory the way a Rubic's Cube illustrates group theory? We could use a picture of a clock (the ring of integers modulo 12) but that seems too prosaic. We could use the multiplication table for the integers mod 4, but that doesn't seem visual enough. Ideas?

Thoughts on merging Ring Theory into this article?

Rick Norwood (talk) 14:14, 30 August 2010 (UTC)

The article Ring theory is supposed to be an article on the advanced topics in ring theory that are not covered in Ring (mathematics). If you see the article Group (mathematics), at the very top of the page you will notice the statement: This article covers basic notions. For advanced topics, see Group theory. An analogous relationship holds between Ring (mathematics) and Ring theory.

At the moment, Ring theory does not have too many advanced topics, but it needs to be there so that mathematicians can visit it. For example, where is the Artin-Wedderburn theorem? The notion of a central simple algebra? The Hilbert basis theorem? What about Nakayama's lemma? Scheme theory? I think these all belong to Ring theory and hence it should not be merged with this article. I suspect more than 80% of people who view this article will have at least an undergraduate knowledge of mathematics and that is the reason most of the emphasis given in most mathematics articles should be on the more advanced mathematics topics. It is near impossible to explain something like Nakayama's lemma (or any of the other topics in ring theory I mentioned) to someone who does not know mathematics. The exception is given in articles like Ring (mathematics) because it is something that even non-mathematicians may come across. That is why we have "dumbed down" this article to the point of triviality. To satisfy the majority of the population who will really need this concept, we have an article on the "advanced topics" in ring theory - Ring theory.

However, the above is just my understanding of the matter. You should probably consult some of the editors at Group (mathematics) - they know more about these things. PST 23:06, 30 August 2010 (UTC)

What you propose sounds reasonable to me, but in that case, somebody in ring theory needs to work on that article. As it stands, it doesn't seem much more advanced than this article. Rick Norwood (talk) 11:50, 31 August 2010 (UTC)

Some impressions

I have looked over the article and it reminded me of Calculus textbooks: large in volume, verbose, glitzy, but often light on content and not terribly accurate, especially beyond the very basics. I think that it was a mistake to imitate the structure of the articles Group (mathematics) and Group theory and, in particular, to hew Ring (mathematics) to the line of Group (mathematics). (I should also point out that there is no general consensus that the so-called "GA process" adds value to our articles — an argument has been made that, on the contrary, it often turns into a charade of satisfying various formal requirements and dumbing down the content to appease the egos of "Category 4" readers in Hans Adler perspicacious classification, at the expense of everyone else.)

Some unintended effects are farcical, e.g. an attempt to pimp ring theory as playing an important role in relativity and quantum chemistry or the list of "ubiqutuos rings" starting with the cohomology ring of a topological space, and some are annoying: anyone who needs a detailed description of the operations on the ring Z₄ would be better off picking up a truly elementary book on ring theory; for the benefit of people wanting to get an encyclopaedic presentation of rings, it's better to talk about many standard examples in a few lines and move onto higher level aspects. Here, for comparison, is an old revision of the article which does not have any of these drawbacks.

It's not clear to me what the division of material should be between Ring (mathematics) and Ring theory, but long sections on principal ideal domains, unique factorization domains and other classes of commutative rings would fit more naturally into Commutative algebra or Commutative ring. Also, summary style, rather than carbon copies of other articles should be employed, especially due to quality control issues. For an illustration, consider the following sentence, which is a part of a paragraph of questionable value lifted wholesale from Noncommutative ring:

The theory of vector spaces is one illustration of a special case of an object studied in noncommutative ring theory.

The theory of vector spaces may have many aspects, but this isn't one of them. Whoever wrote this, got the relation between vector spaces and modules over general noncommutative rings backwards. What he was likely trying to express was that the notion of a field has a generalization in the form of noncommutative division rings. Now this mess needs to be fixed twice.

The use of the picture at the beginning of the article is another issue that needs to be carefully thought through. As pointed out above, the picture should be self-explanatory and to illustrate the concept well; graphs of functions fail on both counts. It is better to have no picture at all than to splash something irrelevant or misleading next to the lead.

I am aware that the bulk of the changes was the work of a single editor. In spite of his best intentions, I think that most of them were counterproductive and if a high quality encyclopaedic article is the goal, they need to be reconsidered and rolled back. Local problems can be fixed, but I see no point in doing it unless we agree on the overall structure, lest the job becomes a fool's errand.

To summarize, I think that this article has moved in the wrong direction. I recommend pruning it by getting rid of verbose explanations, checking for mathematical accuracy, and using the summary style. Arcfrk (talk) 19:48, 6 September 2010 (UTC)

Why a "ring"?

I sort of glanced at this article and while I'm sure it is soundly written, I expected to see an explanation of why these sets are called "rings" (unless there is one and I missed it, in which case I'm sorry for the intrusion). I mean this in a very intuitive, matter-of-factly way. Why were they named "rings" in the first place? There must have been a reason, or else they could have been named "chicken soup", or "carburettor", or "bedroom", or whatever. It may sound silly, and it may actually be silly, but as a lay reader I think that such an explanation should be forthcoming in the article, maybe accompanied by a graphic if applicable. Thanks in anticipation. — Preceding unsigned comment added by 62.1.136.208 (talk) 02:45, 17 December 2011 (UTC)

See the second paragraph of the history section. RobHar (talk) 06:51, 17 December 2011 (UTC)

Thanks, but it's not very informative. I mean, a number, when subjected to an operation, produces (tadaaaaaaaa) another number. Of course. What else would it produce, a Caesar salad? There has to be a niftier explanation for the nomenclature. — Preceding unsigned comment added by 62.1.130.172 (talk) 13:22, 17 December 2011 (UTC)

The explanation, while it could be better, can only be as nifty as the actual reason is. Also, I could probably come up with an operation to perform on numbers that yields caesar salads. Anyway, rings didn't exist back then and it seems that the idea is that he ways looking at certain submodules of C such that if you took powers of any element you would end up back in the original submodule. This is of course not true of any given submodule of C, but is true of rings of integers, which is what he was looking at. RobHar (talk) 01:23, 18 December 2011 (UTC)

There is a very intuitive reason: a ring is like an annulus (mathematics) in that it has a central emptiness: the group of units excludes the zero of the ring and perhaps much more. The case of a field (mathematics) has the least emptiness, just the single element, while a generic ring can be expected to have a significant opening. For some reason authors on ring theory gloss over this obvious feature.Rgdboer (talk) 19:46, 18 December 2011 (UTC)

Erroneous statement in Second example?

The statement "Associativity and commutativity of addition in Z₄ follow from associativity and commutativity of addition in the set of all integers." is less than obvious to me.

How can associativity of mod 4 addition in Z₄ follow from associativity of plain addition in the set of all integers? — Preceding unsigned comment added by Tbtietc (talk • contribs) 06:35, 17 April 2012 (UTC)

The map x → x mod 4 transforms addition and multiplication into addition and multiplication mod 4 (it is a ring homomorphism). It follows that the image by this homomorphism of the addition and multiplication identities are similar identities. D.Lazard (talk) 15:40, 17 April 2012 (UTC)

The presentation of polynomial and matrix rings

I don't find the presentation and notation of these two sections to be very useful, and I'm pretty sure they detract from the topic more than they contribute at the moment. I'm recommending abbreviating them to be more conceptual and less notational. The main articles can take care of the notation. Any thoughts? Rschwieb (talk) 14:56, 30 August 2012 (UTC)

I couldn't agree more. A polynomial ring is just the set of polynomials with the obvious addition and multiplication that high school students know. Saying it is something else is not only misleading unhelpful but wrong :). The same goes to the matrix ring. It's more important to stress their practical usages. For example, the importance of matrix rings, in the major part, comes from its appearance in structure theorems in non-commutative ring theory (which you know much more than I). Also, there could be more than one variable in a polynomial ring. In particular, there could be infinitely many variables, and that's a very important source of manufacturing good (or bad) examples. -- Taku (talk) 16:02, 30 August 2012 (UTC)

You overestimate my ability with commutative algebra, but I know what you're getting at! I plan on taking a look at it in a few days if nothing is done, but don't let that deter you from starting if you have a good idea to improve it. Rschwieb (talk) 16:21, 30 August 2012 (UTC)

Rewrites to the introduction (continued)

I have tried to make the lead shorter and more accessable, by leaving technical details for the body of the article. I would also like to move the last three sentence of the lead down into the body of the article, but I don't want to make too many changes at one time. Comments? Rick Norwood (talk) 13:21, 3 September 2012 (UTC)

It's my impression that "binary operation" is almost always defined as a function, and so mentioning well-definedness here would be overcomplicated. I'm going to have to find a little time to look into the numbers, but it would be nice if everyone chipped in here on that point, too. Rschwieb (talk) 14:27, 4 September 2012 (UTC)

The previous version said something about a ring being "closed" under addition and multiplication. The word "closed" seemed inaccessable to a non-mathematician, while "well-defined", while not strictly necessary, says the same thing (and more) and has an intuitive meaning that is not unlike the mathematical meaning. Rick Norwood (talk) 22:53, 4 September 2012 (UTC)

"closed" and "well-defined" are completely different concepts. "Closed" means that the binary operation on S×S is into S. Well defined means that the relation is a functional relation. IMO, the most common definition of "binary operation" assumes both of these, that it is a function into S. Rschwieb (talk) 16:03, 5 September 2012 (UTC)

One opinion: I think this is a huge improvement, thanks Rick. — Preceding unsigned comment added by AllCluesKey (talk • contribs) 04:43, 5 September 2012 (UTC)

Rewriting of the entire article

I've thinking of rewriting the entire article in order to (i) eliminate repetitions (some introduced by myself) (ii) to improve the structure of the article (in my opinion, this article is very confusing, structurally speaking. for example, basic examples should be discussed before advanced ones) More precisely, I propose the article roughly has the following structure.

• Introduction (This will not be the expanded lede but a place to introduce elementary concepts (e.g., division algorithm) with examples, historically or otherwise. I understand we don't want to give ring axioms right after the lede, and it makes sense to have some prose intended to the general math-minded readers.)
• Definition/basic examples (we should keep the current ones?)
• History
• Basic concepts: ideals, homomorphisms, etc.
• Basic examples: Polynomial rings, PID, UFD, etc.
• Advanced concepts: tensor products, localizations, completions etc
• Advanced examples (what is now "examples of the ubiquity of rings"); we need the ring of differential operators to be discussed.
• Application to the algebraic number theory (a good illustration of the commutative ring theory)
• Application to the representation theory of a finite group (a good illustration of the non-commutative ring theory)
• Relations to geometry: in the commutative case, maybe mention schemes. In in the noncommutative case, the noncommutative geometry (the idea alluded in the lede.) (Many concepts in commutative algebra have geometric interpretations (e.g., Eisenbud) I don't think this is a place to go into this. So, this section comes at very end.)

Thoughts? -- Taku (talk) 14:22, 13 November 2012 (UTC)

I have been seeing all the things go in, and I have been feeling like it's too much "stuff." I think it's already too big. I think the article needs to get "back to basics," and we need to avoid trying to mention the hundreds of elementary things that are possible to mention. Rschwieb (talk) 14:42, 13 November 2012 (UTC)

A part of the purpose of rewriting is to cut down verbosity; that should create more spaces. Also, the article is meant to be elementary; we need to mention the very basic stuff familiar to algebrists like the kernel of a homomorphism is an ideal or define zero-divisors; this was not done before. -- Taku (talk) 15:36, 13 November 2012 (UTC)

I'm on board with the cutting down on verbosity but I don't think it's a good idea at all to mention things as granular as zero divisors if the article is this big (but ideals are certainly fair game). What it really needs is a good sampling of the uses/origins of rings, good examples, and a historical overview with pointers towards subtopics. Rschwieb (talk) 18:30, 13 November 2012 (UTC)

topology seems to be a reasonable model for what I mean. At any rate, please start removing some of the cruft. I see some really hyperspecific stuff in there that isn't worth keeping. Rschwieb (talk) 18:35, 13 November 2012 (UTC)

I think we disagree on the fundamental level. To me, "zero-divisor" is non-negotiable; without defining it how can you introduce domains? In my view the article has to define "all" the basic concepts carefully. I mean where can you/should you find such anywhere but here? What you're asking seems like a survey article for those who already know abstract algebra; ring theory is the article for that purpose. For example, we can streamline much of the discussion of ideals if we use the language of modules. (basically we don't have to do a thing). But I think there is a reason why we don't that here. In any case, I'm not going to make drastic changes soon. -- Taku (talk) 20:35, 13 November 2012 (UTC)

Actually, when describing a domain to a layperson, I couldn't imagine why it would be useful to define a zero divisor. I would just say (and do say) "In such a ring, nonzero things multiply to nonzero things." At any rate, there should probably be a lot more people than just the two of us thinking about this, so it might be good to advertise this project at WPMath. Rschwieb (talk) 14:31, 14 November 2012 (UTC)

I think zero-divisor is an important elementary concept and easily understood. A simple example might be that in military time, 2 time 1200 hours = 0000 hours, so 2 and 1200 are zero divisors. Rick Norwood (talk) 15:50, 14 November 2012 (UTC)

Well there may well be room for it, but the edit on preimages of prime ideals is a case in point of what I mean. That preimages of prime ideas are prime ideals is one of 175 billion random facts that could exist in the article. It is not even directly a fact about rings: it's a fact about two features of rings (prime ideals and homomorphisms.) Can we draw the lines a little more clearly before this collection grows larger? Rschwieb (talk) 19:24, 14 November 2012 (UTC)

And in this case I agree. Preimages of prime ideals can go. Rick Norwood (talk) 20:54, 14 November 2012 (UTC)

ok, so where is the line? The pre-image of an idea an ideal can be in? But not a prime ideal? (In my view it's an important fact since it implies f induces the map between the sets of prime ideals.) -- Taku (talk) 22:25, 14 November 2012 (UTC)

My suggestion is to do what you think best. Rick Norwood (talk) 23:03, 14 November 2012 (UTC)

When I get a chance I'll try to help filter. There is, of course, no easy rule to give you, but I'm going to suggest: "If it's not about rings directly, maybe it shouldn't go in." Rschwieb (talk) 00:48, 15 November 2012 (UTC)

I'm not brave enough to have a shot at the commutative algebra section. Right now it looks like a cut-and-paste of the domain article, and is sorely in need of trimming. Domains should definitely be mentioned, but it really just does not seem like going all the way into details about UFD's and PIDS. It would be nice to talk about them, but using a bird's-eye view instead of the detail currently there. I also want to renew my complaint about having all the structure details about polynomials and matrix rings spelled out. The main articles do that just fine: short prompts would really be better here. Rschwieb (talk) 01:50, 15 November 2012 (UTC)

I think I'm beginning to see the point you have been trying to make. But needless to say I prefer my version: I don't understand why you removed an argument with Euclidean division. That kind of a short argument is the best way to illustrate the concept. (it also generalizes very easily) Also, characteristic: again the point is to illustrate the isomorphism theorem. A very typical example of a bad exposition is to state theorems or define concepts but never use them in concrete ways. We can't just define principal ideals or state isomorphisms without concrete illustrations. -- Taku (talk) 03:12, 15 November 2012 (UTC)

I don't think it is a good idea to derive that the ideals of the integers are all principal. Using the Euclidean algorithm is an excursion we can do without. It is probably done in a more appropriate article.

We definitely need characteristic back in (I said that in an edit summary.) The only thing I disagreed with was its placement. The isomorphism theorem already is demonstrated in Homomorphism, so I don't think there is any lack of demonstration going on here. It can be introduced in a simpler way in the subring section. Rschwieb (talk) 18:08, 15 November 2012 (UTC)

Again I disagree :) I think it's about one's taste. Why is for example "The integers, however, form a Noetherian ring which is not Artinian." in here? I don't see why it's not one of 175 billion. (why 175?) In any rate, we do agree some other parts of the article are more problematic, so I will move on. (The current ideal section is much closer to my version than it was originally; it definitely covers basics.) -- Taku (talk) 19:24, 15 November 2012 (UTC)

The Artinian-Noetherian thing is appropriate by both our standards. It illustrates important ring concepts (my standard) and it is a concrete simple example (your standard.)

I don't mean to sound rude when I point out some of your contributions. Lots of your contributions in articles are just great! However, I also saw a lot that looked like random facts that happened to be on your mind which leapt onto the page before you considered if they fit or not. In an article as long as ring (mathematics), this is more important to keep under control. When an article becomes like War and Peace it is very hard to read... At any rate I'm not trying to discourage you from editing it: I'm just trying to help keep the quality under control as changes grow. Rschwieb (talk) 22:09, 19 November 2012 (UTC)

Hey, if you two are going through the article anyway, how about adding some inline citations? Cesiumfrog (talk) 23:38, 19 November 2012 (UTC)

That's a good point. I have Lang's algebra in my bookshelf; it should work perfectly as a referenece (although I didn't really read it myself). I think we should avoid old books like Jacobson's basic algebra (again I haven't read it; I'm simply assuming it's too old.) -- Taku (talk) 00:03, 20 November 2012 (UTC)

Toward a shorter article

Despite recent extensive activity (ahm who did that?), the current article still remains too long in my opinion. Thus, I really want to cut down some nice but tangent materials; in particular, I would like to suggest we get rid of the Lie ring section and some para in the finite ring section and some verbosity from "Noncommutative rings" rings and "Definition and illustration". Objections? -- Taku (talk) 00:03, 20 November 2012 (UTC)

Someone might have pointed this out already, but: WP:NOTPAPER. I think the question should not be whether the article is longer than one is accustomed to, but whether the content is appropriate (and if so then where). If you really think those materials are nice for the encyclopedia and yet unsuited to the page, two other options are to preserve them here on the talk page or spin them out into another article. It's the wholesale obliteration of content that is most likely to draw opposition (i.e. rollbacks), so usually you likely can boldly go ahead making major edits unopposed provided you leave opportunities for other editors to rescue without difficulty any content they considered useful. Cesiumfrog (talk) 00:37, 20 November 2012 (UTC)

Anything that is nice but tangental should be moved to a more technical article, with a link in this article. Rick Norwood (talk) 12:47, 20 November 2012 (UTC)

OK, well I have a concrete recommendation I'm going to start with, but I'm going to run it by everyone here first. It is mainly about organization. Firstly, it is nonsensical to put "domain" in the "basic concepts" section, which mostly covers ring constructions and features. Secondly, Prufer domains are too specific to have a heading. Thirdly putting ring types in "concepts" sections is haphazard. It's like calling a section "stuff" and putting random things in it. Lastly it doesn't seem like a basic/advanced split is very necessary.

My solution is to collapse these things under "features", "constructions" and "types" headers, probably in that order. Things like products, localization and completion would go under constructions. "Types" would contain prominent types with mention of important subtypes. The types will take some discussing. IMO, the representative types at least contain headers for: field, commutative ring, integral domain, semisimple ring, nonassociative ring. Important subtypes (e.g. UFD, Dedekind or Prufer domains) could be mentioned beneath these headers, but again some restraint needs to be shown to prevent this from becoming a phonebook of ring types. Looking for feedback on what parts of this shift would be OK.

As for things that could maybe disappera, can we consider "Rings in context as algebraic structures" and its table, and "finite ring"? Rschwieb (talk) 18:05, 20 November 2012 (UTC)

Let me amend my suggestion above about "advanced/basic" splits by saying that like the "advanced examples" part, because it looks like a great place to mention ring examples which show how far-flung rings are into other fields of mathematics. I would also like to oppose any major trimming of noncommutative rings until the bigger issue of verbosity in the initial example paragraphs is tackled. I'm not saying noncommutative rings don't need trimming, but I think there are longer sections which would be easier to trim first. Rschwieb (talk) 18:15, 20 November 2012 (UTC)

The reason basic/advanced split is to make prerequisites clearer for the readers. This article is to cover all basic topics and some advanced ones. The tones and style of writing them cannot be the same; some advanced topics would assume substantial backgrounds in the module theory for instance, while basic topics don't. (if the readers know modules, there is no point getting into the details in ideals.) Also, an article like this is expected to be read linearly: from the easy stuff to the difficult ones. So, it makes sense and is in fact necessarily to discuss basic examples before introduced more advanced concepts. -- Taku (talk) 19:06, 21 November 2012 (UTC)

Yeah that makes sense. Let's remove enough of the cruft so that someone would dare to read it linearly :) Rschwieb (talk) 00:57, 22 November 2012 (UTC)

Rings?

Ok so are there any purely theoretical rings in common use, as in algebraic rings that are not identical to modular arithmetic systems or complex numbers, integers, real numbers and the like? — Preceding unsigned comment added by 86.185.99.78 (talk) 19:48, 11 December 2012 (UTC)

Yes, extensively. Matrices, exterior algebra and Clifford algebra (including geometric algebra) immediately come to mind. But the list will be huge, with examples scattered over many areas of mathematics. — Quondum 20:03, 11 December 2012 (UTC)

Hmmm interesting! Are there any purely theoretical rings that have no bearing on our number system that are currently in use? — Preceding unsigned comment added by 86.185.99.78 (talk) 20:13, 11 December 2012 (UTC)

I do not understand what you mean as a "theoretical ring", nor by "have no bearing on our number system". In any case, all modern mathematics (including the modern presentation of Euclidean geometry) are based on set theory, and natural numbers are immediately defined from sets as cardinal numbers or ordinal numbers. Therefore, there is impossible to provide any mathematics notion that is not related, in some way, to numbers. On the other hand, if you want rings whose elements are not some kind of numbers nor finite sets of numbers, the simplest example is probably the ring of the real functions and all its variants (ring of continuous functions, of differentiable functions, of analytic functions, ... D.Lazard (talk) 22:56, 11 December 2012 (UTC)

"identity" vs. "identity element"

Not being as versed in the lingo (like the average reader) and this being an encyclopoedia rather than a mathematics textbook, calling an identity element "an identity" doesn't sound so natural, and to some extent triggers an association with the concept where the word refers to an equation ("a ring with an identity" can be interpreted ambiguously). I would prefer the use of the more long-winded phrase "identity element" in general (except in stock phrases such as "rings with identity"). Comment? — Quondum 16:04, 28 December 2012 (UTC)